diff --git "a/txt/2105.11519.txt" "b/txt/2105.11519.txt" new file mode 100644--- /dev/null +++ "b/txt/2105.11519.txt" @@ -0,0 +1,4395 @@ +Noname manuscript No. +(will be inserted by the editor) +The advent and fall of a vocabulary learning bias from +communicative eciency +David Carrera-Casado Ramon +Ferrer-i-Cancho +Received: date / Accepted: date +Abstract Biosemiosis is a process of choice-making between simultaneously alter- +native options. It is well-known that, when suciently young children encounter +a new word, they tend to interpret it as pointing to a meaning that does not +have a word yet in their lexicon rather than to a meaning that already has a word +attached. In previous research, the strategy was shown to be optimal from an infor- +mation theoretic standpoint. In that framework, interpretation is hypothesized to +be driven by the minimization of a cost function: the option of least communication +cost is chosen. However, the information theoretic model employed in that research +neither explains the weakening of that vocabulary learning bias in older children or +polylinguals nor reproduces Zipf's meaning-frequency law, namely the non-linear +relationship between the number of meanings of a word and its frequency. Here +we consider a generalization of the model that is channeled to reproduce that law. +The analysis of the new model reveals regions of the phase space where the bias +disappears consistently with the weakening or loss of the bias in older children or +polylinguals. The model is abstract enough to support future research on other +levels of life that are relevant to biosemiotics. In the deep learning era, the model is +a transparent low-dimensional tool for future experimental research and illustrates +the predictive power of a theoretical framework originally designed to shed light +on the origins of Zipf's rank-frequency law. +Keywords biosemiosisvocabulary learning mutual exclusivity Zip an laws +information theory quantitative linguistics +David Carrera-Casado & Ramon Ferrer-i-Cancho +Complexity and Quantitative Linguistics Lab +LARCA Research Group +Departament de Ci encies de la Computaci o +Universitat Polit ecnica de Catalunya +Campus Nord, Edi ci Omega +Jordi Girona Salgado 1-3 +08034 Barcelona, Catalonia, Spain +E-mail: david.carrera@estudiantat.upc.edu,rferrericancho@cs.upc.eduarXiv:2105.11519v3 [cs.CL] 20 Jul 20212 David Carrera-Casado, Ramon Ferrer-i-Cancho +Contents +1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 +2 The mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 +3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 +4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 +A The mathematical model in detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 +B Form degrees and number of links . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 +C Complementary heatmaps for other values of . . . . . . . . . . . . . . . . . . . . . 48 +D Complementary gures with discrete degrees . . . . . . . . . . . . . . . . . . . . . . 61 +1 Introduction +Biosemiotics can be de ned as a science of signs in living systems (Kull, 1999, p. +386). Here we join the e ort of developing such a science. Focusing on the problem +of \learning" new signs, we hope to contribute (i) to place choice at the core of +semiotic theory of learning (Kull, 2018) and (ii) to make biosemiotics compatible +with the information theoretic perspective that is regarded as currently dominant +in physics, chemistry, and molecular biology (Deacon, 2015). +Languages use words to convey information. From a semantic perspective, +words stand for meanings (Fromkin et al., 2014). Correlates of word meaning +have been investigated in other species (e.g. Hobaiter and Byrne, 2014; Genty and +Zuberb uhler, 2014; Moore, 2014). From a neurobiological perspective, words can +be seen as the counterparts of cell assemblies with distinct cortical topographies +(Pulvermuller, 2001; Pulverm uller, 2013). From a formal standpoint, the essence +of that research is some binding between a sign or a form, e.g., a word or an ape +gesture, and a counterpart, e.g. a 'meaning' or an assembly of cortical cells. Math- +ematically, that binding can be formalized as a bipartite graph where vertices are +forms and their counterparts (Fig. 1). Such abstract setting allows for a powerful +exploration of natural systems across levels of life, from the mapping of animal +vocal or gestural behaviors (Fig. 2 (a)) into their \meanings" down to the map- +ping from codons into amino acids (Figure 2 (b)) while allowing for a comparison +against \arti cial" coding systems such as the Morse code (Fig. 2 (c)) or those +emerging in arti cial naming games (Hurford, 1989; Steels, 1996). In that setting, +almost connectedness has been hypothesized to be the mathematical condition re- +quired for the emergence of a rudimentary form of syntax and symbolic reference +(Ferrer-i-Cancho et al., 2005; Ferrer-i-Cancho, 2006). By symbolic reference, we +mean here Deacon's revision of Pierce's view (Deacon, 1997). The almost connect- +edness condition is met when it is possible to reach practically any other vertex of +the network by starting a walk from any possible vertex (as in Fig. 1 (a)-(b) but +not in Figs. 1 (c)-(d)). +Since the pioneering research of G. K. Zipf (1949), statistical laws of language +have been interpreted as manifestations of the minimization of cognitive costs +(Zipf, 1949; Ellis and Hitchcock, 1986; Ferrer-i-Cancho and D az-Guilera, 2007; +Gustison et al., 2016; Ferrer-i-Cancho et al., 2019). Zipf argued that the law of +abbreviation, the tendency of more frequent words to be shorter, resulted from a +minimization of a cost function involving, for every word, its frequency, its \mass" +and its \distance", which in turn implies the minimization of the size of words +(Zipf, 1949, p.59). Recently, it as been shown mathematically that the minimiza- +tion of the average of the length of words (the mean code length in the languageThe advent and fall of a vocabulary learning bias from communicative eciency 3 +(a) (b) +(c) (d) +Fig. 1 A bipartite graph linking forms (white circles) with their counterparts (black circles). +(a) a connected graph (b) an almost connected graph (c) a one-to-one mapping between forms +and counterparts (d) a mapping where only one form is linked with counterparts. +of information theory) predicts a correlation between frequency and duration that +cannot be positive, extending and generalizing previous results from information +theory (Ferrer-i-Cancho et al., 2019). The framework addresses the general prob- +lem of assigning codes as short as possible to counterparts represented by distinct +numbers while warranting certain constraints, e.g., that every number will receive +a distinct code (e.g. non-singular coding in the language of information theory). If +the counterparts are word types from a vocabulary, it predicts the law of abbre- +viation as it occurs in the vast majority of languages (Bentz and Ferrer-i-Cancho, +2016). If these counterparts are meanings, it predicts that more frequent mean- +ings should tend to be assigned smaller codes (e.g., shorter words) as found in real +experiments (Kanwal et al., 2017; Brochhagen, 2021). Table 1 summarizes these +and other predictions of compression.4 David Carrera-Casado, Ramon Ferrer-i-Cancho +(a) (b) +(c) +Fig. 2 Real bipartite graphs linking forms (white circles) with their counterparts (black +circles). (a) Chimpanzee gestures and their meaning (Hobaiter and Byrne, 2014, Table S3). +This table was chosen for its broad coverage of gesture types (see other tables satisfying other +constraints, e.g. only gesture-meaning associations employed by a suciently large number of +individuals). (b) Codon translation into amino acids, where forms are 64 codons and counter- +parts are 20 amino acids (c) The international Morse code, where forms are strings of dots +and dashed and the counterparts are letters of the English alphabet ( A;B;:::;Z ) and digits +(0;1;:::;9).The advent and fall of a vocabulary learning bias from communicative eciency 5 +linguistic laws ! principles ! predictions +(K ohler, 1987; Altmann, 1993) +Zipf's law of abbreviation !compression !Menzerath's law +(Gustison et al., 2016; Ferrer-i-Cancho et al., 2019) +!Zipf's rank-frequency law +(Ferrer-i-Cancho, 2016a) +!\shorter words" for more frequent \meanings" +(Ferrer-i-Cancho et al., 2019; Kanwal et al., 2017; Brochhagen, 2021) +Zipf's rank-frequency law !mutual information maximization ++ +surprisal minimization!a vocabulary learning bias +(Ferrer-i-Cancho, 2017a) +!the principle of contrast +(Ferrer-i-Cancho, 2017a) +!range or variation of +(Ferrer-i-Cancho, 2005a, 2006) +Table 1 The application of the scienti c method in quantitative linguistics (italics) with various concrete examples (roman). is the exponent of Zipf's +rank-frequency law (Zipf, 1949). The prediction that is the target of the current article is shown in boldface.6 David Carrera-Casado, Ramon Ferrer-i-Cancho +1.1 A family of probabilistic models +The bipartite graph of form-counterpart associations is the skeleton (Figs. 1 and +2) on which a family of models of communication has been built (Ferrer-i-Cancho +and D az-Guilera, 2007; Ferrer-i-Cancho and Vitevitch, 2018). The target of the + rst of these models (Ferrer-i-Cancho and Sole, 2003) was Zipf's rank-frequency +law, that de nes the relationship between the frequency of a word fand its rank +i, approximately as +fi : +These early models were aimed at shedding light on mainly three questions: +1. The origins of this law (Ferrer-i-Cancho and Sole, 2003; Ferrer-i-Cancho, 2005b). +2. The range of variation of in human language (Ferrer-i-Cancho, 2005a, 2006). +3. The relationship between and the syntactic and referential complexity of a +communication system (Ferrer-i-Cancho et al., 2005; Ferrer-i-Cancho, 2006). +The main assumption of these models is that word frequency is an epiphenomenon +of the structure of the skeleton or the probability of the meanings. Following the +metaphor of the skeleton, the models are bodies whose esh are probabilities that +are calculated from the skeleton. The rst models de ned p(sijrj), the probabil- +ity that a speaker produces sigiven a counterpart rj, as the same for all words +connected to rj. In the language of mathematics, +p(sijrj) =aij +!j; (1) +whereaijis a boolean (0 or 1) that indicates if siandrjare connected and !jis +the degree of rj, namely the number of connections of rjwith forms, i.e. +!j=X +iaij: +These models are often portrayed as models of the assignment of meanings to forms +(Futrell, 2020; Piantadosi, 2014) but this description falls short because: +{They are indeed models of production as they de ne the probability of pro- +ducing a form given some counterparts (as in Eq. 1) or simply the marginal +probability of a form. The claim that theories of language production or discourse +do not explain the law (Piantadosi, 2014) has no basis and raises the questions +of which theories of language production are deemed acceptable. +{They are also models of understanding, as they de ne symmetric conditional +probabilities such as p(rjjsi), the probability that a listener interprets rjwhen +receivingsi. +{The models are exible. In addition to \meaning", other counterparts were +deemed possible from their birth. See for instance the use of the term \stimuli" +(e.g. Ferrer-i-Cancho and D az-Guilera, 2007), as a replacement for meaning +that was borrowed from neurolinguistics (Pulvermuller, 2001). +{The models t in the distributional semantics framework (Lund and Burgess, +1996) for two reasons: their exibility, as counterparts can be dimensions in +some hidden space, and also because of representing a form as a vector of their +joint or conditional probabilities with \counterparts" that is inferred from the +network structure, as we have already explained (Ferrer-i-Cancho and Vite- +vitch, 2018).The advent and fall of a vocabulary learning bias from communicative eciency 7 +Contrary to the conclusions of (Piantadosi, 2014), there are derivations of Zipf's +law that do account for psychological processes of word production, especially the +intentionality of choosing words in order to convey a desired meaning. +The family of models assume that the skeleton that determines all the prob- +abilities, the bipartite graph, is shaped by a combination of minimization of the +entropy (or surprisal) of words ( H) and the maximization of the mutual infor- +mation between words and meanings ( I), two principles that are cognitively mo- +tivated and that capture speaker and listener's requirements (Ferrer-i-Cancho, +2018). When only the entropy of words is minimized, con gurations where only +one form is linked as in Fig. 1 (d) are predicted. When only the mutual informa- +tion between forms and counterparts is maximized, one-to-one mappings between +forms and counterparts are predicted (when the number of forms and counter- +parts is the same) as in Figure 1 (c) or Fig. 2 (d). Real language is argued to be +in-between these two extreme con gurations (Ferrer-i-Cancho and D az-Guilera, +2007). Such a trade-o between simplicity (Zipf's uni cation) and e ective com- +munication (Zipf's diversi cation) is also found in information theoretic models +of communication based on the information bottleneck approach (see Zaslavsky +et al. (2021) and references there in). +In quantitative linguistics, scienti c theory is not possible without taking into +consideration language laws (K ohler, 1987; Debowski, 2020). Laws are seen as +manifestations of principles (also referred as \requirements" by K ohler (1987)), +which are key components of explanations of linguistic phenomena. As part of +the scienti c method cycle, novel predictions are key aim (Altmann, 1993) and +key to validation and re nement of theory (Bunge, 2001). Table 1 synthesizes this +general view as chains of the form: laws,principles that are inferred from them, +and predictions that are made from those principles, giving concrete examples from +previous research. +Although one of the initial goals of the family of models was to shed light on +the origins of Zipf's law for word frequencies, a member of the family of mod- +els turned out to generate a novel prediction on vocabulary learning in children +and the tendency of words to contrast in meaning (Ferrer-i-Cancho, 2017a): when +encountering a new word, children tend to infer that it refers to a concept that +does not have a word attached to it (Markman and Wachtel, 1988; Merriman +and Bowman, 1989; Clark, 1993). The nding is cross-linguistically robust: it has +been found in children speaking English (Markman and Wachtel, 1988), Canadian +French (Nicoladis and Laurent, 2020), Japanese (Haryu, 1991), Mandarin Chinese +(Byers-Heinlein and Werker, 2013; Hung et al., 2015), Korean (Eun-Nam, 2017). +These languages correspond to four distinct linguistic families (Indo-European, +Japonic, Sino-Tibetan, Koreanic). Furthermore, the nding has also been repli- +cated in adults (Hendrickson and Perfors, 2019; Yurovsky and Yu, 2008) and +other species Kaminski et al. (2004). This phenomenon is a example of biosemio- +sis, namely a process of choice-making between simultaneously alternative options +(Kull, 2018, p. 454). +As an explanation for vocabulary learning, the information theoretic model +su ers from some limitations that motivate the present article. The rst one is that +the vocabulary learning bias weakens in older children (Kalashnikova et al., 2016; +Yildiz, 2020) or in polylinguals (Houston-Price et al., 2010; Kalashnikova et al., +2015), while the current version of the model predicts the vocabulary learning bias8 David Carrera-Casado, Ramon Ferrer-i-Cancho +Casea(b) Vertex degrees do not exceed one +Casea(a) Counterpart degrees do not exceed one +µk= 2 µk= 1ωj= 1 ωj= 1 +Caseb +Caseb +Fig. 3 Strategies for linking a new word to a meaning. Strategy aconsists of linking a word to +a free meaning, namely an unlinked meaning. Strategy bconsists of linking a word to a meaning +that is already linked. We assume that the meaning that is already linked is connected to a +single word of degree k. Two simplifying assumptions are considered. (a) Counterpart degrees +do not exceed one, implying k1. (b) Vertex degrees do not exceed one, implying k= 1. +only provided that mutual information maximization is not neglected (Ferrer-i- +Cancho, 2017a). +The second limitation is inherited from the family of models, where the de - +nition of the probabilities over the bipartite graph skeleton leads to a linear rela- +tionship between the frequency of a form and its number of counterparts (Ferrer-i- +Cancho and Vitevitch, 2018). However, this is inconsistent with Zipf's prediction, +namely that the number of meanings a word of frequency fshould follow (Zipf, +1945) +f; (2) +with= 0:5. Eq. 2 is known as Zipf's meaning-frequency law (Zipf, 1949). To over- +come such a limitation, Ferrer-i-Cancho and Vitevitch (2018) proposed di erent +ways of modifying the de nition of the probabilities from the skeleton. Here we +borrow a proposal of de ning the joint probability of a form and its counterpart +as +p(si;rj)/aij(i!j); (3) +whereis a parameter of the model and iand!jare, respectively, the degree +(number of connections) of the form siand the counterpart rj. Previous research +on vocabulary learning in children with these models (Ferrer-i-Cancho, 2017a) +assumed= 0, which leads to = 1 (Ferrer-i-Cancho, 2016b). When = 1, the +system is channeled