7Supplemental analysis (not shown) shows that the structure of this network evolves somewhat over time. At some time points (e.g., nighttime) there is more centralization in both groups. But the degree of centralization tends to be greater among younger adults than it is among older adults in the morning hours (partly because older adults tend to wake up earlier), and this pattern shifts at midday. Thus, a disaggregation of the network estimates by time would reveal significant heterogeneity in levels of observed difference by age.

activities at different times, fewer transitions between different activity-times, are more centralized around fewer and more dominant sequence pathways, and involve less switching between different activities. In light of what we know about the health benefits of complex activity (e.g., Adams et al. 2011), these findings suggest a new area of concern for social gerontology and research on well-being. These findings suggest that older adults inhabit less complex action systems, which may have negative consequences for their social connectedness as well as their cognitive functioning. On the other hand, it is suggestive of greater levels of organization and predictability in everyday life (Zerubavel 1981), which may provide psychological comfort (Giddens 1984) in an otherwise difficult period of the life course. These substantive findings call for further study. But more work is needed to identify which network measures and methods are most useful for examining sequential social action.

I close by reiterating that the network methods that are discussed here are not substitutes for sequence analysis—they are supplements. These methods are a first step in an emerging effort to develop new approaches for understanding how sets of sequences intersect in larger, complex systems. There are several conventional sequence methods already in use which can be used to analyze some of the sequence-related attributes that I have analyzed in this paper. But this paper proposes a new language and analytic framework for the study of sequential phenomena which has as-yet-unknown levels of correspondence with existing sequence methods. I argue that the most insightful analyses of sequential phenomena will come when researchers combine network and sequence methods together.

**Acknowledgements** Early versions of this chapter were presented at colloquia hosted by the Departments of Sociology at Duke University, Columbia University, and the University of Oxford in 2015 and 2016. I thank the many members of those audiences who commented and asked questions—and especially Peter Bearman, Erin York Cornwell, Jonathan Gershuny, Sang Kyung Lee, James Moody, Gilbert Ritschard, Matthias Studer, Oriel Sullivan, Giacomo Vagni, and four anonymous reviewers—for suggesting changes that improved this chapter.

#### **References**

Wasserman, S., & Faust, K. (1994). *Social network analysis*. New York: Cambridge.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

## **Relational Sequence Networks as a Tool for Studying Gendered Mobility Patterns**

**Klaus Hamberger**

#### **1 Introduction**

In recent years social network analysis and sequence analysis have become increasingly attentive to one another. This mutual interest stems from a profound affinity in their objects of study. In fact, sequences and networks are not so much separate domains as complementary dimensions of the same social reality—they belong together as time and space (see Abbott 1997). On the one hand, sequences are not just isolated linear series of events. A given event may participate simultaneously in several sequences, and a given type of event may occur several times in the same sequence. Put in context, sequences turn out to be mutually intersecting and potentially nonlinear walks in a comprehensive network of events or event types. On the other hand, social networks are more than static compilations of relations between individuals. Social relations are essentially processual, being created, transformed and dissolved in time, and each of these transformations constitutes an event in one or more sequences. Considered in their dynamic aspect, social networks turn out to be the cumulative results of a series of interconnected sequences. Sequence analysis thus logically leads to network analysis and vice versa.

One reason for the long-standing separation of the two approaches resides in the fact that they do not involve the same type of elements. The networks constructed from sequences are networks of events linked by ties of causation or succession. These event networks have been introduced in the study of narrative (Bearman et al. 1999; Bearman and Stovel 2000), in life course studies (Bison 2014; Fitzhugh et al. 2015) and in time use studies (Cornwell and Watkins 2015; Cornwell 2015, 2018). By contrast, the historical core domain of social network analysis has

K. Hamberger (-)

Laboratoire d'Anthropologie Sociale, Ecole de Hautes Etudes en Sciences Sociales, Paris, France e-mail<EMAIL_ADDRESS>
G. Ritschard, M. Studer (eds.), *Sequence Analysis and Related Approaches*, Life Course Research and Social Policies 10, https://doi.org/10.1007/978-3-319-95420-2\_8

been constituted by networks of (individual or collective) agents linked by ties of friendship, kinship, exchange, and so on. While sequence analysis has been recently applied to the study of the dynamics of agent networks (Stark and Vedres 2006, 2012), the sequences thus constructed consist of network positions (or positional states) rather than of events.

Clearly, the two approaches just described are complementary. Sequences consist of events that are linked to each other in the experience or narration of an agent, and social networks consist of agents who are connected to each other via their participation in a common event. Event networks and agent networks thus can be derived from one and the same bimodal network of agents and events. An integrated approach should analyze both kinds of networks simultaneously and study their mutual interaction, as each event transforms the agents' social environment, which in turn shapes the path of future events.

A natural candidate for exploring the potential of this integrated method is the domain of migration and mobility. On the one hand, mobility events are among the classical subjects of life course studies and sequence analysis. On the other hand, social network approaches, including longitudinal perspectives, are increasingly used in migration studies (Lubbers et al. 2010). However, these approaches generally deal with the relevance of migrants' social networks to social integration or transnational support, rather than with the migration process itself. Studies of the ways in which social networks influence migration processes (Liu 2013) remain rare, and rarer still are studies that examine how social networks and mobility interact (Wissink and Mazzucato 2018).

This paper adopts an integrated method which uses social network analysis both to define the alphabet of the sequences (as a set of types of relations) and to represent the sequences themselves (as oriented networks of relational types).

We will apply this method to examine the impact of gender on the patterns of local and regional mobility in South-east Togo. In this region of West Africa, both men and women are highly mobile, though for different reasons. The mobility of Togolese women is closely connected to their traditional role in trade (Cordonnier 1987) and the long-standing practice of migration of girls and young women to the capital Lomé as foster children and/or as domestic workers (Pilon and Ségniagbéto 2014). This pattern of non-matrimonial female mobility is complemented and reinforced by a virilocal post-marital residence pattern combined with traditionally high divorce and remarriage rates (Locoh and Thiriat 1995). Male mobility, on the other hand, has historically taken the form of circular labor migration, be it as farmhands, tenant farmers or lumberjacks to the plantations and forests of Western Togo, Ghana or Ivory Coast (in the case of the older generation), or more recently as unskilled workers to urban centres on the west African coast between Abidjan and Lagos. Even when rural emigration becomes long-term, the paternal home in the village frequently remains a refuge in case of unemployment, illness, and for retirement in old age, a pattern that is widespread in colonial and postcolonial West Africa (see e.g. Cordell et al. 1996).

The network approach proposed in this paper is intended to improve understanding of the logic underlying these broad tendencies by examining the micromorphology of male and female migration trajectories and relational environments.

#### **2 Method**

#### *2.1 Basic Concepts*

Every event sequence can be represented as a path in a network, where nodes represent events and arcs represent immediate temporal succession. In Fig. 1 (left), for example, two simultaneous sequences (blue and red), are represented as paths through five events of four different types (A-D), two of which (B2 and B5) are common to both. By gathering similar events into larger classes, any such path will induce a sequence network, the nodes of which are not single events but event classes. Sequence networks of this sort are a variant of the networks used in life course analysis, such as life history graphs (Butts and Pixley 2004) or state transition graphs (Bison 2014). Since events of a given class can occur more than once in these networks, sequences now will no longer be linear paths but potentially nonlinear walks. Thus, in Fig. 1 (right), the networks corresponding to the sequences ABCBB (blue) and CBDAB (red) contain loops (BB), mutual dyads (BCB) and circuits (BDAB).

Events can be classified according to many different criteria, each giving rise to another kind of sequence network. For example, a classification by place of destination will result in the geographical network of migration routes. Here we are interested in people's movement in social space, so we shall classify events according to the relations that link the individuals involved.