to reproduce Zipf's meaning-frequency law, i.e. Eq. 2 with += 0:5 (Ferrer-i-Cancho and Vitevitch, 2018). +1.2 Overview of the present article +It has been argued that there cannot be meaning without interpretation (Eco, +1986). As Kull (2020) puts it, \ Interpretation (which is the same as primitive decision- +making) assumes that there exists a choice between two or more options. The options +can be described as di erent codes applicable simultaneously in the same situation. " +The main aim to of this article is to shed light on the choice between strategy a,The advent and fall of a vocabulary learning bias from communicative eciency 9 +i.e. attaching the new form to a counterpart that is unlinked, and strategy b, i.e. +attaching the new form to a counterpart that is already linked (Fig. 3). +The remainder of the article is organized as follows. Section 2 considers a model +of a communication system that has three components: +1. A skeleton that is de ned by a binary matrix Athat indicates the form- +counterpart connections. +2. A esh that is de ned over the skeleton with Eq. 3, +3. A cost function , that de nes the cost of communication as + +=I+ (1)H; (4) +whereis a parameter that regulates the weight of mutual information ( I) +maximization and word entropy ( H) minimization such that 0 1.Iand +Hare inferred from matrix Aand Eq. 3 (further details are given in Section +2). +This section introduces , i.e. the di erence in the cost of communication between +strategyaand strategy baccording to +(Fig. 3). < 0 indicates that the cost +of communication of strategy ais lower than that of b. Our main hypothesis is +that interpretation is driven by the +cost function and that a receiver will choose +the option that minimizes the resulting +. By doing this, we are challenging the +longstanding and limiting belief that information theory is dissociated from semi- +otics and not concerned about meaning (e.g. Deacon, 2015). This article is a just +one counterexample (see also Zaslavsky et al. (2018)). Information theory, as any +abstract powerful mathematical tool, can serve applications that do not assume +meaning (or meaning-making processes) as in the original setting of telecommu- +nication where it was developed by Shannon, as well as others that do, although +they were not his primary concern for historical and sociological reasons. +In general, the formula of is complex and the analysis of the conditions where +ais advantageous (namely <0) requires making some simplifying assumptions. +If= 0, then one obtains that Ferrer-i-Cancho (2017a) +=(!j+ 1) log(!j+ 1)!jlog(!j) +M+ 1; (5) +whereMis the number of edges in the skeleton and !jis the degree of the al- +ready linked counterpart that is selected in strategy b(Fig. 3). Eq. 5 indicates that +strategyawill be advantageous provided that mutual information maximization +matters (i.e.  >0) and its advantage will increase as mutual information max- +imization becomes more important (i.e. for larger ), the linked counterpart has +more connections (i.e. larger !j) or when the skeleton has less connections (i.e. +smallerM). To be able to analyze the case >0, we will examine two classes of +skeleta that are presented next. +Counterpart degrees do not exceed one. In this class, the degrees of counterparts +are restricted to not exceed one, namely a counterpart can only be disconnected +or connected to just one form. If meanings are taken as counterparts, this class +matches the view that \no two words ever have exactly the same meaning" (Fromkin +et al., 2014, p. 256), based on the notion of absolute synonymy (Dangli and Abazaj, +2009). This class also mirrors the linguistic principle that any two words should10 David Carrera-Casado, Ramon Ferrer-i-Cancho +contrast in meaning (Clark, 1987). Alternatively, if synonyms are deemed real +to some extent, this class may capture early stages of language development in +children or early stages in the evolution of languages where synonyms have not +been learned or developed. From a theoretical standpoint, this class is required +by the maximization of the mutual information between forms and counterparts +when the number of forms does not exceed that of counterparts (Ferrer-i-Cancho +and Vitevitch, 2018). +We usekto refer to degree of the word that will be connected to meaning +selected in strategy b(Fig. 3). We will show that, in this class, is determined by +,,kand the degree distribution of forms, namely the vector of form degrees +~ = (1;:::;i;:::n). +Vertex degrees do not exceed one. In this class, the degrees of any vertex are re- +stricted to not exceed one, namely a form (or a meaning) can only be discon- +nected or connected to just one counterpart (just one form). This class is narrower +than the previous one because it imposes that degrees do not exceed one both for +forms and counterparts. Words lack homonymy (or polysemy). We believe that this +class would correspond to even earlier stages of language development in children +(where children have learned at most one meaning of a word) or earlier stages +in the evolution of languages (where the communication system has not devel- +oped any homonymy). From a theoretical stand point, that class is a requirement +of maximizing mutual information between forms and counterparts when n=m +(Ferrer-i-Cancho and Vitevitch, 2018). We will show that is determined just by +,andM, the number of links of the bipartite skeleton. +Notice that meanings with synonyms have been found in chimpanzee gestures +(Hobaiter and Byrne, 2014), which suggests that the two classes above do not +capture the current state of the development of form-counterpart mappings in +adults of other species. Section 2 presents the formulae of for each classes. Section +3 uses this formulae to explore the conditions that determine when strategy ais +more advantageous, namely  < 0, for each of the two classes of skeleta above, +that correspond to di erent stages of the development of language in children. +While the condition = 0 implies that strategy ais always advantageous when +>0, we nd regions of the space of parameters where this is not the case when +>0 and>0. In the more restrictive class, where vertex degrees do not exceed +one, we nd a region where ais not advantageous when is suciently small and +Mis suciently large. The size of that region increases as increases. From a +complementary perspective, we nd a region where ais not advantageous ( 0) +whenis suciency small and is suciently large; the size of the region increases +asMincreases. As Mis expected to be larger in older children or in polylinguals +(if the forms of each language are mixed in the same skeleton), the model predicts +the weakening of the bias in older children and polylinguals (Liittschwager and +Markman, 1994; Kalashnikova et al., 2016; Yildiz, 2020; Houston-Price et al., 2010; +Kalashnikova et al., 2015, 2019). To ease the exploration of the phase space for +the class where the degrees of counterparts do not exceed one, we will assume +that word frequencies follow Zipf's rank-frequency law. Again, regions where a +is not advantageous ( 0) also appear but the conditions for the emergence +of this regions are more complex. Our preliminary analyses suggest that the bias +should weaken in older children even for this class. Section 4 discusses the ndings,The advent and fall of a vocabulary learning bias from communicative eciency 11 +suggests future research directions and reviews the research program in light of +the scienti c method. +2 The mathematical model +Below we give more details about the model that we use to investigate the learning +of new words and outlines the arguments that take from Eq. 3 to concrete formulae +of. Section 2.1 just presents the concrete formulae for each of the two classes +of skeleta. Full details are given in Appendix A. The model has four components +that we review next. +Skeleton (A=aij).A bipartite graph that de nes the associations between nforms +andmcounterparts that are de ned by an adjacency matrix A=faijg. +Flesh (p(si;rj)).The esh consist of a de nition of p(si;rj), the joint probability +of a form (or word) and a counterpart (or meaning) and a series of probability +de nitions stemming from it. Probabilities depart from previous work (Ferrer-i- +Cancho and Sole, 2003; Ferrer-i-Cancho, 2005b) by the addition of the parameter +. Eq. 3 de nes p(si;rj) as proportional to the product of the degrees of the form +and the counterpart to the power of , which is a parameter of the model. By +normalization, namely +nX +i=1mX +j=1p(si;rj) = 1; +Eq. 3 leads to +p(si;rj) =1 +Maij(i!j); (6) +where +M=nX +i=1mX +j=1aij(i!j): (7) +From these expressions, the marginal probabilities of a form p(si) and a counter- +partp(rj) are obtained easily thanks to +p(si) =mX +j=1p(si;rj) +p(rj) =nX +i=1p(si;rj): +The cost of communication ( +).The cost function is initially de ned in Eq. 4 as in +previous research (e.g. Ferrer-i-Cancho and D az-Guilera, 2007). In more detail, + +=I(S;R) + (1)H(S); (8) +whereI(S;R) is the mutual information between forms from a repertoire Sand +counterparts from a repertoire R, andH(S) is the entropy (or surprisal) of forms12 David Carrera-Casado, Ramon Ferrer-i-Cancho +from a repertoire S. Knowing that I(S;R) =H(S) +H(R)H(S;R) Cover and +Thomas (2006), the nal expression for the cost function in this article is + +() = (12)H(S)H(R) +H(S;R): (9) +The entropies H(S),H(R) andH(S;R) are easy to calculate applying the de ni- +tions ofp(si),p(rj) andp(si;rj), respectively. +The di erence in the cost of learning a new word ( ).There are two possible strate- +gies to determine the counterpart with which a new form (a previously unlinked +form) should connect (Fig. 3): +a. Connect the new form to a counterpart that is not already connected to any +other forms. +b. Connect the new form to a counterpart that is connected to at least one other +form. +The question we intend to answer is \when does strategy aresult in a smaller cost +than strategy b?" Or, in the terminology of child language research, \for which +strategy is the assumption of mutual exclusivity more advantageous?" To answer +these questions, we de ne , as a the di erence between the cost of each strategy. +More precisely, +() = +0 +a() +0 +b(); (10) +where +0a() and +0 +b() are the new value of +when a new link is created using +strategyaorbrespectively. Then, our research question becomes \When is  < +0?". +Formulae for +0a() and +0 +b() are derived in two steps. First, analyzing a +general problem, i.e. +0, the new value of +after producing a single mutation in +A(Appendix A.2). Second, deriving expressions for the case where that mutation +results from linking a new form (an unlinked form) to a counterpart, that can be +linked or unlinked (Appendix A.3). +2.1in two classes of skeleta +In previous work, the value of was already calculated for = 0, obtaining +expressions equivalent to Eq. 5 (see Appendix A.3.1 for a derivation). The next +sections just summarize the more complex formulae that are obtained for each +class of skeleta for 0 (see Appendix A for details on the derivation). +2.1.1 Vertex degrees do not exceed one +Here forms and counterparts both either have a single connection or are discon- +nected. Mathematically, this can be expressed as +i2f0;1gfor eachisuch that 1in +!j2f0;1gfor eachjsuch that 1jm: +Fig. 3 (b) o ers a visual representation of a bipartite graph of this class. In case b, +the counterpart we connect the new form to is connected to only one form ( !j= 1)The advent and fall of a vocabulary learning bias from communicative eciency 13 +and that form is connected to only one counterpart ( k= 1). Under this class,  +becomes +() = (12) +log +1 +2(21) +M+ 1 ++2+1log(2) +M+ 2+11 +2+1log(2) +M+ 2+11;(11) +which can be rewritten as linear function of , i.e. +() =a+b; +with +a= 2 log +1 +2(21) +M+ 1 +(2+ 1)2+1log(2) +M+ 2+11 +b=log +1 +2(21) +M+ 1 ++2+1log(2) +M+ 2+11: +Importantly, notice that this expression of is determined only by ,andM(the +total number of links in the model). See Appendix A.3.3 for thorough derivations. +2.1.2 Counterpart degrees do not exceed one +This class of skeleta is a relaxation of the previous class. Counterparts are either +connected to a single form or disconnected. Mathematically, +!j2f0;1gfor eachjsuch that 1jm: +Fig. 3 (a) o ers a visual representation of a bipartite graph of this class. The +number of forms the counterpart in case bis connected to is still 1 ( !j= 1) but +this form may be connected to any number of counterparts; khas to satisfy +1km. +Under this class, becomes +() = (12)( +log +M+ 1 +M+(21) +k+ 2! ++1 +M+(21) +k+ 2" +(+ 1)X(S;R)(21)( +k+ 1) +M+ 1 +2log(2) + +kh +log(k)(k+) +(k1 + 2) log(k1 + 2)i#) +1 +M+(21) +k+ 2" + + +k+ 12log + +k+ 1 +(1)2 +klog(k)# +;(12)14 David Carrera-Casado, Ramon Ferrer-i-Cancho +where +X(S;R) =nX +i=1+1 +ilogi (13) +M=nX +i=1+1 +i: (14) +Eq. 12 can also be expressed as a linear function of as +() =a+b; +with +a= 2 log +M+ (21) +k+ 2 +M+ 1! +1 +M+ (21) +k+ 2( +2h +( +k+ 1) log( +k+ 1) + +klog(k)i ++2h +(+ 1)X(S;R)(21) +k+ 1 +M+ 1 ++2log(2) +klog(k)(k+)(k1 + 2) log(k1 + 2)i) +b=log +M+ (21) +k+ 2 +M+ 1! ++1 +M+ (21) +k+ 2( +2 +klog(k)(+ 1)X(S;R)(21) +k+ 1 +M+ 1 ++2log(2) +kh +log(k)(k+)(k1 + 2) log(k1 + 2)i) +: +Being a relaxation of the previous class, the resulting expressions of are more +complex than those of the previous class, which are an in turn more complex than +those of the case = 0 (Eq. 5). See Appendix A.3.2 for further details on the +derivation of . +Notice that X(S;R) (Eq. 13) and M(Eq. 14) are determined by the degrees +of the forms ( i's). To explore the phase space with a realistic distribution of i's, +we assume, without any loss of generality, that the i's are sorted decreasingly, +i.e.12:::ii+1:::n. In addition, we assume +1.n= 0, because we are investigating the problem of linking and unlinked form +with counterparts. +2.n1= 1. +3. Form degrees are continuous. +4. The relationship between iand its frequency rank is a right-truncated power- +law, i.e. +i=ci(15) +for 1in1.The advent and fall of a vocabulary learning bias from communicative eciency 15 +Appendix B shows that forms then follow Zipf's rank-frequency law, i.e. +p(si) =c0i +with + =(+ 1) +c0=(n1) +M: +The value of is determined by ,,kand the sequence of degrees of the +forms, which we have parameterized with andn. When= ++1= 0, namely +when = 0 or when !1 , we recover the class where vertex degrees do not +exceed one but with just one form that is unlinked. +A continuous approximation to the number of edges gives (Appendix B) +M= (n1) ++1n1X +i=1i ++1: (16) +We aim to shed some light on the possible trajectory that children will describe +on Fig. 4 as they become older. One expects that Mtends to increase as children +become older, due to word learning. It is easy to see that Eq. 16 predicts that, if  +and remain constant, Mis expected to increase as nincreases (Fig. 4). Besides, +whennremains constant, a reduction of implies a reduction of Mwhen= 0 +but that e ect vanishes for >0 (Fig. 4). Obviously, ntends to increase as a child +becomes older (Saxton, 2010) and thus children's trajectory will be from left to +right in Fig. 4. As for the temporal evolution of , there are two possibilities. Zipf's +pioneering investigations suggest that remains close to 1 over time in English +children (Zipf, 1949, Chapter IV). In contrast, a wider study reported a tendency of + to decrease over time in suciently old children of di erent languages (Baixeries +et al., 2013) but the study did not determine the actual number of children where +that trend was statistically signi cant after controlling for multiple comparisons. +Then children, as they become older, are likely to move either from left to right, +keeping constant, or from the left-upper corner (high , lown) to the bottom- +right corner (low , highn) within each panel of Fig. 4. When is suciently +large, the actual evolution of some children (decrease of jointly with an increase +ofn) is dominated by the increase of Mthat the growth of nimplies in the long +run (Fig. 4). +When exploring the space of parameters, we must warrant that kdoes not +exceed the maximum degree that n,and yield, namely k1, where1is +de ned according to Eq. 15 with i= 1, i.e. +k1 +=c += (n1) += (n1) ++1: (17)16 David Carrera-Casado, Ramon Ferrer-i-Cancho +0.00.51.01.52.0 +0 250 500 750 1000 +nα +0246log10 Mφ = 0(a) +0.00.51.01.52.0 +0 250 500 750 1000 +nα +01234log10 Mφ = 0.5(b) +0.00.51.01.52.0 +0 250 500 750 1000 +nα +0123log10 Mφ = 1(c) +0.00.51.01.52.0 +0 250 500 750 1000 +nα +0123log10 Mφ = 1.5(d) +0.00.51.01.52.0 +0 250 500 750 1000 +nα +0123log10 Mφ = 2(e) +0.00.51.01.52.0 +0 250 500 750 1000 +nα +0123log10 Mφ = 2.5(f) +Fig. 4 log10M, the logarithm of the number of links M, as a function of n(x-axis) and + (y-axis) according to Eq. 16. log10Mis used instead of Mto capture changes in order of +magnitude of M. (a)= 0, (b)= 0:5, (c)= 1, (d)= 1:5, (e)= 2 and (f) = 2:5. +3 Results +Here we will analyze , that takes a negative value when strategy a(linking a +new form to a new counterpart) is more advantageous than strategy blinking a +new form to an already connected counterpart), and a positive value otherwise. +jjindicates the strength of the bias towards strategy aif<0; towards strategyThe advent and fall of a vocabulary learning bias from communicative eciency 17 +bif>0. Therefore, when <0, the smaller the value of , the higher the bias +for strategy awhereas when >0, the greater the value of , the higher the bias +for