Agents can participate in events in a variety of roles (such as "migrant", "host", and so on), and may be related to each other by relations that are polyadic ("A is child of B with C") and composite ("father's sister", "mother's employer" etc.). Complicated as they may appear, networks of this sort are not unfamiliar to

**Fig. 1** Event networks (left) and sequence networks constructed from them (right). Nodes represent events (left) or event types (right), arcs represent immediate succession. Sequences are represented as (linear) paths in networks of events and as (nonlinear) walks in networks of event classes (sequence networks)

social network analysis: kinship networks are commonly conceptualized as bimodal networks where "individuals" are linked to "families" by a variety of roles such as "father", "mother" and "child" (Batagelj and Mrvar 2008; Hamberger et al. 2011, 2014), and kinship relations comprise composite chains of parent-child or marriage ties. Now what we call "families" in kinship network theory are not so much sociological units but series of life course events (marriages and births) related to the same couple. We thus can conceive of "family events" (to which individuals are linked as "parent" or "child") in the same way as of "mobility events" (to which they are linked as "host" or "migrant") and vice versa. This similarity between kinship and mobility networks is more than formal. Marriage and divorce usually involve a change of residence for one of the partners, and every birth is in a sense a mobility event creating the starting point of an itinerary. Kinship and migration networks can accordingly be modeled as parts of an integrated bimodal network (see Fig. 2), where individuals and (mobility or family) events are represented as two different sets of nodes, and where the various roles that individuals play in these events are represented by different sets of edges.

The unimodal projections of this integrated bimodal network yield the one-mode networks usually studied in life course and kinship studies: networks of events linked to each other by the involved individuals, and networks of individuals linked to each other by shared events. For a given individual (ego), we can in this manner derive both the event sequence that constitutes his or her itinerary and the personal network that results from this itinerary.

The anchorage of both personal networks and event sequences in a single, multiplex bimodal network makes it possible to classify individuals and events by means of the relational circuits that emerge as mobility links combine with chains of other links (kinship, friendship, employment, and so). For example, individuals C and H in Fig. 2 are linked to each other as migrant and host (via mobility event 2), but also as brother's wife and husband's sibling (via birth events 3a and 3b and marriage event 2), thus forming a relational circuit (marked blue in Fig. 2). Thus, we can first classify the individuals of ego's personal network by the relational chains that link them to ego (for example, host H as "brother-in-law"), and then classify the events of ego's itinerary by the relational classes that characterize individuals in certain roles (for example, mobility event 2 as "move to brotherin-law"). This relational classification of events in turn allows us to merge similar nodes of ego's itinerary, thus transforming it into a sequence network. Since the alphabet of sequence networks of this kind consists in classes of relations rather than in exogenous attributes (such as the motive for mobility events or the social class of the host), we distinguish them as relational sequence networks.

The specificity of relational sequence networks resides in using network techniques both for analyzing and for constructing sequences. They can be applied to any kind of sequences whose elements are pertinently characterized in terms of relations. In addition to migration studies, they may, for example, prove useful in the sociology of interactions, the morphology of folktales or the anthropology of ritual. The fact that we shall use them to study the impact of what may at first sight appear as an attribute of the individual—gender—does not contradict this

**Fig. 2** An integrated kinship-mobility network. White nodes represent individuals; green and yellow relational nodes represent mobility and family events, respectively (family event nodes can be merged to form conventional family nodes, light-yellow). Edges linking individual to relational nodes correspond to the individual's roles in the event. Itineraries are represented as (dotted red) paths through successive mobility events

thoroughly relational orientation. Actually, the results of our analysis corroborate the now widely accepted approach that views gender itself in relational terms.