strategy b. Each class of skeleta is analyzed separately, beginning by the most +restrictive class. +3.1 Vertex degrees do not exceed one +In this class of skeleta, corresponding to younger children, depends only on ,M +and. We will explore the phase space with the help of two-dimensional heatmaps +ofwhere thex-axis is always and they-axis isMor. +Figs. 5 and 6 reveal regions where strategy ais more advantageous (red) and +regions where bis more advantageous (blue) according to . The extreme situation +is found when = 0 where a single red region covers practically all space except for += 0 (Fig. 5, top-left) as expected from previous work (Ferrer-i-Cancho, 2017a) +and Eq. 5. Figs. 7 and 8 summarize these nding of regions, displaying the curve +that de nes the boundary between strategies aandb(= 0). +Figs. 7 and 8 show that strategy bis the optimal only if is suciently low, +namely when the weight of entropy minimization is suciently high compared to +that of mutual information maximization. Fig. 7 shows that the larger the value of +the larger the number of links ( M) that is required for strategy bto be optimal. +Fig. 7 also indicates that the larger the value of , the broader the blue region +wherebis optimal. From a symmetric perspective, Fig. 8 shows that the larger the +value ofthe larger the value of that is required for strategy bto be optimal and +also that the larger the number of links ( M), the broader the blue region where b +is optimal. +3.2 Counterpart degrees do not exceed one +For this class of skeleta, corresponding to older children, we have assumed that +word frequencies follow Zipf's rank-frequency law, namely the relationship between +the probability of a form (the number of counterparts connected to each form) and +its frequency rank follows a right-truncated power-law with exponent (Section +2). Thendepends only on (the exponent of the right-truncated power law), +n(the number of forms), k(the degree of the form linked to the counterpart +in strategy bas shown in Fig. 3), and. We will explore the phase space with +the help of two-dimensional heatmaps of where the x-axis is always and the +y-axis isk, orn. While in the class where vertex degrees do not exceed one +we have found only one blue region (a region where  > 0 meaning that bis +more advantageous), this class yields up to two distinct blue regions located in +opposite corners of the heatmap while keeping always a red region as show in +Figs. 10, 12 and 14 for = 1 from di erent perspectives. For the sake of brevity, +this section only presents heatmaps of for= 0 or= 1 (see Appendix C for +the remainder). A summary of exploration of the parameter space follows. +Heatmaps of as a function of andk.The heatmaps of for di erent com- +binations of parameters in Figs. 9, 10, 16, 17, 18 and 19 are summarized in Fig. +11, showing the frontiers between regions where = 0. Notice how, for = 0,18 David Carrera-Casado, Ramon Ferrer-i-Cancho +0255075100 +0.00 0.25 0.50 0.75 1.00 +λM +-0.6-0.4-0.2Δ < 0 +0Δ ≥ 0φ = 0(a) +0255075100 +0.00 0.25 0.50 0.75 1.00 +λM +-0.6-0.4-0.2Δ < 0 +0.0050.010Δ ≥ 0φ = 0.5(b) +0255075100 +0.00 0.25 0.50 0.75 1.00 +λM +-0.6-0.4-0.2Δ < 0 +0.000.010.020.030.040.05Δ ≥ 0φ = 1(c) +0255075100 +0.00 0.25 0.50 0.75 1.00 +λM +-0.6-0.4-0.2Δ < 0 +0.0250.0500.0750.1000.125Δ ≥ 0φ = 1.5(d) +0255075100 +0.00 0.25 0.50 0.75 1.00 +λM +-0.6-0.4-0.2Δ < 0 +0.000.050.100.150.20Δ ≥ 0φ = 2(e) +0255075100 +0.00 0.25 0.50 0.75 1.00 +λM +-0.8-0.6-0.4-0.2Δ < 0 +0.10.20.3Δ ≥ 0φ = 2.5(f) +Fig. 5, the di erence between the cost of strategy aand strategy b, as a function of M, the +number of links and , the parameter that controls the balance between mutual information +maximization and entropy minimization, when vertex degrees do not exceed one (Eq. 11). Red +indicates that strategy ais more advantageous while blue indicates that bis more advantageous. +The lighter the red, the stronger the bias for strategy a. The lighter the blue, the stronger the +bias for strategy b. (a)= 0, (b)= 0:5, (c)= 1, (d)= 1:5, (e)= 2 and (f) = 2:5.The advent and fall of a vocabulary learning bias from communicative eciency 19 +0.02.55.07.510.0 +0.00 0.25 0.50 0.75 1.00 +λφ +-1.00-0.75-0.50-0.25Δ < 0 +0.00.10.20.30.4Δ ≥ 0M = 2(a) +0.02.55.07.510.0 +0.00 0.25 0.50 0.75 1.00 +λφ +-1.0-0.5Δ < 0 +0.00.20.40.6Δ ≥ 0M = 3(b) +0.02.55.07.510.0 +0.00 0.25 0.50 0.75 1.00 +λφ +-1.5-1.0-0.5Δ < 0 +0.000.250.500.751.00Δ ≥ 0M = 5(c) +0.02.55.07.510.0 +0.00 0.25 0.50 0.75 1.00 +λφ +-2.0-1.5-1.0-0.5Δ < 0 +0.00.40.81.21.6Δ ≥ 0M = 10(d) +0.02.55.07.510.0 +0.00 0.25 0.50 0.75 1.00 +λφ +-3-2-1Δ < 0 +0123Δ ≥ 0M = 50(e) +0.02.55.07.510.0 +0.00 0.25 0.50 0.75 1.00 +λφ +-4-3-2-1Δ < 0 +0123Δ ≥ 0M = 150(f) +Fig. 6, the di erence between the cost of strategy aand strategy b, as a function of , +the parameter that de nes how the esh of the model from the skeleton, and , the parameter +that controls the balance between mutual information maximization and entropy minimization +(Eq. 11). Red indicates that strategy ais more advantageous while blue indicates that bis +more advantageous. The lighter the red, the stronger the bias for strategy a. The lighter the +blue, the stronger the bias for strategy b. (a)M= 2, (b)M= 3, (c)M= 5, (d)M= 10, (e) +M= 50 and (f) M= 150.20 David Carrera-Casado, Ramon Ferrer-i-Cancho +0255075100 +0.00 0.25 0.50 0.75 1.00 +λMφ +0 +0.5 +1 +1.5 +2 +2.5 +Fig. 7 Summary of the boundaries between positive and negative values of when vertex +degrees do not exceed one (Fig. 5). Each curve shows the points where = 0 (Eq. 12) as a +function of andMfor distinct values of . +strategyais optimal for all values of >0, as one would expect from Eq. 5. The +remainder of the gures show how the shape of the two areas changes with each +of the parameters. For small nand , a single blue region indicates that strategy +bis more advantageous than awhenis closer to 0 and kis higher. For higher +nor an additional blue region appears indicating that strategy bis also optimal +for high values of and low values of k. +Heatmaps of as a function of and .The heatmaps of for di erent combi- +nations of parameters in Figs. 12, 20, 21, 22 and 23 are summarized in Fig. 13, +showing the frontiers between regions. There is a single region where strategy bis +optimal for small values of kand, but for larger values a second blue region +appears. +Heatmaps of as a function of andn.The heatmaps of for di erent combina- +tions of parameters in Figs. 14, 24, 25, 26 and 27 are summarized in Fig. 15. Again, +one or two blue regions appear depending on the combination of parameters. +See Appendix D for the impact of using discrete form degrees on the results +presented in this section.The advent and fall of a vocabulary learning bias from communicative eciency 21 +0.02.55.07.510.0 +0.00 0.25 0.50 0.75 1.00 +λφM +2 +3 +5 +10 +50 +150 +Fig. 8 Summary of the boundaries between positive and negative values of when vertex +degrees do not exceed one (Fig. 6). Each curve shows the points where = 0 (Eq. 12) as a +function of andfor distinct values of M.22 David Carrera-Casado, Ramon Ferrer-i-Cancho +1.01.52.02.53.0 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.075-0.050-0.025Δ < 0 +0Δ ≥ 0φ = 0 α = 0.5 n = 10(a) +2.55.07.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.05-0.04-0.03-0.02-0.01Δ < 0 +0Δ ≥ 0φ = 0 α = 1 n = 10(b) +01020 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.025-0.020-0.015-0.010-0.005Δ < 0 +0Δ ≥ 0φ = 0 α = 1.5 n = 10(c) +2.55.07.510.0 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.006-0.004-0.002Δ < 0 +0Δ ≥ 0φ = 0 α = 0.5 n = 100(d) +0255075100 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.0025-0.0020-0.0015-0.0010-0.0005Δ < 0 +0Δ ≥ 0φ = 0 α = 1 n = 100(e) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λμk +-5e-04-4e-04-3e-04-2e-04-1e-04Δ < 0 +0Δ ≥ 0φ = 0 α = 1.5 n = 100(f) +0102030 +0.00 0.25 0.50 0.75 1.00 +λμk +-6e-04-4e-04-2e-04Δ < 0 +0Δ ≥ 0φ = 0 α = 0.5 n = 1000(g) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.00015-0.00010-0.00005Δ < 0 +0Δ ≥ 0φ = 0 α = 1 n = 1000(h) +0100002000030000 +0.00 0.25 0.50 0.75 1.00 +λμk +-1.6e-05-1.2e-05-8.0e-06-4.0e-06Δ < 0 +0Δ ≥ 0φ = 0 α = 1.5 n = 1000(i) +Fig. 9, the di erence between the cost of strategy aand strategy b, as a function of k, +the degree of the form linked to the counterpart in strategy bas shown in Fig. 3, the number of +links and, the parameter that controls the balance between mutual information maximization +and entropy minimization, when the degrees of counterparts do not exceed one (Eq. 11) and += 0. Red indicates that strategy ais more advantageous while blue indicates that bis more +advantageous. The lighter the red, the stronger the bias for strategy a. The lighter the blue, +the stronger the bias for strategy b. Each heatmap corresponds to a distinct combination of +nand . The heatmaps are arranged, from left to right, with = 0:5;1;1:5 and, from top to +bottom, with n= 10;100;1000. (a) = 0:5 andn= 10, (b) = 1 andn= 10, (c) = 1:5 +andn= 10, (d) = 0:5 andn= 100, (e) = 1 andn= 100, (f) = 1:5 andn= 100, (g) + = 0:5 andn= 1000, (h) = 1 andn= 1000, (i) = 1:5 andn= 1000.The advent and fall of a vocabulary learning bias from communicative eciency 23 +1.01.21.41.6 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.25-0.20-0.15-0.10-0.05Δ < 0 +0.020.040.06Δ ≥ 0φ = 1 α = 0.5 n = 10(a) +1.01.52.02.53.0 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.20-0.15-0.10-0.05Δ < 0 +0.020.040.06Δ ≥ 0φ = 1 α = 1 n = 10(b) +12345 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.10-0.05Δ < 0 +0.010.020.030.040.05Δ ≥ 0φ = 1 α = 1.5 n = 10(c) +1.01.52.02.53.0 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.04-0.03-0.02-0.01Δ < 0 +0.0050.0100.0150.0200.025Δ ≥ 0φ = 1 α = 0.5 n = 100(d) +2.55.07.510.0 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.04-0.03-0.02-0.01Δ < 0 +0.010.020.03Δ ≥ 0φ = 1 α = 1 n = 100(e) +0102030 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.020-0.015-0.010-0.005Δ < 0 +0.0050.0100.015Δ ≥ 0φ = 1 α = 1.5 n = 100(f) +12345 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.0100-0.0075-0.0050-0.0025Δ < 0 +0.0020.0040.006Δ ≥ 0φ = 1 α = 0.5 n = 1000(g) +0102030 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.010-0.005Δ < 0 +0.00250.00500.00750.01000.0125Δ ≥ 0φ = 1 α = 1 n = 1000(h) +050100150 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.004-0.003-0.002-0.001Δ < 0 +0.0010.0020.0030.004Δ ≥ 0φ = 1 α = 1.5 n = 1000(i) +Fig. 10 Same as in Fig. 9 but with = 1.24 David Carrera-Casado, Ramon Ferrer-i-Cancho +1.21.51.82.1 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +1 +1.5 +2 +2.5α = 0.5 n = 10(a) +234 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +1 +1.5 +2 +2.5α = 1 n = 10(b) +2.55.07.5 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +1 +1.5 +2 +2.5α = 1.5 n = 10(c) +1234 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +1 +1.5 +2 +2.5α = 0.5 n = 100(d) +05101520 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +1 +1.5 +2 +2.5α = 1 n = 100(e) +0255075100 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +1 +1.5 +2 +2.5α = 1.5 n = 100(f) +2.55.07.510.0 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +1 +1.5 +2 +2.5α = 0.5 n = 1000(g) +0255075100 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +1 +1.5 +2 +2.5α = 1 n = 1000(h) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +1 +1.5 +2 +2.5α = 1.5 n = 1000(i) +Fig. 11 Summary of the boundaries between positive and negative values of when the +degrees of counterparts do not exceed one ( gures 9, 10, 16, 17, 18 and 19). Each curve shows +the points where = 0 (Eq. 12) as a function of andkfor distinct values of . (a) = 0:5 +andn= 10, (b) = 1 andn= 10, (c) = 1:5 andn= 10, (d) = 0:5 andn= 100, (e) = 1 +andn= 100, (f) = 1:5 andn= 100, (g) = 0:5 andn= 1000, (h) = 1 andn= 1000, (i) + = 1:5 andn= 1000.The advent and fall of a vocabulary learning bias from communicative eciency 25 +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.10-0.05Δ < 0 +0.0050.0100.0150.020Δ ≥ 0φ = 1 μk = 1 n = 10(a) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.0100-0.0075-0.0050-0.0025Δ < 0 +0.0010.0020.0030.004Δ ≥ 0φ = 1 μk = 1 n = 100(b) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.00075-0.00050-0.00025Δ < 0 +1e-042e-043e-044e-045e-04Δ ≥ 0φ = 1 μk = 1 n = 1000(c) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.3-0.2-0.1Δ < 0 +0.0250.0500.0750.100Δ ≥ 0φ = 1 μk = 2 n = 10(d) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.025-0.020-0.015-0.010-0.005Δ < 0 +0.0020.0040.006Δ ≥ 0φ = 1 μk = 2 n = 100(e) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.0020-0.0015-0.0010-0.0005Δ < 0 +1e-042e-043e-044e-045e-04Δ ≥ 0φ = 1 μk = 2 n = 1000(f) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.6-0.4-0.2Δ < 0 +0.10.20.30.4Δ ≥ 0φ = 1 μk = 4 n = 10(g) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.06-0.04-0.02Δ < 0 +0.010.020.030.04Δ ≥ 0φ = 1 μk = 4 n = 100(h) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.006-0.004-0.002Δ < 0 +0.0010.0020.003Δ ≥ 0φ = 1 μk = 4 n = 1000(i) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-1.0-0.5Δ < 0 +0.30.60.9Δ ≥ 0φ = 1 μk = 8 n = 10(j) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.16-0.12-0.08-0.04Δ < 0 +0.050.10Δ ≥ 0φ = 1 μk = 8 n = 100(k) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.016-0.012-0.008-0.004Δ < 0 +0.00250.00500.00750.01000.0125Δ ≥ 0φ = 1 μk = 8 n = 1000(l) +Fig. 12, the di erence between the cost of strategy aand strategy b, as a function of , the +exponent of the rank-frequency law, and , the parameter that controls the balance between +mutual information maximization and entropy minimization, when the degrees of counterparts +do not exceed one (Eq. 11) and = 1. Red indicates that strategy ais more advantageous +while blue indicates that bis more advantageous. The lighter the red, the stronger the bias for +strategya. The lighter the blue, the stronger the bias for strategy b. Each heatmap corresponds +to a distinct combination of nandk. The heatmaps are arranged, from left to right, with +n= 10;100;1000 and, from top to bottom, with k= 1;2;4;8. Gray indicates regions where +kexceeds the maximum degree according to other parameters (Eq. 17). (a) k= 1 and +n= 10, (b)k= 1 andn= 100, (c)k= 1 andn= 1000, (d) k= 2 andn= 10, (e)k= 2 +andn= 100, (f) k= 2 andn= 1000, (g) k= 4 andn= 10, (h)k= 4 andn= 100, +(i)k= 4 andn= 1000, (j) k= 8 andn= 10, (k)k= 8 andn= 100, (l)k= 8 and +n= 1000.26 David Carrera-Casado, Ramon Ferrer-i-Cancho +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +1 +1.5 +2 +2.5μk = 1 n = 10(a) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2 +2.5μk = 1 n = 100(b) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2 +2.5μk = 1 n = 1000(c) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2 +2.5μk = 2 n = 10(d) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2 +2.5μk = 2 n = 100(e) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2 +2.5μk = 2 n = 1000(f) +0.60.81.01.21.4 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1μk = 4 n = 10(g) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2 +2.5μk = 4 n = 100(h) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2 +2.5μk = 4 n = 1000(i) +1.01.11.21.31.41.5 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5μk = 8 n = 10(j) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2μk = 8 n = 100(k) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2 +2.5μk = 8 n = 1000(l) +Fig. 13 Summary of the boundaries between positive and negative values of when the +degrees of counterparts do not exceed one (Figs. 12, 20, 21, 22 and 23). Each curve shows +the points where = 0 (Eq. 12) as a function of and for distinct values of . Points are +restricted to combinations of parameters where kdoes not exceed the maximum (Eq. 17). +Each distinct heatmap corresponds to a distinct combination of kandn. (a)k= 1 and +n= 10, (b)k= 1 andn= 100, (c)k= 1 andn= 1000, (d) k= 2 andn= 10, (e)k= 2 +andn= 100, (f) k= 2 andn= 1000, (g) k= 4 andn= 10, (h)k= 4 andn= 100, +(i)k= 4 andn= 1000, (j) k= 8 andn= 10, (k)k= 8 andn= 100, (l)k= 8 and +n= 1000.The advent and fall of a vocabulary learning bias from communicative eciency 27 +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.10-0.05Δ < 0φ = 1 μk = 1 α = 0.5(a) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.06-0.04-0.02Δ < 0 +0.0010.0020.003Δ ≥ 0φ = 1 μk = 1 α = 1(b) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.06-0.04-0.02Δ < 0 +0.0050.0100.0150.020Δ ≥ 0φ = 1 μk = 1 α = 1.5(c) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.3-0.2-0.1Δ < 0 +0.0250.0500.0750.100Δ ≥ 0φ = 1 μk = 2 α = 0.5(d) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.09-0.06-0.03Δ < 0 +3e-046e-049e-04Δ ≥ 0φ = 1 μk = 2 α = 1(e) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.06-0.04-0.02Δ < 0 +0.00250.00500.00750.01000.0125Δ ≥ 0φ = 1 μk = 2 α = 1.5(f) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.6-0.4-0.2Δ < 0 +0.10.20.30.4Δ ≥ 0φ = 1 μk = 4 α = 0.5(g) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.3-0.2-0.1Δ < 0 +0.040.080.120.16Δ ≥ 0φ = 1 μk = 4 α = 1(h) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.08-0.06-0.04-0.02Δ < 0 +0.0020.0040.006Δ ≥ 0φ = 1 μk = 4 α = 1.5(i) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-1.0-0.5Δ < 0 +0.30.60.9Δ ≥ 0φ = 1 μk = 8 α = 0.5(j) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.6-0.4-0.2Δ < 0 +0.10.20.30.40.5Δ ≥ 0φ = 1 μk = 8 α = 1(k) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.2-0.1Δ < 0 +0.050.100.15Δ ≥ 0φ = 1 μk = 8 α = 1.5(l) +Fig. 14, the di erence between the cost of strategy aand strategy b, as function of n, the +number of forms, and , the parameter that controls the balance between mutual information +maximization and entropy minimization, when the degrees of counterparts do not exceed one +(Eq. 11) and = 1. We are taking values of nfrom 10 onwards (instead of one onwards) to see +more clearly the light regions that are re ected on the color scales. Red indicates that strategy +ais more advantageous while blue indicates that bis more advantageous. The lighter the red, +the stronger the bias for strategy a. The lighter the blue, the stronger the bias for strategy b. +Each heatmap corresponds to a distinct combination of kand . The heatmaps are arranged, +from left to right, with = 0:5;1;1:5 and, from top to bottom, with k= 1;2;4;8. Gray +indicates regions where kexceeds the maximum degree according to other parameters (Eq. +17). (a)k= 1 and = 0:5, (b)k= 1 and = 1, (c)k= 1 and = 1:5, (d)k= 2 and + = 0:5, (e)k= 2 and = 1, (f)k= 2 and = 1:5, (g)k= 4 and = 0:5, (h)k= 4 +and = 1, (i)k= 4 and = 1:5, (j)k= 8 and = 0:5, (k)k= 8 and = 1, (l)k= 8 +and = 1:5.28 David Carrera-Casado, Ramon Ferrer-i-Cancho +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +1.5 +2 +2.5μk = 1 α = 0.5(a) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2 +2.5μk = 1 α = 1(b) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2 +2.5μk = 1 α = 1.5(c) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2 +2.5μk = 2 α = 0.5(d) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2 +2.5μk = 2 α = 1(e) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2 +2.5μk = 2 α = 1.5(f) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1μk = 4 α = 0.5(g) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2 +2.5μk = 4 α = 1(h) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2 +2.5μk = 4 α = 1.5(i) +2505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5μk = 8 α = 0.5(j) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2μk = 8 α = 1(k) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2 +2.5μk = 8 α = 1.5(l) +Fig. 15 Summary of the boundaries between positive and negative values of when the +degrees of counterparts do not exceed one (Figs. 14, 24, 25, 26 and 27). Each curve shows +the points where = 0 (Eq. 12) as a function of andnfor distinct values of . Points are +restricted to combinations of parameters where kdoes not exceed the maximum (Eq. 17). +Each distinct heatmap corresponds to a distinct combination of kand . (a)k= 1 and + = 0:5, (b)k= 1 and = 1, (c)k= 1 and = 1:5, (d)k= 2 and = 0:5, (e)k= 2 +and = 1, (f)k= 2 and = 1:5, (g)k= 4 and = 0:5, (h)k= 4 and = 1, (i)k= 4 +and = 1:5, (j)k= 8 and = 0:5, (k)k= 8 and = 1, (l)k= 8 and = 1:5.The advent and fall of a vocabulary learning bias from communicative eciency 29 +4 Discussion +4.1 Vocabulary learning +In previous research with = 0, we predicted that the vocabulary learning bias +(strategya) would be present provided that mutual information minimization is +not disabled (  > 0) (Ferrer-i-Cancho, 2017a) as show in Eq. 5. However, the +\decision" on whether assigning a new label to a linked or to an unlinked object +is in uenced by the age of a child and his/her degree of polylingualism. As for the +e ect of the latter, polylingual children tend to pick familiar objects more often +than monolingual children, violating mutual exclusivity. This has been found for +younger children below two years of age (17-22 months old in one study, 17-18 +in another) (Houston-Price et al., 2010; Byers-Heinlein and Werker, 2013). From +three years onward, the di erence between polylinguals and monolinguals either +vanishes, namely both violate mutual exclusivity similarly (Nicoladis and Laurent, +2020; Frank and Poulin-Dubois, 2002), or polylingual children are still more willing +to accept lexical overlap (Kalashnikova et al., 2015). One possible explanation for +this phenomenon is the lexicon structure hypothesis (Byers-Heinlein and Werker, +2013), which suggests that children that already have many multiple-word-to- +single-object mappings may be more willing to suspend mutual exclusivity. +As for the e ect of age on monolingual children, the so-called mutual exclusivity +bias has been shown to appear at an early age and, as time goes on, it is more easily +suspended. Starting at 17 months old, children tend to look at a novel object rather +than a familiar one when presented with a new word while 16-month-olds do not +show a preference (Halberda, 2003). Interestingly, in the same study, 14-month-olds +systematically look at a familiar object instead of a newer one. Reliance on mutual +exclusivity is shown to improve between 18 and 30 months (Bion et al., 2013). +Starting at least at 24 months of age, children may suspend mutual exclusivity to +learn a second label for an object (Liittschwager and Markman, 1994). In a more +recent study, it has been shown that three year old children will suspend mutual +exclusivity if there are enough social cues present (Yildiz, 2020). Four to ve year +old children continue to apply mutual exclusivity to learn new words but are able +to apply it exibly, suspending it when given appropriate contextual information +(Kalashnikova et al., 2016) in order to associate multiple labels to the same familiar +object. As seen before, at 3 years of age both monolingual and polylingual children +have similar willingness to suspend mutual exclusivity (Nicoladis and Laurent, +2020; Frank and Poulin-Dubois, 2002), although polylinguals may still have a +greater tendency to accept multiple labels for the same object (Kalashnikova et al., +2015). +Here we have made an important contribution with respect to the precursor +of the current model (Ferrer-i-Cancho, 2017a): we have shown that the bias is not +theoretically inevitable (even when  > 0) according a more realistic model. In +a more complex setting, research on deep neural networks has shed light on the +architectures, learning biases and pragmatic strategies that are required for the +vocabulary learning bias to emerge (e.g. Gandhi and Lake, 2020; Gulordava et al., +2020). In section 3, we have discovered regions of the space of parameters where +strategyais not advantageous for two classes of skeleta. In the restrictive class, +where one where vertex degrees do no exceed one, as expected in the earliest stages +of vocabulary learning in children, we have unveiled the existence of a region of the30 David Carrera-Casado, Ramon Ferrer-i-Cancho +phase space where strategy ais not advantageous (Figs. 7 and 6). In the broader +class of skeleta where the degree of counterparts does not exceed one we have +found up to two distinct regions where ais not advantageous (Figs. 11 and 13). +Crucially, our model predicts that the bias should be lost in older children. +The argument is as follows. Suppose a child that has not learned a word yet. +Then his skeleton belongs to the class where vertex degrees do not exceed one. +Then suppose that the child learns a new word. It could be that he/she learns +it following strategy aorb. If he applies bthen the bias is gone at least for this +word. Let us suppose that the child learns words adhering to strategy afor as long +as possible. By doing this, he/she will increasing the number of links ( M) of the +skeleton keeping as invariant a one-to-one mapping between words and meanings +(Figs. 1 (c) and 2 (d)), which satis es that vertex degrees do not exceed one. Then +Figs. 7 and 8 predict that the longer the time strategy ais kept (when >0) the +larger the region of the phase space where ais not advantageous. Namely, as times +goes on, it will become increasingly more dicult to keep aas the best option. +Then it is not surprising that the bias weakens either in older children (e.g., Yildiz, +2020; Kalashnikova et al., 2016), as they are expected to have more links (larger M) +because of their continued accretion of new words (Saxton, 2010), or in polylinguals +(e.g., Nicoladis and Secco, 2000; Greene et al., 2013), where the mapping of words +into meanings combining all their languages, is expected to yield more links than +in monolinguals. Polylinguals make use of code-mixing to compensate for lexical +gaps, as reported for from one-year-olds onward (Nicoladis and Secco, 2000) as +well as in older children ( ve year olds) (Greene et al., 2013). As a result, the +bipartite skeleton of a polylingual integrates the words and association in all the +languages spoken and thus polylinguals are expected to have a larger value of M. +Children who know more translation equivalents (words from di erent languages +but with same meaning), adhere to mutual exclusivity less than other children +(Byers-Heinlein and Werker, 2013). Therefore, our theoretical framework provides +an explanation for the lexicon structure hypothesis (Byers-Heinlein and Werker, +2013), but shedding light on the possible origin of the mechanism, that is not the +fact that there are already synonyms but rather the large number of links (Fig. +8) as well as the capacity of words of higher degree to attract more meanings, a +consequence of Eq. 3 with >0 in the vocabulary learning process (Fig. 3). Recall +the stark contrast between Fig. 10 for = 1 and the Fig. 9 with = 0, where +such attraction e ect is missing. Our models o er a transparent theoretical tool +to understand the failure of deep neural networks to reproduce the vocabulary +learning bias (Gandhi and Lake, 2020): in its simpler form (vertex degrees do not +exceed one), whether it is due to an excessive (Fig. 7) or an excessive M(Fig. +8). +We have focused on the loss of the bias in older children. However, there is +evidence that the bias is missing initially in children, by the age of 14 months +(Halberda, 2003). We speculate that this could be related to very young children +having lower values of or larger values of as suggested by Figs. 7 and 6. This +issue should be the subject of future research. Methods to estimate andin real +speakers should be investigated. +Now we turn our attention to skeleta where only the degree of the counterparts +does not exceed one, that we believe to be more appropriate for older children. +Whereas,andMsuced for the exploration of the phase space when vertex +degrees do not exceed one, the exploration of that kind of skeleta involved manyThe advent and fall of a vocabulary learning bias from communicative eciency 31 +parameters: ,,n,kand . The more general class exhibits behaviors that we +have already seen in the more restrictive class. While an increase in Mimplies a +widening of the region where ais not advantageous in the more restrictive class, +the more general class experiences an increase of Mwhennis increased but and +remain constant (Section 2.1.2). Consistently with the more restrictive class, +such increase of Mleads to a growth of the regions where ais not advantageous as +it can be seen in Figs. 16, 10, 17, 18 and 19 when selecting a column (thus xing + and) and moving from the top to the bottom increasing n. The challenge is +that may not remain constant in real children as they become older and how to +involve the remainder of the parameters in the argument. In fact, some of these +parameters are known to be correlated with child's age: +{ntends to increase over time in children, as children are learning new words +over time (Saxton, 2010). We assume that the loss of words can be neglected +in children. +{Mtends to increase over time in children. In this class of skeleta, the growth +ofMhas two sources: the learning of new words as well as the learning of new +meanings for existing words. We assume that the loss of connections can be +neglected in children. +{The ambiguity of the words that children learn over time tends to increase over +time (Casas et al., 2018). This does not imply that children are learning all +the meanings of the word according to some online dictionary but rather than +as times go on, children are able to handle words that have more meanings +according to adult standards. +{ remains stable over time or tends to decrease over time in children depending +on the individual (Baixeries et al., 2013; Zipf, 1949, Chapter IV). +For other parameters, we can just speculate on their evolution with child's age. +The growth of Mand the increase in the learning of ambiguous words over time +leads to expect that the maximum value of kwill be larger in older children. It +is hard to tell if older children will have a chance to encounter larger values of +k. We do not know the value of in real language but the higher diversity of +vocabulary in older children and adults (Baixeries et al., 2013) suggests that  +may tend to increase over time, because the lower the value of , the higher the +pressure to minimize the entropy of words (Eq. 4), namely the higher the force +towards uni cation in Zipf's view (Zipf, 1949). We do not know the real value of  +but a reasonable choice for adult language is = 1 (Ferrer-i-Cancho and Vitevitch, +2018). +Given the complexity of the space of parameters in the more general class +of skeleta where only the degrees of counterparts cannot exceed one, we cannot +make predictions that are as strong as those stemming from the class where vertex +degrees cannot exceed one. However, we wish to make some remarks suggesting +that a weakening of the vocabulary learning bias is also expected in older children +for this class (provided that >0). The combination of increasing nand a value +of that is stable over time suggests a weakening of the strategy aover time from +di erent perspectives +{Children evolve on a column of panels (constant ) of the matrix of panels +in Figs. 16, 10, 17, 18 and 19, moving from top (low n) to the bottom (large +n). That trajectory implies an increase of the size of the blue region, where +strategyais not advantageous.32 David Carrera-Casado, Ramon Ferrer-i-Cancho +{We do not know the temporal evolution of kbut oncekis xed, namely a +row of panels is selected in Figs. 20, 12, 21, 22 and 23, children evolve from +left (lower n) to right (higher n), which implies an increase of the size of the +blue region where strategy ais not advantageous as children become older. +{Within each panel in Figs. 24, 14, 25, 26 and 27, an increase of n, as a results +of vocabulary learning over time, implies a widening of the blue region. +In the preceding analysis we have assumed that remains stable over time. We +wish to speculate on the combination of increasing nand decreasing as time goes +on in certain children. In that case, children would evolve close to the diagonal of +the matrix of panels, starting from the right-upper corner (low n, high , panel +(c)) towards the lower-left corner (high n, low , panel (g)) in Figs. 16, 10, 17, +18 and 19, which implies an increase of the size of the blue region where strategy +ais not advantageous. Recall that we have argued that a combined increase of n +and decrease of is likely to lead in the long run to an increase of M(Fig. 4). We +suggest that the behavior "along the diagonal" of the matrix is an extension of +the weakening of the bias when Mis increased in the more restrictive class (Fig. +8). +In our exploration of the phase space for the class of the skeleta where the +degrees of counterparts do not exceed one, we assumed a right-truncated power-law +with two parameters, andnas a model for Zipf's rank-frequency law. However, +distributions giving a better t have been considered (Li et al., 2010) and function +(distribution) capturing the shape of the law of what Piotrowski called saturated +samples (Piotrowski and Spivak, 2007) should be considered in future research. Our +exploration of the phase space was limited by a brute force approach neglecting +the negative correlation between nand that is expected in children where +and time are negatively correlated: as children become older, nincreases as a +result of word learning (Saxton, 2010) but decreases (Baixeries et al., 2013). A +more powerful exploration of the phase space could be performed with a realistic +mathematical relationship of the expected correlation between nand , which +invites to empirical research. Finally, there might be deeper and better ways of +parameterizing the class of skeleta. +4.2 Biosemiotics +Biosemiotics is concerned about building bridges between biology, philosophy, lin- +guistics, and the communication sciences as announced in the front page of this +journal https://www.springer.com/journal/12304 . As far as we know, there is lit- +tle research on the vocabulary learning bias in other species. Its con rmation in +a domestic dog suggests that \ the perceptual and cognitive mechanisms that may +mediate the comprehension of speech were already in place before early humans began +to talk " (Kaminski et al., 2004). We hypothesize that the cost function +cap- +tures the essence of these mechanisms. A promising target for future research are +ape gestures, where there has been signi cant progress recently on their meaning +(Hobaiter and Byrne, 2014). As far as we know, there is no research on that bias +in other domains that also fall into the scope of biosemiotics, e.g., in unicellu- +lar organisms such as bacteria. Our research has established some mathematical +foundations for research on the accretion and interpretation of signs across theThe advent and fall of a vocabulary learning bias from communicative eciency 33 +living world, not only among great apes, a key problem in research program of +biosemiotics (Kull, 2018). +The remainder of the discussion section is devoted to examine general chal- +lenges that are shared by biosemiotics and quantitative