#### *2.2 Data*

The study is based on data collected in a field survey conducted in rural Southeast Togo between 2010 and 2015.1 The survey was centered on Afagnan-Gbléta, a large village whose core area has a population of around 4800 inhabitants (census

1Most of the interviews were conducted by the author in collaboration with Toussaint Yakobi and Komi Malou Kakanou during four stays (2009-10, 2010, 2012, and 2014-15). Forty-two interviews during autumn 2010 have been conducted by Ibitola Tchitou (URD, Lomé University) and the same collaborators.

conducted by author in 2014).2 It is located in the maritime region of Togo, a very densely populated area (while it covers 10% of the national territory, 42% of the Togolese population live there). The region falls within the zone of West Africa with the highest intra-regional migration. Afagnan-Gbléta is situated close to the border of Benin, and the capital Lomé, at 81 km distance, can be reached within two hours by "bush taxi". The village is located close to a major West African migration route connecting the cities of Abidjan, Accra, Lomé, Cotonou, and Lagos. Short geographical distances, affordable transport, easy border crossings in the ECOWAS region and extended social and kinship networks all facilitate high transnational mobility.

The sample has been established in a two-step procedure. The first 60 interviews were conducted with persons drawn at random from the adult (*>*15 years) residents of Afagnan-Gbléta (based on the village census effectuated in 2004/5) who were still alive and resident in Togo at the beginning of the study in 2010.3 In the second step, we conducted interviews with all persons mentioned by the initial interviewees as having played a role in their migration biographies or as having been hosted by them, as far as these persons were alive, reachable within the perimeter of Southeast Togo, and willing to talk with us.  The total snowball sample thus contains 509 migration biographies (260 male and 249 female).

Migration biographies were collected through retrospective semi-directive interviews. We asked interviewees to recall all migratory events leading to a change of residence for more than three months4 from birth to the present (time of the interview, last update of information in 2014/15), and to describe the context, motive and course of each migratory event. In particular, we collected information (name, relation and contact) on the persons who received ego (hosts), accompanied ego (co-migrants), initiated ego's displacement (initiators) and financed ego's journey (financers). All other persons mentioned in the context of the migration event were equally noted (others). This set of names was complemented by a list of the interviewee's parents, spouses and children. Finally, we asked each interviewee to name all those persons for whom he or she had acted as a host.

This dataset is completed by a genealogical and residential dataset (about 50,000 individuals including the deceased) collected during three censuses of the village of Afagnan-Gbléta (2004/5, 2009/10 and 2014/15) as part of ongoing ethnographic

2According to 2010 official census data, Afagnan-Gbléta counts 12,411 inhabitants (5916 males and 6495 females) when all its hamlets are included.

3Information on present life status and residence were obtained after the draw, so that there were 76 draws in all (35 women/41 men) because 6 (3/3) had died, 5 (1/4) had moved abroad, one woman refused the interview, and 4 young men appeared in the sample due to errors concerning their age.

4This restriction was meant to exclude short-term stays for funerals, visits or holidays. In fact, the problem was rather that even year-long stays, for example at the natal home after childbirth, were often considered by interviewees as temporary absences not to be mentioned, and discovered only afterwards when dealing with the children's itineraries.

fieldwork (see Hamberger 2011). Most of the remote kinship relations (such as "father's mother's brother's daughter") and mediated non-kinship relations (such as "friend's employer") were thus not (only) directly reported by interviewees but computed from data stemming from numerous different oral sources. The anonymized dataset is available on the Kinsources repository (permanent link: https://www.kinsources.net/kidarep/dataset-275-watchi-2017.xhtml).

The total snowball sample has only an auxiliary value for the analysis. Besides the bias in favor of local and circular migration trajectories owing to the geographical limits of the survey, the snowball technique introduces a systematic bias in favor of connected and mobile persons, who have a higher chance of playing a role in the itineraries of the first interviewees. All comparative data concerning the overall sample thus are likely to overestimate mobility and network cohesion. Moreover, not being composed of independent observations, the snowball sample cannot be analyzed with standard statistical techniques. Most of our analyses are therefore restricted to the random sample of 60 itineraries. The remaining 449 itineraries serve essentially to construct the ego-centered social spaces and sequence networks of these 60 cases. For the purpose of illustration, four of these cases will be considered in further detail below.