linguistics, a eld that, as +biosemiotics, aspires to contribute to develop a science beyond human communi- +cation. +4.3 Science and its method +It has been argued that a problem of research on the rank-frequency is law is the +The absence of novel predictions... which has led to a very peculiar situation in the +cognitive sciences, where we have a profusion of theories to explain an empirical phe- +nomenon, yet very little attempt to distinguish those theories using scienti c methods. +(Piantadosi, 2014). As we have already shown the predictive power of a model +whose original target was the rank-frequency laws here and in previous research +(Ferrer-i-Cancho, 2017a), we take this criticism as an invitation to re ect on sci- +ence and its method (Altmann, 1993; Bunge, 2001). +4.3.1 The generality of the patterns for theory construction +While in psycholinguisics and the cognitive sciences a major source of evidence are +often experiments involving restricted tasks or sophisticated statistical analyses +covering a handful of languages (typically English and a few other Indo-European +languages), quantitative linguistics aims to build theory departing from statistical +laws holding in a typologically wide range of languages (K ohler, 1987; Debowski, +2020) as re ected in Fig. 1. In addition, here we have investigated a speci c vocab- +ulary learning phenomenon that is, however, supported cross-linguistically (recall +Section 1). A recent review on the eciency of languages, only pays attention to +the law of abbreviation (Gibson et al., 2019) in contrast with the body of work +that has been developed in the last decades linking laws with optimization princi- +ples (Fig. 1), suggesting that this law is the only general pattern of languages that +is shaped by eciency or that linguistic laws are secondary for deep theorizing +on eciency. In other domains of the cognitive sciences, the importance of scaling +laws has been recognized (Chater and Brown, 1999; Kello et al., 2010; Baronchelli +et al., 2013). +4.3.2 Novel predictions +In section 4.1, we have checked predictions of our information theoretic framework +that matches knowledge on the vocabulary learning bias from past research. Our +theoretical framework allows the researcher to play the game of science in another +direction: use the relevant parameters to guide the design of new experiments with +children or adults where more detailed predictions of the theoretical framework +can be tested. For children who have about the same nand , and= 1, our +model predicts that strategy awill be discarded if (Fig. 10) +(1)is low and k(Fig.3) is large enough. +(2)is high and kis suciently low.34 David Carrera-Casado, Ramon Ferrer-i-Cancho +Interestingly, there is a red horizontal band in Fig. 10, and even for other values of +such that6= 1 but keeping >0 (Figs. 16, 17, 18, 19), indicating the existence +of some value of kor a range of kwhere strategy ais always advantageous (notice +however, that when >1, the band may become too narrow for an integer kto + t as suggested by Figs. 31, 32, 33 in Appendix D). Therefore the 1st concrete +prediction is that, for a given child, there is likely to be some range or value of k +where the bias (strategy a) will be observed. The 2nd concrete prediction that can +be made is on the conditions where the bias will not be observed. Although the +true value of is not known yet, previous theoretical research with = 0 suggests +that1=2 in real language (Ferrer-i-Cancho and Sole, 2003; Ferrer-i-Cancho, +2005b, 2006, 2005a), which would imply that real speakers should satisfy only (1). +Child or adult language researchers may design experiments where kis varied. +If successful, that would con rm the lexicon structure hypothesis (Byers-Heinlein +and Werker, 2013) but providing a deeper understanding. These are just examples +of experiments that could be carried out. +4.3.3 Towards a mathematical theory of language eciency +Our past and current research on the eciency are supported by a cost function +and a (analytical or numerical) mathematical procedure that links the minimiza- +tion of the cost function with the target phenomena, e.g., vocabulary learning, +as in research on how pressure for eciency gives rise to Zipf's rank-frequency +law, the law of abbreviation or Menzerath's law (Ferrer-i-Cancho, 2005b; Gusti- +son et al., 2016; Ferrer-i-Cancho et al., 2019). In the cognitive sciences, such a +cost function and the mathematical linking argument are sometimes missing (e.g., +Piantadosi et al., 2011) and neglected when reviewing how languages are shaped +by eciency (Gibson et al., 2019). A truly quantitative approach in the context +of language eciency is two-fold: it has to comprise either a quantitative descrip- +tion of the data and a quantitative theorizing, i.e. it has to employ both statistical +methods of analysis and mathematical methods to de ne the cost and the how cost +minimization leads to the expected phenomena. Our framework relies on standard +information theory (Cover and Thomas, 2006) and its extensions (Ferrer-i-Cancho +et al., 2019; Debowski, 2020). The psychological foundations of the information +theoretic principles postulated in that framework and the relationships between +them have already been reviewed (Ferrer-i-Cancho, 2018). How the so-called noisy- +channel \theory" or noisy-channel hypothesis explains the results in (Piantadosi +et al., 2011), others reviewed recently (Gibson et al., 2019) or language laws in a +broad sense has not yet shown, to our knowledge, with detailed enough information +theory arguments. Furthermore, the major conclusions of the statistical analysis +of (Piantadosi et al., 2011) have recently been shown to change substantially after +improving the methods: e ects attributable to plain compression are stronger than +previously reported (Meylan and Griths, 2021). Theory is crucial to reduce false +positives and replication failures (Stewart and Plotkin, 2021). In addition, higher +order compression can explain more parsimoniously phenomena that are central +in noisy-channel \theorizing" (Ferrer-i-Cancho, 2017b).The advent and fall of a vocabulary learning bias from communicative eciency 35 +4.3.4 The trade-o between parsimony and perfect t. +Our emphasis is on generality and parsimony over perfect t. Piantadosi (2014) +makes emphasis on what models of Zipf's rank-frequency law apparently do not +explain while our emphasis is on what the models do explain and the many predic- +tions they make (Table 1), in spite of their simple design. It is worth reminding a +big lesson from machine learning, i.e. a perfect t can be obtained simply by over- + tting the data and another big lesson from the philosophy of science to machine +learning and AI: sophisticated models (specially deep learning ones) are in most +cases black boxes that imitate complex behavior but neither explain nor yield un- +derstanding. In our theoretical framework, the principle of contrast (Clark, 1987) +or the mutual exclusivity bias (Markman and Wachtel, 1988; Merriman and Bow- +man, 1989) are not principles per se (or core principles) but predictions of the prin- +ciple of mutual information maximization involved in explaining the emergence of +Zipf's rank-frequency law (Ferrer-i-Cancho and Sole, 2003; Ferrer-i-Cancho, 2005b) +and word order patterns (Ferrer-i-Cancho, 2017b). Although there are computa- +tional models that are able to account for that vocabulary learning bias and other +phenomena (Frank et al., 2009; Gulordava et al., 2020), ours is much simpler, +transparent (in opposition to black box modeling) and to the best our knowledge, +the rst to predict that the bias will weaken over time providing a preliminary +understanding of why this could happen. +Acknowledgements We are grateful to two anonymous reviewers for their valuable feeback +and recommendations to improve the article. We are also grateful to A. Hern andez-Fern andez +and G. Boleda for their revision of the article and many recommendations to improve it. The +article has bene ted from discussions with T. Brochhagen, S. Semple and M. Gustison. Finally, +we thank C. Hobaiter for her advice and inspiration for future research. DCC and RFC are +supported by the grant TIN2017-89244-R from MINECO (Ministerio de Econom a, Industria +y Competitividad). 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Section A.1 details the expressions for probabilities and +entropies introduced in Section 2. Section A.2 addresses the general problem of the dynamic +calculation of +(Eq. 8) when a cell of the adjacency matrix is mutated, deriving the formulae +to update these entropies once a single mutation has taken place. Finally, Section A.3 applies +these formulae to derive the expressions for presented in Section 2.1.40 David Carrera-Casado, Ramon Ferrer-i-Cancho +A.1 Probabilities and entropies +In section 2, we obtained an expression for the joint probability of a form and a counterpart (Eq. +6) and the corresponding normalization factor, M(Eq. 7). Notice that M0is the number of +edges of the bipartite graph. i.e. M=M0. To ease the derivation of the marginal probabilities, +we de ne +;i=mX +j=1aij! +j(18) +!;i=nX +i=1aij +i: (19) +Notice that ;iand!;jshould not be confused with iand!i(the degree of the form iand +of the counterpart jrespectively). Indeed, i=0;iand!j=!0;j. From the joint probability +(Eq. 6), we obtain the marginal probabilities +p(si) =mX +j=1p(si;rj) += +i;i +M(20) +p(rj) =nX +i=1p(si;rj) +=! +j!;j +M: (21) +To obtain expressions for the entropies, we use the rule +X +ixi +Tlogxi +T= logT1 +TX +ixilogxi; (22) +which holds whenP +ixi=T. +We can now derive the entropies H(S;R),H(S) andH(R). Applying Eq. 6 to +H(S;R) =nX +i=1mX +j=1p(si;rj) logp(si;rj); +we obtain +H(S;R) = logM +MnX +i=1mX +j=1aij(i!j)log(i!j): +Applying Eq. 20 and the rule in Eq. 22, +H(S) =nX +i=1p(si) logp(si) +becomes +H(S) = logM1 +MnX +i=1 + +i;i +log + +i;i +: +By symmetry, equivalent formulae for H(R) can be derived easily using Eq. 21, obtaining +H(R) = logM1 +MmX +j=1 +! +j!;j +log +! +j!;j +:The advent and fall of a vocabulary learning bias from communicative eciency 41 +Interestingly, when = 0, the entropies simplify as +H(S;R) = logM0 +H(S) = logM01 +M0nX +i=1ilogi +H(R) = logM01 +M0mX +j=1!ilog!i +as expected from previous work (Ferrer-i-Cancho, 2005b). Given the formulae for H(S;R), +H(S) andH(R) above, the calculation of +() (Eq. 9) is straightforward. +A.2 Change in entropies after a single mutation in the adjacency matrix +Here we investigate a general problem: the change in the entropies needed to calculate +when +there is a single mutation in the cell ( i;j) of the adjacency matrix, i.e. when a link between +a formiand a counterpart jis added (aijbecomes 1) or deleted ( aijbecomes 0). The goal +of this analysis is to provide the mathematical foundations for research on the evolution of +communication and in particular, the problem of learning of a new word, i.e. linking a form +that was previously unlinked (Appendix A.3), which is a particular case of mutation where +aij= 0 andi= 0 before the mutation ( aij= 1 andi= 1 after the mutation). +Firstly, we express the entropies compactly as +H(S;R) = logM +MX(S;R) (23) +H(S) = logM1 +MX(S) (24) +H(R) = logM1 +MX(R) (25) +with +X(S;R) =X +(i;j)2Ex(si;rj) +X(S) =nX +i=1x(si) +X(R) =mX +j=1x(rj) (26) +x(si;rj) = (i!j)log(i!j) (27) +x(si) = + +i;i +log + +i;i +(28) +x(rj) = +! +j!;j +log +! +j!;j +: (29) +We will use a prime mark to indicate the new value of a certain measure once a mutation has +been produced in the adjacency matrix. Suppose that aijmutates. Then +a0 +ij= 1aij +0 +i=i+ (1)aij (30) +!0 +j=!j+ (1)aij: (31) +We de neS(i) as the set of neighbors of siin the graph and, similarly, R(j) as the set of +neighbors of rjin the graph. Then 0 +;kcan only change if k=iork2R(j) (recall Eq. 18)42 David Carrera-Casado, Ramon Ferrer-i-Cancho +and!0 +;lcan only change if l=jorl2R(i) (Eq. 19). Then, for any ksuch that 1kn, +we have that +0 +;k=8 +>< +>:;kaij! +j+ (1aij)!0 +jifk=i +;k! +j+!0 +jifk2R(j) +;k otherwise:(32) +Likewise, for any lsuch that 1lm, we have that +!0 +;l=8 +< +:!;laij +i+ (1aij)0 +iifl=j +!;l +i+0 +iifl2S(i) +!;l otherwise:(33) +We then aim to calculate M0 +andX0(S;R) fromMandX(S;R) (Eq. 7 and Eq. 23) respec- +tively. Accordingly, we focus on the pairs ( sk;rl), shortly (k;l), such that 0 +k!0 +l=k!lmay +not hold. These pairs belong to E(i;j)[(i;j), whereE(i;j) is the set of edges having sior +rjat one of the ends. That is, E(i;j) is the set of edges of the form ( i;l) wherel2S(i) or +(k;j) wherek2R(j). Then the new value of Mwill be +M0 +=M2 +4X +(k;l)2E(i;j)(k!l)3 +5aij(i!j) ++2 +4X +(k;l)2E(i;j)(0 +k!0 +l)3 +5+ (1aij)(0 +i!0 +j):(34) +Similarly, the new value of X(S;R) will be +X0(S;R) =X(S;R)2 +4X +(k;l)2E(i;j)x(sk;rl)3 +5aijx(si;rj) ++2 +4X +(k;l)2E(i;j)x0(sk;rl)3 +5+ (1aij)x0(si;rj):(35) +x0(si;rj) can be obtained by applying 0 +iand!0 +j(Eqs. 30 and 31) to x(si;rj) (Eq. 27). The +value ofH0(S;R) is then obtained applying M0 +(Eq. 34) and X(S;R)0(Eq. 35) to H(S;R) +(Eq. 23). +As forH0(S), notice that x0(sk) can only di er from x(sk) if0 +kand0 +;kchange, namely +whenk=iork2R(j). Therefore +X0(S) =X(S)2 +4X +k2R(j)x(sk)3 +5aijx(si) +2 +4X +k2R(j)x0(sk)3 +5+ (1aij)x0(si):(36) +Similarly,x0(si) can be obtained by applying i(Eq. 30) and ;i(Eq. 32) to x(si) (Eq 28). +ThenH0(S) is obtained by applying MandX0(S) (Eqs. 34 and 36) to H(S) (Eq. 24). By +symmetry, +X0(R) =X(R)2 +4X +l2S(i)x(rl)3 +5aijx(rj) +2 +4X +l2S(i)x0(rl)3 +5+ (1aij)x0(rj); (37) +wherex0(rj) andH0(R) are obtained similarly, applying !j(Eq. 33)x(rj) (Eq. 29) and nally +H(R) (Eq. 25).The advent and fall of a vocabulary learning bias from communicative eciency 43 +A.3 Derivation of  +Following from the previous sections, we set o to obtain expressions for for each of the +skeleton classes we set out to study. As before, we denote the value of a variable after applying +either strategy with a prime mark, meaning that it is a modi ed value after a mutation in the +adjacency matrix. We also use a subindex aorbto indicate the vocabulary learning strategy +corresponding to the mutation. A value without prime mark then denotes the state of that +variable before applying either strategy. +Firstly, we aim to obtain an expression for that depends on the new values of the +entropies after either strategy aorbhas been chosen. Combining () (Eq. 10) with +() +(Eq. 9), one obtains +() = (12)(H0 +a(S)H0 +b(S))(H0 +a(R)H0 +b(R)) +(H0 +a(S;R)H0 +b(S;R)): +The application of H(S;R) (Eq. 23), H(S) (Eq. 24) and H(R) (Eq. 25), yields +() = (12) logM0 +a +M0 +b1 +M0 +aM0 +b +(12)X(S)X(R)+X(S;R) +(38) +with +X(S)=M0 +bX0 +a(S)M0 +aX0 +b(S) +X(R)=M0 +bX0 +a(R)M0 +aX0 +b(R) +X(S;R)=M0 +bX0 +a(S;R)M0 +aX0 +b(S;R): +Now we nd expressions for M0 +a,X0 +a(S;R),X0 +a(S),X0 +a(R),M0 +b,X0 +b(S;R),X0 +b(S),X0 +b(R). +To obtain generic expressions for M0 +,X0(S;R),X0(S) andX0(R) via Eqs. 34, 35, 36 and 37, +we de ne mathematically the state of the bipartite matrix before and after applying either +strategyaorbwith the following restrictions +{aija=aijb= 0. Formiand counterpart jare initially unconnected. +{ia=ib= 0. Formihas initially no connections. +{0 +ia=0 +ib= 1. Formiwill have one connection afterwards. +{!ja= 0. In case a, counterpart jis initially disconnected. +{!jb=!j>0. In caseb, counterpart jhas initially at least one connection. +{!0 +ja= 1. In case a, counterpart jwill have one connection afterwards. +{!0 +jb=!j+ 1. In case b, counterpart jwill have one more connection afterwards. +{Sa(i) =Sb(i) =?. Formihas initially no neighbors. +{Ra(j) =?. In casea, counterpart jhas initially no neighbors. +{Rb(j)6=?. In caseb, counterpart jhas initially some neighbors. +{Ea(i;j) =?. In casea, there are no links with iorjat one of their ends. +{Eb(i;j) =f(k;j)jk2R(j)g. In caseb, there are no links with iat one of their ends, only +withj. +We can apply these restrictions to x(si;rj),x(si) andx(rj) (Eqs. 27, 28 and 29) to obtain +expressions of x0 +a(si),x0 +b(si),x0 +b(rj) andx0 +b(si;rj) that depend only on the initial values of !j +and!;j +x0(si) =!0 +jlog!0 +j +x0 +a(si) = 0 (39) +x0 +a(rj) = 0 (40) +x0 +b(si) =(!j+ 1)log(!j+ 1) (41) +x0 +b(rj) = (!j+ 1)(!;j+ 1) log((!j+ 1)(!;j+ 1)) (42) +x0 +a(si;rJ) = 0 (43) +x0 +b(si;rj) = (!j+ 1)log(!j+ 1): (44)44 David Carrera-Casado, Ramon Ferrer-i-Cancho +Additionally, for any forms sksuch thatk2Rb(j) (that is, for every form that counterpart +jis connected to), we can also obtain expressions that depend only on the initial values of !j, +!;j,kand;kusing the same restrictions and equations +xb(sk;rj) =! +j( +klogk) + (! +jlog!j) +k(45) +x0 +b(sk;rj) = (!j+ 1)( +klogk) +h +(!j+ 1)log(!j+ 1)i + +k(46) +x0 +b(sk) =n + +k;k+ +kh +! +j+ (!j+ 1)io +log" +( +k;k) +;k! +j+ (!j+ 1) +;k!# +=sb(sk) +h +(!j+ 1)! +ji + +klogn + +kh +;k! +j+ (!j+ 1)io ++ +k;klog +;k! +j+ (!j+ 1) +;k! +:(47) +Applying the restrictions to M0 +(Eq. 34), we can also obtain an expression that depends only +on some initial values +M0 +a=M+ 1 (48) +M0 +b=M+h +(!j+ 1)! +ji +!;j+ (!j+ 1): (49) +Applying now the expressions for x0 +a(si;rj) (Eq. 43), x0 +b(si;rj) (Eq. 44), xb(sk;rj) (Eq. 45) +andx0 +b(sk;rj) (Eq. 46) to X0(S;R) (Eq. 35), along with the restrictions, we obtain +X0 +a(S;R) =X(S;R) (50) +X0 +b(S;R) =X(S;R) +h +(!j+ 1)! +jiX +k2R(j) +klogk ++!;jh +(!j+ 1)log(!j+ 1)! +jlog(!j)i ++ (!j+ 1)log(!j+ 1):(51) +Similarly, we apply x0 +a(si) (Eq. 39), x0 +b(si) (Eq. 41) and x0 +b(sk) (Eq. 47) to X0(S) (Eq. 36) as +well as the restrictions and obtain +X0 +a(S) =X(S) (52) +X0 +b(S) =X(S) +(!j+ 1)log(!j+ 1) ++h +(!j+ 1)! +jiX +k2R(j) +klogn + +kh +;k! +j+ (!j+ 1)io ++X +k2R(j) +k;klog +;k! +j+ (!j+ 1) +;k! +:(53) +We applyx0 +a(rj) (Eq. 40) and x0 +b(rj) (Eq. 42) to X0(R) (Eq. 37) along with the restrictions +and obtain +X0 +a(R) =X(R) (54) +X0 +b(R) =X(R)! +j!;jlog(! +j!;j) ++ (!j+ 1)(!;j+ 1) logh +(!j+ 1)(!;j+ 1)i +:(55) +At this point we could attempt to build an expression for for the most general case. However, +this expression would be extremely complex. Instead, we study the expression of in three +simplifying conditions: the case = 0 and the two classes of skeleta.The advent and fall of a vocabulary learning bias from communicative eciency 45 +A.3.1 The case = 0 +The condition = 0 corresponds to a model that is a precursor of the current model Ferrer- +i-Cancho (2017a), and that we use to ensure our that our general expressions are correct. We +apply= 0 to the expressions in Section A.3. M0 +aandM0 +b(Eqs. 48 and 49) both simplify as +M0 +a=M0 +b=M+ 1: (56) +X0 +a(S;R) andX0 +b(S;R) (Eqs. 50 and 51) simplify as +X0 +a(S;R) =X(S;R) (57) +X0 +b(S;R) =X(S;R) + (!j+ 1) log(!j+ 1)!jlog(!j): (58) +X0 +a(S) andX0 +b(S) (Eqs. 52 and 53) both simplify as +X0 +a(S) =X0 +b(S) =X(S): (59) +X0 +a(R) andX0 +b(R) (Eqs. 54 and 55) simplify as +X0 +a(R) =X(R) (60) +X0 +b(R) =X(R)!jlog(!j) + (!j+ 1) log(!j+ 1): (61) +The application of Eqs. 56, 57, 58, 59, 60 and 61 into the expression of (Eq. 38) results in +the expression for (Eq. 5) presented in Section 1. +A.3.2 Counterpart degrees do not exceed one +In this case we assume that !j2f0;1gfor everyrjand further simplify the expressions from +A.3 under this assumption. This is the most relaxed of the conditions and so these expressions +remain fairly complex. +M0 +aandM0 +b(Eqs. 48 and 49) simplify as +M0 +a=M+ 1 (62) +M0 +b=M+ (21) +k+ 2(63) +with +M=nX +i=1+1 +i; +X0 +a(S;R) andX0 +b(S;R) (Eqs. 50 and 51) simplify as +X0 +a(S;R) =X(S;R) (64) +X0 +b(S;R) =X(S;R) + (21) +klogk+ ( +k+ 1)2log 2 (65) +with +X(S;R) =nX +i=1+1 +ilogi: (66) +X0 +a(S) andX0 +b(S) (Eqs. 52 and 53) simplify as +X0 +a(S) =X(S) (67) +X0 +b(S) =X(S) + (21) +klogh + +k(k1 + 2)i +++1 +klogk1 + 2 +k ++2log(2)(68)46 David Carrera-Casado, Ramon Ferrer-i-Cancho +with +X(S) =nX +i=1 +i;ilog( +i;i) +=nX +i=1 +iilog( +ii) += (+ 1)nX +i=1+1 +ilogi += (+ 1)X(S;R): +X0 +a(R) andX0 +b(R) (Eqs. 54 and 55) simplify as +X0 +a(R) =X(R) (69) +X0 +b(R) =X(R) +klog(k) + 2( +k+ 1) logh +2( +k+ 1)i +(70) +with +X(R) =X(S;R): (71) +The previous result on X(R) deserves a brief explanation as it is not straightforward. Firstly, +we apply the de nition of x(rj) (Eq. 29) to that of X(R) (Eq. 26) +X(R) =mX +j=1! +j!;jlog(! +j!;j): +As counterpart degrees are one, !j= 1 and!;j= +i j, wherei jis used to indicate that +we refer to the form ithat the counterpart jis connected to (see Eq. 19). That leads to +X(R) =mX +j=1 +i jlog(i j): +In order to change the summation over each j(every counterpart) to a summation over each +i(every form) we must take into account that when summing over j, we accounted for each +formia total ofitimes. Therefore we need to multiply by iin order for the summations to +be equivalent, as otherwise we would be accounting for each form ionly once. This leads to +X(R) =nX +i=1+1 +ilogi +and eventually Eq. 71 thanks to Eq. 66. +The application of Eqs. 62, 63, 64, 65, 67, 68, 69 and 70 into the expression of (Eq. 38) +results in the expression for (Eq. 12) presented in Section 1. If we apply the two extreme +values of, i.e.= 0 and= 1, to that equation, we obtain the following expressions +(0) = log +M+ 1 +M+ (21) +k+ 2! ++1 +M+ (21) +k+ 2( +2 +klog(k) +h +(+ 1)X(S;R)(21)( +k+ 1) +M+ 12log(2) ++ +kh +log(k)(k+)(k1 + 2) log(k1 + 2)ii)The advent and fall of a vocabulary learning bias from communicative eciency 47 +(1) =log +M+ 1 +M+ (21) +k+ 2! +1 +M+ (21) +k+ 2( +( +k+ 1)2log( +k+ 1) +h +(+ 1)X(S;R)(21)( +k+ 1) +M+ 12log(2) ++ +kh +log(k)(k+)(k1 + 2) log(k1 + 2)ii) +: +A.3.3 Vertex degrees do not exceed one +As seen in Section 2.1, for this class we are working under the two conditions that !j2f0;1g +for everyrjandi2f0;1gfor everysi. We can simplify the expressions from A.3. M0 +aand +M0 +b(Eqs. 62 and 63) simplify as +M0 +a=M+ 1 (72) +M0 +b=M+ 2+11; (73) +whereM=M0=M, the number of edges in the bipartite graph. X0 +a(S;R) andX0 +b(S;R) +(Eqs. 64 and 65) simplify as +X0 +a(S;R) = 0 (74) +X0 +b(S;R) = 2+1log 2: (75) +X0 +a(S) andX0 +b(S) (Eqs. 67 and 68) simplify as +X0 +a(S) = 0 (76) +X0 +b(S) =2+1log 2: (77) +X0 +a(R) andX0 +b(R) (Eqs. 69 and 70) simplify as +X0 +a(R) = 0 (78) +X0 +b(R) = (+ 1)2+1log 2: (79) +Combining Eqs. 72, 73, 74, 75, 76, 77, 78, 79 into the equation for (Eq. 38) results in the +expression for (Eq.11) presented in Section 1. When the extreme values, i.e. = 0 and += 1, are applied to this equation, we obtain the following expressions +(0) =log +1 +2(21) +M+ 1 ++2+1log(2) +M+ 2+11 +(1) = log +1 +2(21) +M+ 1 +2+1(+ 1) log(2) +M+ 2+11: +B Form degrees and number of links +Here we develop the implications of Eq. 15 with n1= 1 andn= 0. Imposing n1= 1, +we get +c= (n1): +Inserting the previous results into the de nition of p(si) when!j1, we have that +p(si) =1 +M+1 +i +=c0i ;48 David Carrera-Casado, Ramon Ferrer-i-Cancho +with + =(+ 1) +c0=(n1) +M: +A continuous approximation to vertex degrees and the number of edges gives +M=nX +i=1i +=cn1X +i=1i += (n1)n1X +i=1i: +Thanks to well-known integral bounds (Cormen et al., 1990, pp. 50-51), we have that +Zn +1idin1X +i=1i1 +Zn1 +1idi: +as0 by de nition. When = 1, one obtains +lognn1X +i=1i11 + log(n1): +When6= 1, one obtains +1 +1 +1n1 +n1X +i=1i1 +1 +1 +1(n1)1 +: +Combining the results above, one obtains +(n1) lognM(n1)[1 + log(n1)] +for= 1 and +(n1)1 +1 +1n1 +M(n1) +1 +1 +1 +1(n1)1 +for6= 1. +C Complementary heatmaps for other values of  +In Section 3, heatmaps were used to analyze takes for distinct sets of parameters. For the +class of skeleta where counterpart degrees do not exceed one, only heatmaps corresponding to += 0 (Fig. 9) and = 1 (Figs. 10, 12 and 14) were presented. The summary gures presented +in that same section (Figs. 11, 13 and 15) already displayed the boundaries between positive +and negative values of for the whole range of values of . Heatmaps for the remainder of +values ofare presented next. +Heatmaps of as a function of andkFigures 16, 17, 18 and 19 vary kon they-axis +(while keeping on thex-axis, as with all others) and correspond to values of = 0:5,= 1:5, += 2 and= 2:5 respectively. +Heatmaps of as a function of and Figures 20, 21, 22 and 23 vary on they-axis +and correspond to values of = 0:5,= 1:5,= 2 and= 2:5 respectively. +Heatmaps of as a function of andnFigures 24, 25, 26 and 27 vary non they-axis +and correspond to values of = 0:5,= 1:5,= 2 and= 2:5 respectively.The advent and fall of a vocabulary learning bias from communicative eciency 49 +1.21.51.82.1 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.16-0.12-0.08-0.04Δ < 0 +0.0040.0080.0120.016Δ ≥ 0φ = 0.5 α = 0.5 n = 10(a) +1234 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.09-0.06-0.03Δ < 0 +0.00250.00500.00750.01000.0125Δ ≥ 0φ = 0.5 α = 1 n = 10(b) +2.55.07.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.06-0.04-0.02Δ < 0 +0.0010.0020.0030.004Δ ≥ 0φ = 0.5 α = 1.5 n = 10(c) +1234 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.020-0.015-0.010-0.005Δ < 0 +0.0020.0040.006Δ ≥ 0φ = 0.5 α = 0.5 n = 100(d) +05101520 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.0125-0.0100-0.0075-0.0050-0.0025Δ < 0 +0.0010.0020.0030.0040.005Δ ≥ 0φ = 0.5 α = 1 n = 100(e) +0255075100 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.003-0.002-0.001Δ < 0 +0.00050.00100.0015Δ ≥ 0φ = 0.5 α = 1.5 n = 100(f) +2.55.07.510.0 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.003-0.002-0.001Δ < 0 +0.00040.00080.0012Δ ≥ 0φ = 0.5 α = 0.5 n = 1000(g) +0255075100 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.0020-0.0015-0.0010-0.0005Δ < 0 +0.00040.00080.0012Δ ≥ 0φ = 0.5 α = 1 n = 1000(h) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λμk +-3e-04-2e-04-1e-04Δ < 0 +1e-042e-04Δ ≥ 0φ = 0.5 α = 1.5 n = 1000(i) +Fig. 16 Same as in Fig. 9 but with = 0:5.50 David Carrera-Casado, Ramon Ferrer-i-Cancho +1.01.21.4 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.4-0.3-0.2-0.1Δ < 0 +0.050.100.15Δ ≥ 0φ = 1.5 α = 0.5 n = 10(a) +1.01.52.0 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.3-0.2-0.1Δ < 0 +0.040.080.120.16Δ ≥ 0φ = 1.5 α = 1 n = 10(b) +123 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.2-0.1Δ < 0 +0.050.10Δ ≥ 0φ = 1.5 α = 1.5 n = 10(c) +1.01.52.02.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.100-0.075-0.050-0.025Δ < 0 +0.020.040.06Δ ≥ 0φ = 1.5 α = 0.5 n = 100(d) +246 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.125-0.100-0.075-0.050-0.025Δ < 0 +0.0250.0500.0750.100Δ ≥ 0φ = 1.5 α = 1 n = 100(e) +481216 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.06-0.04-0.02Δ < 0 +0.020.040.06Δ ≥ 0φ = 1.5 α = 1.5 n = 100(f) +1234 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.020-0.015-0.010-0.005Δ < 0 +0.0050.0100.015Δ ≥ 0φ = 1.5 α = 0.5 n = 1000(g) +481216 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.05-0.04-0.03-0.02-0.01Δ < 0 +0.010.020.030.04Δ ≥ 0φ = 1.5 α = 1 n = 1000(h) +0204060 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.020-0.015-0.010-0.005Δ < 0 +0.0050.0100.0150.020Δ ≥ 0φ = 1.5 α = 1.5 n = 1000(i) +Fig. 17 Same as in Fig. 9 but with = 1:5.The advent and fall of a vocabulary learning bias from communicative eciency 51 +1.01.11.21.31.4 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.5-0.4-0.3-0.2-0.1Δ < 0 +0.10.2Δ ≥ 0φ = 2 α = 0.5 n = 10(a) +1.21.51.82.1 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.5-0.4-0.3-0.2-0.1Δ < 0 +0.10.20.3Δ ≥ 0φ = 2 α = 1 n = 10(b) +1.01.52.02.53.0 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.4-0.3-0.2-0.1Δ < 0 +0.10.2Δ ≥ 0φ = 2 α = 1.5 n = 10(c) +1.001.251.501.752.00 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.15-0.10-0.05Δ < 0 +0.050.10Δ ≥ 0φ = 2 α = 0.5 n = 100(d) +1234 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.25-0.20-0.15-0.10-0.05Δ < 0 +0.050.100.150.20Δ ≥ 0φ = 2 α = 1 n = 100(e) +2.55.07.510.0 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.15-0.10-0.05Δ < 0 +0.050.100.15Δ ≥ 0φ = 2 α = 1.5 n = 100(f) +1.01.52.02.53.0 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.05-0.04-0.03-0.02-0.01Δ < 0 +0.010.020.030.04Δ ≥ 0φ = 2 α = 0.5 n = 1000(g) +2.55.07.510.0 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.125-0.100-0.075-0.050-0.025Δ < 0 +0.0250.0500.0750.1000.125Δ ≥ 0φ = 2 α = 1 n = 1000(h) +0102030 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.06-0.04-0.02Δ < 0 +0.020.040.06Δ ≥ 0φ = 2 α = 1.5 n = 1000(i) +Fig. 18 Same as in Fig. 9 but with = 2.52 David Carrera-Casado, Ramon Ferrer-i-Cancho +1.01.11.21.3 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.8-0.6-0.4-0.2Δ < 0 +0.10.20.30.4Δ ≥ 0φ = 2.5 α = 0.5 n = 10(a) +1.001.251.501.75 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.6-0.4-0.2Δ < 0 +0.10.20.30.4Δ ≥ 0φ = 2.5 α = 1 n = 10(b) +1.01.52.02.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.6-0.4-0.2Δ < 0 +0.10.20.30.4Δ ≥ 0φ = 2.5 α = 1.5 n = 10(c) +1.001.251.501.75 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.3-0.2-0.1Δ < 0 +0.050.100.150.200.25Δ ≥ 0φ = 2.5 α = 0.5 n = 100(d) +123 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.4-0.3-0.2-0.1Δ < 0 +0.10.20.30.4Δ ≥ 0φ = 2.5 α = 1 n = 100(e) +246 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.3-0.2-0.1Δ < 0 +0.10.20.3Δ ≥ 0φ = 2.5 α = 1.5 n = 100(f) +1.01.52.02.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.075-0.050-0.025Δ < 0 +0.020.040.060.08Δ ≥ 0φ = 2.5 α = 0.5 n = 1000(g) +246 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.2-0.1Δ < 0 +0.050.100.150.200.25Δ ≥ 0φ = 2.5 α = 1 n = 1000(h) +5101520 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.15-0.10-0.05Δ < 0 +0.050.100.15Δ ≥ 0φ = 2.5 α = 1.5 n = 1000(i) +Fig. 19 Same as in Fig. 9 but with = 2:5.The advent and fall of a vocabulary learning bias from communicative eciency 53 +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.100-0.075-0.050-0.025Δ < 0φ = 0.5 μk = 1 n = 10(a) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.006-0.004-0.002Δ < 0 +0.000250.000500.000750.00100Δ ≥ 0φ = 0.5 μk = 1 n = 100(b) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-6e-04-4e-04-2e-04Δ < 0 +0.000050.000100.00015Δ ≥ 0φ = 0.5 μk = 1 n = 1000(c) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.12-0.08-0.04Δ < 0 +0.0050.010Δ ≥ 0φ = 0.5 μk = 2 n = 10(d) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.012-0.009-0.006-0.003Δ < 0 +0.000250.000500.00075Δ ≥ 0φ = 0.5 μk = 2 n = 100(e) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.00100-0.00075-0.00050-0.00025Δ < 0 +5e-051e-04Δ ≥ 0φ = 0.5 μk = 2 n = 1000(f) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.20-0.15-0.10-0.05Δ < 0 +0.020.040.06Δ ≥ 0φ = 0.5 μk = 4 n = 10(g) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.015-0.010-0.005Δ < 0 +0.0010.0020.0030.004Δ ≥ 0φ = 0.5 μk = 4 n = 100(h) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.0015-0.0010-0.0005Δ < 0 +1e-042e-043e-04Δ ≥ 0φ = 0.5 μk = 4 n = 1000(i) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.3-0.2-0.1Δ < 0 +0.050.100.15Δ ≥ 0φ = 0.5 μk = 8 n = 10(j) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.03-0.02-0.01Δ < 0 +0.0050.010Δ ≥ 0φ = 0.5 μk = 8 n = 100(k) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.002-0.001Δ < 0 +3e-046e-049e-04Δ ≥ 0φ = 0.5 μk = 8 n = 1000(l) +Fig. 20 The same as in Fig. 12 but with = 0:5.54 David Carrera-Casado, Ramon Ferrer-i-Cancho +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.20-0.15-0.10-0.05Δ < 0 +0.010.020.030.04Δ ≥ 0φ = 1.5 μk = 1 n = 10(a) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.016-0.012-0.008-0.004Δ < 0 +0.0020.0040.006Δ ≥ 0φ = 1.5 μk = 1 n = 100(b) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.0016-0.0012-0.0008-0.0004Δ < 0 +0.000250.000500.000750.00100Δ ≥ 0φ = 1.5 μk = 1 n = 1000(c) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.6-0.5-0.4-0.3-0.2-0.1Δ < 0 +0.10.20.3Δ ≥ 0φ = 1.5 μk = 2 n = 10(d) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.06-0.04-0.02Δ < 0 +0.010.020.03Δ ≥ 0φ = 1.5 μk = 2 n = 100(e) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.006-0.005-0.004-0.003-0.002-0.001Δ < 0 +0.0010.002Δ ≥ 0φ = 1.5 μk = 2 n = 1000(f) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-1.5-1.0-0.5Δ < 0 +0.250.500.751.001.25Δ ≥ 0φ = 1.5 μk = 4 n = 10(g) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.20-0.15-0.10-0.05Δ < 0 +0.050.100.150.20Δ ≥ 0φ = 1.5 μk = 4 n = 100(h) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.020-0.015-0.010-0.005Δ < 0 +0.0050.0100.0150.020Δ ≥ 0φ = 1.5 μk = 4 n = 1000(i) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-3-2-1Δ < 0 +12Δ ≥ 0φ = 1.5 μk = 8 n = 10(j) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.8-0.6-0.4-0.2Δ < 0 +0.20.40.6Δ ≥ 0φ = 1.5 μk = 8 n = 100(k) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.075-0.050-0.025Δ < 0 +0.0250.0500.075Δ ≥ 0φ = 1.5 μk = 8 n = 1000(l) +Fig. 21 The same as in Fig. 12 but with = 1:5.The advent and fall of a vocabulary learning bias from communicative eciency 55 +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.3-0.2-0.1Δ < 0 +0.020.040.06Δ ≥ 0φ = 2 μk = 1 n = 10(a) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.03-0.02-0.01Δ < 0 +0.0030.0060.0090.012Δ ≥ 0φ = 2 μk = 1 n = 100(b) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.0025-0.0020-0.0015-0.0010-0.0005Δ < 0 +0.00040.00080.0012Δ ≥ 0φ = 2 μk = 1 n = 1000(c) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-1.00-0.75-0.50-0.25Δ < 0 +0.20.40.6Δ ≥ 0φ = 2 μk = 2 n = 10(d) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.12-0.08-0.04Δ < 0 +0.0250.0500.0750.100Δ ≥ 0φ = 2 μk = 2 n = 100(e) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.015-0.010-0.005Δ < 0 +0.00250.00500.00750.0100Δ ≥ 0φ = 2 μk = 2 n = 1000(f) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-2.0-1.5-1.0-0.5Δ < 0 +0.51.01.52.0Δ ≥ 0φ = 2 μk = 4 n = 10(g) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.6-0.4-0.2Δ < 0 +0.20.40.6Δ ≥ 0φ = 2 μk = 4 n = 100(h) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.075-0.050-0.025Δ < 0 +0.020.040.060.08Δ ≥ 0φ = 2 μk = 4 n = 1000(i) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-3-2-1Δ < 0 +123Δ ≥ 0φ = 2 μk = 8 n = 10(j) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-2.5-2.0-1.5-1.0-0.5Δ < 0 +0.51.01.52.02.5Δ ≥ 0φ = 2 μk = 8 n = 100(k) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.4-0.3-0.2-0.1Δ < 0 +0.10.20.30.4Δ ≥ 0φ = 2 μk = 8 n = 1000(l) +Fig. 22 The same as in Fig. 12 but with = 2.56 David Carrera-Casado, Ramon Ferrer-i-Cancho +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.4-0.3-0.2-0.1Δ < 0 +0.050.10Δ ≥ 0φ = 2.5 μk = 1 n = 10(a) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.05-0.04-0.03-0.02-0.01Δ < 0 +0.0040.0080.0120.016Δ ≥ 0φ = 2.5 μk = 1 n = 100(b) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.004-0.003-0.002-0.001Δ < 0 +0.00050.00100.00150.0020Δ ≥ 0φ = 2.5 μk = 1 n = 1000(c) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-1.0-0.5Δ < 0 +0.30.60.9Δ ≥ 0φ = 2.5 μk = 2 n = 10(d) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.3-0.2-0.1Δ < 0 +0.10.2Δ ≥ 0φ = 2.5 μk = 2 n = 100(e) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.03-0.02-0.01Δ < 0 +0.010.020.03Δ ≥ 0φ = 2.5 μk = 2 n = 1000(f) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-2-1Δ < 0 +12Δ ≥ 0φ = 2.5 μk = 4 n = 