#### *2.3 Software Tools*

All analyses used in this paper have been made using the open source software Puck version 2.3.50 (Hamberger et al. 2014), which can be downloaded at http:// www.kintip.net (source code at http://sourceforge.net/projects/tip-puck/). Initially developed for the study of kinship networks, Puck contains, from its 2.2 version onwards, a package (org.tip.puck.sequences) for the study of longitudinal data. The networks of Figs. 3, 6, 7, and 8 have been visualized with the software Pajek (http:// mrvar.fdv.uni-lj.si/pajek/) from files produced by Puck. The detailed procedures for reproducing the analyses of the paper are described in the document accompanying the Watchi dataset on the Kinsources repository (see above).

#### **3 Results**

#### *3.1 Personal Networks*

In order to introduce the analyses presented in the remainder of this paper, let us begin by considering one empirical case example. We have chosen the itinerary of Betty,5 a middle-aged female domestic worker with transnational experience (Benin

5First names of the four example individuals have been changed.

**Table 1** Betty's (10683) itinerary as sequence of successive mobility events. Relational roles of hosts and co-migrants are indicated in brackets. Village and quarter names have been replaced by numbers

and Senegal, most recently Mali). Table 1 represents the sequence of successive mobility events, each of which links her to one or more hosts,6 some of them also to co-migrants. Their respective identity numbers are listed in the last two columns of the table.

In order to study the social networks shaping (and shaped by) individual trajectories, we first have to look at the way in which the relations created by a mobility event (linking a migrant to his or her hosts, co-migrants, and so on)

6While birth mothers constitute by convention the starting points of all itineraries (and sequence networks), they have not been explicitly coded as hosts at birth so as to avoid trivial results of relational censuses and personal network analyses.

coincide with other direct or indirect relations between these individuals.7 We have considered seven main types of relations: (1) kinship ties up to the fourth canonic degree for consanguines (ex. third cousins) and up to the first degree for affines (ex. fathers-in-law); (2) non-genealogical kinship (including both kinship ties for which the genealogical chain is unknown and those for which probably no such chain exists, such as the "father's child" relationship established by reference to a common locality of origin); (3) friendship (including relations between colleagues, fellow workers and classmates); (4) apprenticeship; (5) initiation (which includes a more or less prolonged stay at a vodu priest's home); (6) employment; and (7) landlord-tenant relationship. Since the latter generally implies the absence of a preexisting social relation, hosts are classified as "unrelated" only when they have accommodated ego without payment.

The relations linking Betty to her hosts and co-migrants are given in brackets in the last two columns of Table 1. As we can see, Betty has been received by her parents (1 event, without counting the birth event), her brother (1), her husband (3), a relative (1), three affines (4), one of whom is also an employer (1) and another a (future) master (2), various unrelated employers (4), her husband's employer (1), and various landlords who rented her a room (3). She has mainly travelled alone, and has only occasionally been accompanied by one or more children (3), a uterine niece (1), and her husband (1).

Table 2 shows the frequencies of these various kinds of host and co-migration relations for all itineraries of the total snowball sample (509 cases). The numbers indicate the number of itineraries in which the migrant-host or the co-migrant relation coincides at least once with a social relation of the respective type.

A look at the types of hosts and co-migrants occurring in male and female itineraries instantly reveals some marked differences (but also some important similarities). The virilocal orientation of residence in Togolese society is clearly evident: almost all adult women have moved to their husbands or affines on at least one occasion while the reverse trajectory concerns only a small minority of men.8

A more remarkable feature is the marked tendency to return to the parental home. This circular pattern holds not only for men who hold land rights in their natal family, but also for women, who systematically return to it before (re)marrying.

For both men and women, relatives outside the immediate family circle are at least as important as parents in the role of hosts—with a slight preference of male and female for agnatic and uterine relatives, respectively.