10(g) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-1.5-1.0-0.5Δ < 0 +0.51.01.5Δ ≥ 0φ = 2.5 μk = 4 n = 100(h) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.3-0.2-0.1Δ < 0 +0.10.20.3Δ ≥ 0φ = 2.5 μk = 4 n = 1000(i) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-3-2-1Δ < 0 +123Δ ≥ 0φ = 2.5 μk = 8 n = 10(j) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-4-3-2-1Δ < 0 +1234Δ ≥ 0φ = 2.5 μk = 8 n = 100(k) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-1.5-1.0-0.5Δ < 0 +0.51.01.5Δ ≥ 0φ = 2.5 μk = 8 n = 1000(l) +Fig. 23 The same as in Fig. 12 but with = 2:5.The advent and fall of a vocabulary learning bias from communicative eciency 57 +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.100-0.075-0.050-0.025Δ < 0φ = 0.5 μk = 1 α = 0.5(a) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.03-0.02-0.01Δ < 0 +0.000050.000100.000150.000200.00025Δ ≥ 0φ = 0.5 μk = 1 α = 1(b) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.03-0.02-0.01Δ < 0 +0.0010.0020.003Δ ≥ 0φ = 0.5 μk = 1 α = 1.5(c) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.12-0.08-0.04Δ < 0 +0.0050.010Δ ≥ 0φ = 0.5 μk = 2 α = 0.5(d) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.06-0.04-0.02Δ < 0 +2.5e-055.0e-057.5e-05Δ ≥ 0φ = 0.5 μk = 2 α = 1(e) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.02-0.01Δ < 0 +0.00050.00100.0015Δ ≥ 0φ = 0.5 μk = 2 α = 1.5(f) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.20-0.15-0.10-0.05Δ < 0 +0.020.040.06Δ ≥ 0φ = 0.5 μk = 4 α = 0.5(g) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.100-0.075-0.050-0.025Δ < 0 +0.0020.0040.0060.008Δ ≥ 0φ = 0.5 μk = 4 α = 1(h) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.02-0.01Δ < 0 +3e-046e-049e-04Δ ≥ 0φ = 0.5 μk = 4 α = 1.5(i) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.3-0.2-0.1Δ < 0 +0.050.100.15Δ ≥ 0φ = 0.5 μk = 8 α = 0.5(j) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.16-0.12-0.08-0.04Δ < 0 +0.010.020.030.040.050.06Δ ≥ 0φ = 0.5 μk = 8 α = 1(k) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.05-0.04-0.03-0.02-0.01Δ < 0 +2e-044e-046e-04Δ ≥ 0φ = 0.5 μk = 8 α = 1.5(l) +Fig. 24 The same as in Fig. 14 but with = 0:5.58 David Carrera-Casado, Ramon Ferrer-i-Cancho +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.20-0.15-0.10-0.05Δ < 0 +0.00050.00100.00150.00200.0025Δ ≥ 0φ = 1.5 μk = 1 α = 0.5(a) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.08-0.06-0.04-0.02Δ < 0 +0.0020.0040.0060.008Δ ≥ 0φ = 1.5 μk = 1 α = 1(b) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.09-0.06-0.03Δ < 0 +0.010.020.030.04Δ ≥ 0φ = 1.5 μk = 1 α = 1.5(c) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.6-0.5-0.4-0.3-0.2-0.1Δ < 0 +0.10.20.3Δ ≥ 0φ = 1.5 μk = 2 α = 0.5(d) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.20-0.15-0.10-0.05Δ < 0 +0.020.040.06Δ ≥ 0φ = 1.5 μk = 2 α = 1(e) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.08-0.06-0.04-0.02Δ < 0 +0.0050.0100.0150.020Δ ≥ 0φ = 1.5 μk = 2 α = 1.5(f) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-1.5-1.0-0.5Δ < 0 +0.250.500.751.001.25Δ ≥ 0φ = 1.5 μk = 4 α = 0.5(g) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.8-0.6-0.4-0.2Δ < 0 +0.20.40.6Δ ≥ 0φ = 1.5 μk = 4 α = 1(h) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.3-0.2-0.1Δ < 0 +0.050.100.15Δ ≥ 0φ = 1.5 μk = 4 α = 1.5(i) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-3-2-1Δ < 0 +12Δ ≥ 0φ = 1.5 μk = 8 α = 0.5(j) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-2.0-1.5-1.0-0.5Δ < 0 +0.51.01.52.0Δ ≥ 0φ = 1.5 μk = 8 α = 1(k) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.9-0.6-0.3Δ < 0 +0.250.500.751.00Δ ≥ 0φ = 1.5 μk = 8 α = 1.5(l) +Fig. 25 The same as in Fig. 14 but with = 1:5.The advent and fall of a vocabulary learning bias from communicative eciency 59 +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.3-0.2-0.1Δ < 0 +0.010.020.030.040.05Δ ≥ 0φ = 2 μk = 1 α = 0.5(a) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.075-0.050-0.025Δ < 0 +0.0030.0060.009Δ ≥ 0φ = 2 μk = 1 α = 1(b) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.15-0.10-0.05Δ < 0 +0.020.040.06Δ ≥ 0φ = 2 μk = 1 α = 1.5(c) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-1.00-0.75-0.50-0.25Δ < 0 +0.20.40.6Δ ≥ 0φ = 2 μk = 2 α = 0.5(d) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.5-0.4-0.3-0.2-0.1Δ < 0 +0.050.100.150.200.25Δ ≥ 0φ = 2 μk = 2 α = 1(e) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.075-0.050-0.025Δ < 0 +0.010.020.03Δ ≥ 0φ = 2 μk = 2 α = 1.5(f) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-2.0-1.5-1.0-0.5Δ < 0 +0.51.01.52.0Δ ≥ 0φ = 2 μk = 4 α = 0.5(g) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-1.5-1.0-0.5Δ < 0 +0.51.01.5Δ ≥ 0φ = 2 μk = 4 α = 1(h) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.8-0.6-0.4-0.2Δ < 0 +0.20.40.6Δ ≥ 0φ = 2 μk = 4 α = 1.5(i) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-3-2-1Δ < 0 +123Δ ≥ 0φ = 2 μk = 8 α = 0.5(j) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-3-2-1Δ < 0 +123Δ ≥ 0φ = 2 μk = 8 α = 1(k) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-2.0-1.5-1.0-0.5Δ < 0 +0.51.01.52.0Δ ≥ 0φ = 2 μk = 8 α = 1.5(l) +Fig. 26 The same as in Fig. 14 but with = 2.60 David Carrera-Casado, Ramon Ferrer-i-Cancho +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.4-0.3-0.2-0.1Δ < 0 +0.050.10Δ ≥ 0φ = 2.5 μk = 1 α = 0.5(a) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.125-0.100-0.075-0.050-0.025Δ < 0 +0.0030.0060.0090.012Δ ≥ 0φ = 2.5 μk = 1 α = 1(b) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.15-0.10-0.05Δ < 0 +0.020.040.06Δ ≥ 0φ = 2.5 μk = 1 α = 1.5(c) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-1.0-0.5Δ < 0 +0.30.60.9Δ ≥ 0φ = 2.5 μk = 2 α = 0.5(d) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.8-0.6-0.4-0.2Δ < 0 +0.10.20.30.40.50.6Δ ≥ 0φ = 2.5 μk = 2 α = 1(e) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.2-0.1Δ < 0 +0.020.040.060.08Δ ≥ 0φ = 2.5 μk = 2 α = 1.5(f) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-2-1Δ < 0 +12Δ ≥ 0φ = 2.5 μk = 4 α = 0.5(g) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-2.0-1.5-1.0-0.5Δ < 0 +0.51.01.52.0Δ ≥ 0φ = 2.5 μk = 4 α = 1(h) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-1.2-0.8-0.4Δ < 0 +0.51.0Δ ≥ 0φ = 2.5 μk = 4 α = 1.5(i) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-4-3-2-1Δ < 0 +1234Δ ≥ 0φ = 2.5 μk = 8 α = 0.5(j) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-3-2-1Δ < 0 +123Δ ≥ 0φ = 2.5 μk = 8 α = 1(k) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-2-1Δ < 0 +12Δ ≥ 0φ = 2.5 μk = 8 α = 1.5(l) +Fig. 27 The same as in Fig. 14 but with = 2:5.The advent and fall of a vocabulary learning bias from communicative eciency 61 +D Complementary gures with discrete degrees +To investigate the class of skeleta such that the degree of counterparts does not exceed one, +we have assumed that the relationship between the degree of a vertex and its rank follows a +power-law (Eq. 15). For the plots of the regions where strategy ais advantageous, we have +assumed, for simplicity, that the degree of a form is a continuous variable. As form degrees are +actually discrete in the model, here we show the impact of rounding form degrees de ned by +Eq. 15 to the nearest integer in previous gures. +The correspondence between the gures in this appendix with rounded form degrees and +the gures in other sections is as follows. Figs. 28, 29, 30, 31, 32 and 33 are equivalent to +Figs. 9, 16, 10, 17, 18 and 19, respectively. These are the gures where is on thex-axis and +kon they-axis of the heatmap. Fig. 34, that summarizes the boundaries of the heatmaps, +corresponds to Fig. 11 after discretization. Figs. 35, 36, 37, 38 and 39 are equivalent to Figs. +20, 12, 21, 22 and 23, respectively. In these gures, is placed on the y-axis instead. Fig. +40 summarizes the boundaries and is the discretized version of Fig. 13. Finally, Fig. 41, 42, +43, 44 and 45 are equivalent to Figs. 24, 14, 25, 26 and 27, respectively. This set places non +they-axis. The boundaries in these last discretized gures are summarized by Fig. 46, that +corresponds to Figure 15. +We have presented two kinds of gures: heatmaps showing the value of and gures +summarizing the boundaries between regions where  > 0 and < 0. Interestingly, the +discretization does not change the presence of regions where < 0 and> 0 and in general, +it does not change the shape of the regions in a qualitative sense except in some cases where +remarkable distortions appear (e.g., Figs. 32 or 33 have one or very few integer values on +they-axis for certain combinations of parameters, forming one dimensional bands that don't +change over that axis; see also the distorted shapes in Figs. 38 and specially 45). In contrast, +123 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.075-0.050-0.025Δ < 0 +0Δ ≥ 0φ = 0 α = 0.5 n = 10(a) +2.55.07.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.05-0.04-0.03-0.02-0.01Δ < 0 +0Δ ≥ 0φ = 0 α = 1 n = 10(b) +01020 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.025-0.020-0.015-0.010-0.005Δ < 0 +0Δ ≥ 0φ = 0 α = 1.5 n = 10(c) +2.55.07.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.006-0.004-0.002Δ < 0 +0Δ ≥ 0φ = 0 α = 0.5 n = 100(d) +0255075100 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.002-0.001Δ < 0 +0Δ ≥ 0φ = 0 α = 1 n = 100(e) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λμk +-5e-04-4e-04-3e-04-2e-04-1e-04Δ < 0 +0Δ ≥ 0φ = 0 α = 1.5 n = 100(f) +0102030 +0.00 0.25 0.50 0.75 1.00 +λμk +-6e-04-4e-04-2e-04Δ < 0 +0Δ ≥ 0φ = 0 α = 0.5 n = 1000(g) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.00015-0.00010-0.00005Δ < 0 +0Δ ≥ 0φ = 0 α = 1 n = 1000(h) +0100002000030000 +0.00 0.25 0.50 0.75 1.00 +λμk +-1.6e-05-1.2e-05-8.0e-06-4.0e-06Δ < 0 +0Δ ≥ 0φ = 0 α = 1.5 n = 1000(i) +Fig. 28 Figure equivalent to Fig. 9 after discretization of the 0 +is.62 David Carrera-Casado, Ramon Ferrer-i-Cancho +0.51.01.52.02.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.15-0.10-0.05Δ < 0 +0.0050.0100.015Δ ≥ 0φ = 0.5 α = 0.5 n = 10(a) +1234 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.09-0.06-0.03Δ < 0 +0.00250.00500.0075Δ ≥ 0φ = 0.5 α = 1 n = 10(b) +2.55.07.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.06-0.04-0.02Δ < 0 +0.0010.0020.0030.0040.005Δ ≥ 0φ = 0.5 α = 1.5 n = 10(c) +1234 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.015-0.010-0.005Δ < 0 +0.0010.0020.0030.004Δ ≥ 0φ = 0.5 α = 0.5 n = 100(d) +05101520 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.0125-0.0100-0.0075-0.0050-0.0025Δ < 0 +0.0010.0020.0030.0040.005Δ ≥ 0φ = 0.5 α = 1 n = 100(e) +0255075100 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.003-0.002-0.001Δ < 0 +0.00050.00100.0015Δ ≥ 0φ = 0.5 α = 1.5 n = 100(f) +2.55.07.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.003-0.002-0.001Δ < 0 +5e-041e-03Δ ≥ 0φ = 0.5 α = 0.5 n = 1000(g) +0255075100 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.0020-0.0015-0.0010-0.0005Δ < 0 +0.00050.00100.0015Δ ≥ 0φ = 0.5 α = 1 n = 1000(h) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λμk +-3e-04-2e-04-1e-04Δ < 0 +1e-042e-04Δ ≥ 0φ = 0.5 α = 1.5 n = 1000(i) +Fig. 29 Figure equivalent to Fig. 16 after discretization of the 0 +is. +the discretization has drastic impact on the summary plots of the boundary curves, where the +curvy shapes of the continuous case are lost and altered substantially in many cases (Fig. 34, +where some curves become one or a few points, or Fig. 40, re ecting the loss of the curvy +shapes).The advent and fall of a vocabulary learning bias from communicative eciency 63 +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.16-0.12-0.08-0.04Δ < 0φ = 1 α = 0.5 n = 10(a) +123 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.20-0.15-0.10-0.05Δ < 0 +0.010.020.030.040.05Δ ≥ 0φ = 1 α = 1 n = 10(b) +12345 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.10-0.05Δ < 0 +0.010.020.030.040.05Δ ≥ 0φ = 1 α = 1.5 n = 10(c) +123 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.05-0.04-0.03-0.02-0.01Δ < 0 +0.0050.0100.0150.020Δ ≥ 0φ = 1 α = 0.5 n = 100(d) +2.55.07.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.03-0.02-0.01Δ < 0 +0.010.02Δ ≥ 0φ = 1 α = 1 n = 100(e) +0102030 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.020-0.015-0.010-0.005Δ < 0 +0.0050.0100.015Δ ≥ 0φ = 1 α = 1.5 n = 100(f) +12345 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.0075-0.0050-0.0025Δ < 0 +0.0020.0040.006Δ ≥ 0φ = 1 α = 0.5 n = 1000(g) +0102030 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.010-0.005Δ < 0 +0.00250.00500.00750.0100Δ ≥ 0φ = 1 α = 1 n = 1000(h) +050100150 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.004-0.003-0.002-0.001Δ < 0 +0.0010.0020.0030.004Δ ≥ 0φ = 1 α = 1.5 n = 1000(i) +Fig. 30 Figure equivalent to Fig. 10 after discretization of the 0 +is.64 David Carrera-Casado, Ramon Ferrer-i-Cancho +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.15-0.12-0.09-0.06Δ < 0φ = 1.5 α = 0.5 n = 10(a) +0.51.01.52.02.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.2-0.1Δ < 0 +0.020.040.06Δ ≥ 0φ = 1.5 α = 1 n = 10(b) +123 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.12-0.09-0.06-0.03Δ < 0 +0.010.020.030.040.05Δ ≥ 0φ = 1.5 α = 1.5 n = 10(c) +0.51.01.52.02.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.06-0.04-0.02Δ < 0 +0.010.02Δ ≥ 0φ = 1.5 α = 0.5 n = 100(d) +246 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.100-0.075-0.050-0.025Δ < 0 +0.0250.0500.075Δ ≥ 0φ = 1.5 α = 1 n = 100(e) +051015 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.06-0.04-0.02Δ < 0 +0.010.020.030.040.050.06Δ ≥ 0φ = 1.5 α = 1.5 n = 100(f) +123 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.015-0.010-0.005Δ < 0 +0.00250.00500.00750.0100Δ ≥ 0φ = 1.5 α = 0.5 n = 1000(g) +051015 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.04-0.03-0.02-0.01Δ < 0 +0.010.020.030.04Δ ≥ 0φ = 1.5 α = 1 n = 1000(h) +0204060 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.020-0.015-0.010-0.005Δ < 0 +0.0050.0100.0150.020Δ ≥ 0φ = 1.5 α = 1.5 n = 1000(i) +Fig. 31 Figure equivalent to Fig. 17 after discretization of the 0 +is.The advent and fall of a vocabulary learning bias from communicative eciency 65 +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.5-0.4-0.3-0.2-0.1Δ < 0 +0.050.100.150.20Δ ≥ 0φ = 2 α = 0.5 n = 10(a) +0.51.01.52.02.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.4-0.3-0.2-0.1Δ < 0 +0.050.100.150.20Δ ≥ 0φ = 2 α = 1 n = 10(b) +123 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.4-0.3-0.2-0.1Δ < 0 +0.050.100.150.200.25Δ ≥ 0φ = 2 α = 1.5 n = 10(c) +0.51.01.52.02.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.15-0.10-0.05Δ < 0 +0.030.060.09Δ ≥ 0φ = 2 α = 0.5 n = 100(d) +1234 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.10-0.05Δ < 0 +0.0250.0500.0750.100Δ ≥ 0φ = 2 α = 1 n = 100(e) +2.55.07.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.125-0.100-0.075-0.050-0.025Δ < 0 +0.030.060.090.12Δ ≥ 0φ = 2 α = 1.5 n = 100(f) +123 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.04-0.03-0.02-0.01Δ < 0 +0.010.020.030.04Δ ≥ 0φ = 2 α = 0.5 n = 1000(g) +2.55.07.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.075-0.050-0.025Δ < 0 +0.020.040.060.08Δ ≥ 0φ = 2 α = 1 n = 1000(h) +0102030 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.06-0.04-0.02Δ < 0 +0.020.040.06Δ ≥ 0φ = 2 α = 1.5 n = 1000(i) +Fig. 32 Figure equivalent to Fig. 18 after discretization of the 0 +is.66 David Carrera-Casado, Ramon Ferrer-i-Cancho +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.6-0.4-0.2Δ < 0 +0.10.20.3Δ ≥ 0φ = 2.5 α = 0.5 n = 10(a) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.15-0.10-0.05Δ < 0φ = 2.5 α = 1 n = 10(b) +0.51.01.52.02.