7Though these relations generally precede the mobility event, there are some cases of relations that have been created after the move (e.g. by marrying a host's relative, or by starting an apprenticeship with the host as master).

8The definition of 'host' is here bound to land and residence rights rather than to residential precedence. Thus, a wife returning to her husband has been classed as returning to her marital home, while a man returning to his wife has been categorized as returning to his own (or parental, or rented) home. This is a reasonable assumption to make in the rural context but may introduce unwarranted male bias into data collected in an urban setting.

**Table 2** Types of relations to hosts and co-migrants occurring in male and female itineraries. Figures indicate the number of (male, female, all) itineraries where a host or co-migrant in the corresponding relation to ego appears in at least one event. Percentages refer to the total number

Residence with a master during apprenticeship is more relevant for boys than for girls, who are more frequently employed as market helpers or domestic workers.9 By contrast, it is mainly girls who as initiates spent a (today much reduced) period of residence with a vodu priest. The male predominance in monetary relations to landlords may partly reflect a tendency among interviewees to represent the husband as the contracting party even if both spouses contribute to the rent (we did not verify who pays). By contrast, stays in state- or NGO-owned facilities (caserns, schools, hospitals) or in hotels are largely restricted to the educated male elite.

Unknown 10 *3*.*8* 12 *4*.*8* 22 *4*.*3* 9 *3*.*5* 2 *0*.*8* 11 *2*.*2*

State or NGO 14 *5*.*4* 5 *2*.*0* 19 *3*.*7* Unrelated 16 *6*.*2* 1 *0*.*4* 17 *3*.*3*

While the gender inflection of host-migrant relations concerns mainly the marital tie, its impact on co-migration becomes manifest in the parental tie: children move with mothers rather than with fathers. By contrast, travelling with non-kin (friends and fellow workers) is more frequent for men than for women. Predictably (since our data include infant migration), siblings are among the most frequent co-migrants.

After this first inspection of the positions that make up the social space in which a person navigates, let us now take a closer look at the structure of this space. We

9In cases where masters and employers were relatives, the work relationship has been given precedence over kinship when constructing sequence networks.

shall confine our analysis to the personal networks of the 60 initial interviewees. Clearly, our picture of these networks is incomplete since neither the dead nor most of those living abroad could be interviewed about their own relations, forcing us to rely on ego's indications for them. This data collection bias reinforces ego's memory bias, so that the resulting networks are in a sense doubly "ego-centered", being most dense in ego's actual spatial and temporal neighborhood.

The nodes of the networks are made up of all persons who played a role as host, co-migrant, initiator or financer in ego's itinerary, but not those persons for whom ego played only the role of a host. Also, the network does not contain those named as "others", whose role consists in providing the general context. Composite kinship links mediated by persons who are themselves in the network have not been represented by direct arcs so as to preserve their mediated character.10

Figure 3 shows four example networks, where relation types are indicated by different colors. Besides Betty's personal network (upper left), we have selected the networks of Omar, a male tailor circulating between Afagnan-Gbléta and the Benin-Nigerian frontier; Sosime, an elderly female trader who has returned to Afagnan-Gbléta after an itinerary including transnational commuting between Lomé and Lagos; and Aloesso, a former goldsmith who spent his youth living in his elder brother's household at Lomé.

We are interested in the importance of different relations for the network structure, be it as intermediaries for indirect relations or as nuclei of locally cohesive clusters. The first aspect can be measured by the betweenness centrality11 of the individuals so related to ego; the second by the fact that these individuals play a locally central role (measured by node degree)12 in one of the connected components into which the network decomposes if ego is removed. In Fig. 3, the individuals (other than ego) with the highest betweenness centrality have been marked by bold borders, and connected components (after removal of ego) have been encircled. As can be seen, Betty's network can be decomposed into 10 components, centered respectively around her parents (1), her husband (1), an employer (3), a relative (2) and a landlord (3), where all but one of the three last mentioned categories constitute single-element components. The largest component (9 nodes) is the marital component centered around the husband, who is also the individual with the highest betweenness centrality (14%) after ego (85%).