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.20-0.15-0.10-0.05Δ < 0 +0.0250.0500.0750.1000.125Δ ≥ 0φ = 2.5 α = 1.5 n = 10(c) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.05-0.04-0.03-0.02-0.01Δ < 0 +0.00250.00500.0075Δ ≥ 0φ = 2.5 α = 0.5 n = 100(d) +123 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.16-0.12-0.08-0.04Δ < 0 +0.050.10Δ ≥ 0φ = 2.5 α = 1 n = 100(e) +246 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.3-0.2-0.1Δ < 0 +0.10.20.3Δ ≥ 0φ = 2.5 α = 1.5 n = 100(f) +0.51.01.52.02.5 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.04-0.03-0.02-0.01Δ < 0 +0.010.020.03Δ ≥ 0φ = 2.5 α = 0.5 n = 1000(g) +246 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.20-0.15-0.10-0.05Δ < 0 +0.050.100.150.20Δ ≥ 0φ = 2.5 α = 1 n = 1000(h) +05101520 +0.00 0.25 0.50 0.75 1.00 +λμk +-0.15-0.10-0.05Δ < 0 +0.050.100.15Δ ≥ 0φ = 2.5 α = 1.5 n = 1000(i) +Fig. 33 Figure equivalent to Fig. 19 after discretization of the 0 +is.The advent and fall of a vocabulary learning bias from communicative eciency 67 +1.001.251.501.752.00 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +2 +2.5α = 0.5 n = 10(a) +2.02.53.03.54.0 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +1 +1.5 +2α = 1 n = 10(b) +2.55.07.5 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +1 +1.5 +2 +2.5α = 1.5 n = 10(c) +1234 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +1 +1.5 +2 +2.5α = 0.5 n = 100(d) +05101520 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +1 +1.5 +2 +2.5α = 1 n = 100(e) +0255075100 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +1 +1.5 +2 +2.5α = 1.5 n = 100(f) +2.55.07.5 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +1 +1.5 +2 +2.5α = 0.5 n = 1000(g) +0255075100 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +1 +1.5 +2 +2.5α = 1 n = 1000(h) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λμkφ +0.5 +1 +1.5 +2 +2.5α = 1.5 n = 1000(i) +Fig. 34 Figure equivalent to Fig. 11 after discretization of the 0 +is. It summarizes Figs. 29, +30, 31, 32 and 33.68 David Carrera-Casado, Ramon Ferrer-i-Cancho +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.09-0.06-0.03Δ < 0φ = 0.5 μk = 1 n = 10(a) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.008-0.006-0.004-0.002Δ < 0 +0.000250.000500.000750.00100Δ ≥ 0φ = 0.5 μk = 1 n = 100(b) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-6e-04-4e-04-2e-04Δ < 0 +0.000050.000100.00015Δ ≥ 0φ = 0.5 μk = 1 n = 1000(c) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.15-0.10-0.05Δ < 0 +0.0050.0100.015Δ ≥ 0φ = 0.5 μk = 2 n = 10(d) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.0125-0.0100-0.0075-0.0050-0.0025Δ < 0 +0.000250.000500.00075Δ ≥ 0φ = 0.5 μk = 2 n = 100(e) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-9e-04-6e-04-3e-04Δ < 0 +5e-051e-04Δ ≥ 0φ = 0.5 μk = 2 n = 1000(f) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.25-0.20-0.15-0.10-0.05Δ < 0 +0.020.040.060.08Δ ≥ 0φ = 0.5 μk = 4 n = 10(g) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.015-0.010-0.005Δ < 0 +0.0010.0020.0030.004Δ ≥ 0φ = 0.5 μk = 4 n = 100(h) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.0015-0.0010-0.0005Δ < 0 +1e-042e-043e-04Δ ≥ 0φ = 0.5 μk = 4 n = 1000(i) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.4-0.3-0.2-0.1Δ < 0 +0.050.100.15Δ ≥ 0φ = 0.5 μk = 8 n = 10(j) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.03-0.02-0.01Δ < 0 +0.0050.010Δ ≥ 0φ = 0.5 μk = 8 n = 100(k) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.002-0.001Δ < 0 +0.000250.000500.000750.00100Δ ≥ 0φ = 0.5 μk = 8 n = 1000(l) +Fig. 35 Figure equivalent to Fig. 20 after discretization of the 0 +is.The advent and fall of a vocabulary learning bias from communicative eciency 69 +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.16-0.12-0.08-0.04Δ < 0 +0.0050.0100.0150.020Δ ≥ 0φ = 1 μk = 1 n = 10(a) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.009-0.006-0.003Δ < 0 +0.0010.0020.0030.004Δ ≥ 0φ = 1 μk = 1 n = 100(b) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.00100-0.00075-0.00050-0.00025Δ < 0 +1e-042e-043e-044e-045e-04Δ ≥ 0φ = 1 μk = 1 n = 1000(c) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.3-0.2-0.1Δ < 0 +0.0250.0500.0750.100Δ ≥ 0φ = 1 μk = 2 n = 10(d) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.02-0.01Δ < 0 +0.0020.0040.006Δ ≥ 0φ = 1 μk = 2 n = 100(e) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.0025-0.0020-0.0015-0.0010-0.0005Δ < 0 +1e-042e-043e-044e-045e-04Δ ≥ 0φ = 1 μk = 2 n = 1000(f) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.6-0.4-0.2Δ < 0 +0.10.20.30.4Δ ≥ 0φ = 1 μk = 4 n = 10(g) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.06-0.04-0.02Δ < 0 +0.010.020.030.04Δ ≥ 0φ = 1 μk = 4 n = 100(h) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.006-0.004-0.002Δ < 0 +0.0010.0020.003Δ ≥ 0φ = 1 μk = 4 n = 1000(i) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-1.5-1.0-0.5Δ < 0 +0.250.500.751.001.25Δ ≥ 0φ = 1 μk = 8 n = 10(j) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.15-0.10-0.05Δ < 0 +0.050.100.15Δ ≥ 0φ = 1 μk = 8 n = 100(k) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.015-0.010-0.005Δ < 0 +0.0050.010Δ ≥ 0φ = 1 μk = 8 n = 1000(l) +Fig. 36 Figure equivalent to Fig. 12 after discretization of the 0 +is.70 David Carrera-Casado, Ramon Ferrer-i-Cancho +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.15-0.10-0.05Δ < 0 +0.010.020.030.040.05Δ ≥ 0φ = 1.5 μk = 1 n = 10(a) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.015-0.010-0.005Δ < 0 +0.0020.0040.0060.008Δ ≥ 0φ = 1.5 μk = 1 n = 100(b) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.0015-0.0010-0.0005Δ < 0 +0.000250.000500.000750.00100Δ ≥ 0φ = 1.5 μk = 1 n = 1000(c) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.5-0.4-0.3-0.2-0.1Δ < 0 +0.050.100.150.200.25Δ ≥ 0φ = 1.5 μk = 2 n = 10(d) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.06-0.04-0.02Δ < 0 +0.010.02Δ ≥ 0φ = 1.5 μk = 2 n = 100(e) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.006-0.004-0.002Δ < 0 +0.0010.002Δ ≥ 0φ = 1.5 μk = 2 n = 1000(f) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-1.0-0.5Δ < 0 +0.30.60.9Δ ≥ 0φ = 1.5 μk = 4 n = 10(g) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.25-0.20-0.15-0.10-0.05Δ < 0 +0.050.100.150.20Δ ≥ 0φ = 1.5 μk = 4 n = 100(h) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.02-0.01Δ < 0 +0.0050.0100.0150.020Δ ≥ 0φ = 1.5 μk = 4 n = 1000(i) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-2-1Δ < 0 +0.51.01.52.02.5Δ ≥ 0φ = 1.5 μk = 8 n = 10(j) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.8-0.6-0.4-0.2Δ < 0 +0.20.40.60.8Δ ≥ 0φ = 1.5 μk = 8 n = 100(k) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.100-0.075-0.050-0.025Δ < 0 +0.0250.0500.0750.100Δ ≥ 0φ = 1.5 μk = 8 n = 1000(l) +Fig. 37 Figure equivalent to Fig. 21 after discretization of the 0 +is.The advent and fall of a vocabulary learning bias from communicative eciency 71 +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.5-0.4-0.3-0.2-0.1Δ < 0 +0.050.100.150.20Δ ≥ 0φ = 2 μk = 1 n = 10(a) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.03-0.02-0.01Δ < 0 +0.00250.00500.00750.01000.0125Δ ≥ 0φ = 2 μk = 1 n = 100(b) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.0025-0.0020-0.0015-0.0010-0.0005Δ < 0 +0.00040.00080.00120.0016Δ ≥ 0φ = 2 μk = 1 n = 1000(c) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-1.0-0.5Δ < 0 +0.250.500.751.00Δ ≥ 0φ = 2 μk = 2 n = 10(d) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.15-0.10-0.05Δ < 0 +0.030.060.09Δ ≥ 0φ = 2 μk = 2 n = 100(e) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.016-0.012-0.008-0.004Δ < 0 +0.00250.00500.00750.0100Δ ≥ 0φ = 2 μk = 2 n = 1000(f) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-2-1Δ < 0 +0.51.01.52.02.5Δ ≥ 0φ = 2 μk = 4 n = 10(g) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.8-0.6-0.4-0.2Δ < 0 +0.20.40.60.8Δ ≥ 0φ = 2 μk = 4 n = 100(h) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.100-0.075-0.050-0.025Δ < 0 +0.0250.0500.075Δ ≥ 0φ = 2 μk = 4 n = 1000(i) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-4-3-2-1Δ < 0 +123Δ ≥ 0φ = 2 μk = 8 n = 10(j) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-2-1Δ < 0 +12Δ ≥ 0φ = 2 μk = 8 n = 100(k) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.5-0.4-0.3-0.2-0.1Δ < 0 +0.10.20.30.40.5Δ ≥ 0φ = 2 μk = 8 n = 1000(l) +Fig. 38 Figure equivalent to Fig. 22 after discretization of the 0 +is.72 David Carrera-Casado, Ramon Ferrer-i-Cancho +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.6-0.4-0.2Δ < 0 +0.10.20.3Δ ≥ 0φ = 2.5 μk = 1 n = 10(a) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.05-0.04-0.03-0.02-0.01Δ < 0 +0.0050.0100.015Δ ≥ 0φ = 2.5 μk = 1 n = 100(b) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.004-0.003-0.002-0.001Δ < 0 +0.00050.00100.00150.0020Δ ≥ 0φ = 2.5 μk = 1 n = 1000(c) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-1.5-1.0-0.5Δ < 0 +0.51.0Δ ≥ 0φ = 2.5 μk = 2 n = 10(d) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.3-0.2-0.1Δ < 0 +0.10.20.3Δ ≥ 0φ = 2.5 μk = 2 n = 100(e) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.04-0.03-0.02-0.01Δ < 0 +0.010.020.03Δ ≥ 0φ = 2.5 μk = 2 n = 1000(f) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-3-2-1Δ < 0 +12Δ ≥ 0φ = 2.5 μk = 4 n = 10(g) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-2.0-1.5-1.0-0.5Δ < 0 +0.51.01.5Δ ≥ 0φ = 2.5 μk = 4 n = 100(h) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-0.3-0.2-0.1Δ < 0 +0.10.20.3Δ ≥ 0φ = 2.5 μk = 4 n = 1000(i) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-3-2-1Δ < 0 +123Δ ≥ 0φ = 2.5 μk = 8 n = 10(j) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-4-3-2-1Δ < 0 +1234Δ ≥ 0φ = 2.5 μk = 8 n = 100(k) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λα +-2.0-1.5-1.0-0.5Δ < 0 +0.51.01.52.0Δ ≥ 0φ = 2.5 μk = 8 n = 1000(l) +Fig. 39 Figure equivalent to Fig. 23 after discretization of the 0 +is.The advent and fall of a vocabulary learning bias from communicative eciency 73 +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +1 +1.5 +2 +2.5μk = 1 n = 10(a) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2 +2.5μk = 1 n = 100(b) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2 +2.5μk = 1 n = 1000(c) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2 +2.5μk = 2 n = 10(d) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2 +2.5μk = 2 n = 100(e) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2 +2.5μk = 2 n = 1000(f) +0.60.81.01.21.4 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1μk = 4 n = 10(g) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2 +2.5μk = 4 n = 100(h) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2 +2.5μk = 4 n = 1000(i) +1.01.11.21.31.41.5 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5μk = 8 n = 10(j) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2μk = 8 n = 100(k) +0.500.751.001.251.50 +0.00 0.25 0.50 0.75 1.00 +λαφ +0 +0.5 +1 +1.5 +2 +2.5μk = 8 n = 1000(l) +Fig. 40 Figure equivalent to Fig. 13 after discretization of the 0 +is. It summarizes Figs. 35, +36, 37, 38 and 39.74 David Carrera-Casado, Ramon Ferrer-i-Cancho +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.09-0.06-0.03Δ < 0φ = 0.5 μk = 1 α = 0.5(a) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.04-0.03-0.02-0.01Δ < 0 +1e-042e-04Δ ≥ 0φ = 0.5 μk = 1 α = 1(b) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.03-0.02-0.01Δ < 0 +0.0010.0020.003Δ ≥ 0φ = 0.5 μk = 1 α = 1.5(c) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.15-0.10-0.05Δ < 0 +0.0050.0100.015Δ ≥ 0φ = 0.5 μk = 2 α = 0.5(d) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.06-0.04-0.02Δ < 0 +2.5e-055.0e-057.5e-051.0e-04Δ ≥ 0φ = 0.5 μk = 2 α = 1(e) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.02-0.01Δ < 0 +0.00050.00100.0015Δ ≥ 0φ = 0.5 μk = 2 α = 1.5(f) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.25-0.20-0.15-0.10-0.05Δ < 0 +0.020.040.060.08Δ ≥ 0φ = 0.5 μk = 4 α = 0.5(g) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.09-0.06-0.03Δ < 0 +0.00250.00500.0075Δ ≥ 0φ = 0.5 μk = 4 α = 1(h) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.03-0.02-0.01Δ < 0 +3e-046e-049e-04Δ ≥ 0φ = 0.5 μk = 4 α = 1.5(i) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.4-0.3-0.2-0.1Δ < 0 +0.050.100.15Δ ≥ 0φ = 0.5 μk = 8 α = 0.5(j) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.15-0.10-0.05Δ < 0 +0.020.040.06Δ ≥ 0φ = 0.5 μk = 8 α = 1(k) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.05-0.04-0.03-0.02-0.01Δ < 0 +2e-044e-046e-04Δ ≥ 0φ = 0.5 μk = 8 α = 1.5(l) +Fig. 41 Figure equivalent to Fig. 24 after discretization of the 0 +is.The advent and fall of a vocabulary learning bias from communicative eciency 75 +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.16-0.12-0.08-0.04Δ < 0φ = 1 μk = 1 α = 0.5(a) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.06-0.04-0.02Δ < 0 +0.0010.0020.0030.004Δ ≥ 0φ = 1 μk = 1 α = 1(b) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.06-0.04-0.02Δ < 0 +0.0050.0100.0150.020Δ ≥ 0φ = 1 μk = 1 α = 1.5(c) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.3-0.2-0.1Δ < 0 +0.0250.0500.0750.100Δ ≥ 0φ = 1 μk = 2 α = 0.5(d) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.100-0.075-0.050-0.025Δ < 0 +5e-041e-03Δ ≥ 0φ = 1 μk = 2 α = 1(e) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.06-0.04-0.02Δ < 0 +0.00250.00500.00750.01000.0125Δ ≥ 0φ = 1 μk = 2 α = 1.5(f) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.6-0.4-0.2Δ < 0 +0.10.20.30.4Δ ≥ 0φ = 1 μk = 4 α = 0.5(g) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.3-0.2-0.1Δ < 0 +0.050.10Δ ≥ 0φ = 1 μk = 4 α = 1(h) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.075-0.050-0.025Δ < 0 +0.0020.0040.006Δ ≥ 0φ = 1 μk = 4 α = 1.5(i) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-1.5-1.0-0.5Δ < 0 +0.250.500.751.001.25Δ ≥ 0φ = 1 μk = 8 α = 0.5(j) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.6-0.4-0.2Δ < 0 +0.10.20.30.40.5Δ ≥ 0φ = 1 μk = 8 α = 1(k) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.3-0.2-0.1Δ < 0 +0.050.100.150.20Δ ≥ 0φ = 1 μk = 8 α = 1.5(l) +Fig. 42 Figure equivalent to Fig. 14 after discretization of the 0 +is.76 David Carrera-Casado, Ramon Ferrer-i-Cancho +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.15-0.10-0.05Δ < 0φ = 1.5 μk = 1 α = 0.5(a) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.075-0.050-0.025Δ < 0 +0.0030.0060.0090.012Δ ≥ 0φ = 1.5 μk = 1 α = 1(b) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.12-0.09-0.06-0.03Δ < 0 +0.010.020.030.040.05Δ ≥ 0φ = 1.5 μk = 1 α = 1.5(c) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.5-0.4-0.3-0.2-0.1Δ < 0 +0.050.100.150.200.25Δ ≥ 0φ = 1.5 μk = 2 α = 0.5(d) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.2-0.1Δ < 0 +0.020.040.06Δ ≥ 0φ = 1.5 μk = 2 α = 1(e) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.100-0.075-0.050-0.025Δ < 0 +0.010.02Δ ≥ 0φ = 1.5 μk = 2 α = 1.5(f) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-1.0-0.5Δ < 0 +0.30.60.9Δ ≥ 0φ = 1.5 μk = 4 α = 0.5(g) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.75-0.50-0.25Δ < 0 +0.20.40.6Δ ≥ 0φ = 1.5 μk = 4 α = 1(h) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.2-0.1Δ < 0 +0.050.10Δ ≥ 0φ = 1.5 μk = 4 α = 1.5(i) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-2-1Δ < 0 +0.51.01.52.02.5Δ ≥ 0φ = 1.5 μk = 8 α = 0.5(j) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-2.0-1.5-1.0-0.5Δ < 0 +0.51.01.52.0Δ ≥ 0φ = 1.5 μk = 8 α = 1(k) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-1.00-0.75-0.50-0.25Δ < 0 +0.250.500.75Δ ≥ 0φ = 1.5 μk = 8 α = 1.5(l) +Fig. 43 Figure equivalent to Fig. 25 after discretization of the 0 +is.The advent and fall of a vocabulary learning bias from communicative eciency 77 +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.5-0.4-0.3-0.2-0.1Δ < 0 +0.050.100.150.20Δ ≥ 0φ = 2 μk = 1 α = 0.5(a) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.125-0.100-0.075-0.050-0.025Δ < 0 +0.010.02Δ ≥ 0φ = 2 μk = 1 α = 1(b) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.15-0.10-0.05Δ < 0 +0.020.040.06Δ ≥ 0φ = 2 μk = 1 α = 1.5(c) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-1.0-0.5Δ < 0 +0.250.500.751.00Δ ≥ 0φ = 2 μk = 2 α = 0.5(d) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.4-0.3-0.2-0.1Δ < 0 +0.050.100.150.20Δ ≥ 0φ = 2 μk = 2 α = 1(e) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.06-0.04-0.02Δ < 0 +0.010.020.03Δ ≥ 0φ = 2 μk = 2 α = 1.5(f) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-2-1Δ < 0 +0.51.01.52.02.5Δ ≥ 0φ = 2 μk = 4 α = 0.5(g) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-1.6-1.2-0.8-0.4Δ < 0 +0.51.0Δ ≥ 0φ = 2 μk = 4 α = 1(h) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.8-0.6-0.4-0.2Δ < 0 +0.20.40.6Δ ≥ 0φ = 2 μk = 4 α = 1.5(i) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-4-3-2-1Δ < 0 +123Δ ≥ 0φ = 2 μk = 8 α = 0.5(j) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-3-2-1Δ < 0 +123Δ ≥ 0φ = 2 μk = 8 α = 1(k) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-2.0-1.5-1.0-0.5Δ < 0 +0.51.01.52.0Δ ≥ 0φ = 2 μk = 8 α = 1.5(l) +Fig. 44 Figure equivalent to Fig. 26 after discretization of the 0 +is.78 David Carrera-Casado, Ramon Ferrer-i-Cancho +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.6-0.4-0.2Δ < 0 +0.10.20.3Δ ≥ 0φ = 2.5 μk = 1 α = 0.5(a) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.15-0.10-0.05Δ < 0 +0.010.020.03Δ ≥ 0φ = 2.5 μk = 1 α = 1(b) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.20-0.15-0.10-0.05Δ < 0 +0.0250.0500.0750.1000.125Δ ≥ 0φ = 2.5 μk = 1 α = 1.5(c) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-1.5-1.0-0.5Δ < 0 +0.51.0Δ ≥ 0φ = 2.5 μk = 2 α = 0.5(d) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.5-0.4-0.3-0.2-0.1Δ < 0 +0.10.20.3Δ ≥ 0φ = 2.5 μk = 2 α = 1(e) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-0.09-0.06-0.03Δ < 0 +0.020.040.06Δ ≥ 0φ = 2.5 μk = 2 α = 1.5(f) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-3-2-1Δ < 0 +12Δ ≥ 0φ = 2.5 μk = 4 α = 0.5(g) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-1.5-1.0-0.5Δ < 0 +0.51.01.5Δ ≥ 0φ = 2.5 μk = 4 α = 1(h) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-1.00-0.75-0.50-0.25Δ < 0 +0.250.500.75Δ ≥ 0φ = 2.5 μk = 4 α = 1.5(i) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-4-3-2-1Δ < 0 +1234Δ ≥ 0φ = 2.5 μk = 8 α = 0.5(j) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-3-2-1Δ < 0 +123Δ ≥ 0φ = 2.5 μk = 8 α = 1(k) +102505007501000 +0.00 0.25 0.50 0.75 1.00 +λn +-2.5-2.0-1.5-1.0-0.5Δ < 0 +0.51.01.52.0Δ ≥ 0φ = 2.5 μk = 8 α = 1.5(l) +Fig. 45 Figure equivalent to Fig. 27 after discretization of the 0 +is.The advent and fall of a vocabulary learning bias from communicative eciency 79 +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +1.5 +2 +2.5μk = 1 α = 0.5(a) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2 +2.5μk = 1 α = 1(b) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2 +2.5μk = 1 α = 1.5(c) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2 +2.5μk = 2 α = 0.5(d) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2 +2.5μk = 2 α = 1(e) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2 +2.5μk = 2 α = 1.5(f) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1μk = 4 α = 0.5(g) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2 +2.5μk = 4 α = 1(h) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2 +2.5μk = 4 α = 1.5(i) +2505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5μk = 8 α = 0.5(j) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2μk = 8 α = 1(k) +02505007501000 +0.00 0.25 0.50 0.75 1.00 +λnφ +0 +0.5 +1 +1.5 +2 +2.5μk = 8 α = 1.5(l) +Fig. 46 Figure equivalent to Fig. 15 after discretization of the 0 +is. It summarizes Figs. 41, +42, 43, 44 and 45. \ No newline at end of file