Table 3 indicates the number of itineraries in which each of the considered relations characterizes the most central alter in the whole network or in one of its components (in addition, we have indicated the mean size and the maximal number of the respective components centered on this relation).

10For example, if ego has migrated with his or her mother and maternal grandmother, the network will contain the grandmother but no direct link between her and ego.

11The betweenness centrality of a node (Freeman 1977) is measured as the normalized number of shortest paths running through it.

12The (in-/out-) degree of a node is defined as the number of nodes to which it is linked (by incoming/outgoing arcs).

**Table 3** Globally and locally central relations. Figures indicate the (absolute and percentage) numbers of personal networks where the individual(s) most central in the network or in one of its components have the corresponding relation with ego. The table also indicates the mean size and the maximal number of such components; if there is more than one, mean component size refers to the largest. Percentages refer to the total number of itineraries concerned (30 male, 30 female, 60 in all)

We now see that the various relations that we previously examined in isolation operate in quite different ways in organizing ego's social space. On the one hand, there are relations that function as connectors: they give rise to other relations which they mediate, and thus become centers and nuclei of large connected components this is the case of kinship relations, in particular the parental ties that mediate all others. On the other hand, there are relations that operate as divisors: they neither integrate themselves into an existing social space nor lead to other relations, but tend to form numerous small components. These "atomizing" relations are usually mediated by money, linking ego to landlords, masters or employers. However, some relations of this kind may mediate relations between fellow apprentices or working colleagues (counted as "friends" in our classification).

The extremes of these two types of relations—one centralizing a singular large component, the other leading to multiple marginal isolates—constitute a polarity which to a certain extent characterizes the male mobility pattern. On the one hand, the kinship core, while important for both genders, is more fundamental for male networks: 50% of male networks, as opposed to 33% of female networks, are centered around the father (who forms the local center of a component in 50% of both male and female networks). On the other hand, the atomized components of rent and wage relations occur both more frequently and in greater number in male than in female networks. The typical male social space is centered around a single large kinship base from which repeated expeditions are undertaken to a variety of rapidly changing social and spatial destinations. The rural-urban circular migration also largely follows this radial pattern, and even those who are in stable employment (teachers, policemen, etc.) are mostly implicated in continuous displacements.

Not all relations, however, fit into this polarity. On the one hand, there are relations that are both unique and isolated, such as the relation of initiate to vodu priest. On the other hand, there is the marital relation, which is both relatively central and gives rise to cohesive components structured around nuclear families, but which, in this region of West Africa, is typically multiple. While men can have several wives at the same time, women generally have several husbands at different times in the course of their lives. Contrary to men, for women each new marriage involves a displacement, and mediates a series of new relations that play a role for their future migrations. Each new husband thus becomes the nucleus of a (frequently large) component of the wife's mobility space.

The gender-specific occurrence of different relations noted previously thus corresponds to different relational logics in the structuring of social space, which is evident in the contrastive overall morphologies of male and female personal networks. On average, female networks are more cohesive than male networks: they contain half as many isolates (1.3 vs 3), their average number of components is lower (2.8 vs 4.1), and their concentration—measured as the Herfindahl index of component shares—is higher (0.58 vs 0.52). However, this concentration does not imply the presence of one dominant giant component, but of two or several components of roughly equal importance. As a corollary, ego's own centrality is generally lower for female than for male egos (0.48 vs 0.62). The more or less (de-)centered position of ego can be expressed more precisely in considering the normalized difference between the (betweenness) centralities of ego and the most central alter. This index of ego's differential centrality ranges from −100% for maximal marginality to +100% for maximal centrality, where 0% indicates a central position shared by ego and one or more central alters. Differential centrality is closely correlated to the concentration of ego's personal network (see Fig. 5). The more disintegrated the network (after removal of ego), the higher ego's centrality (with the migrant teacher as an extreme example), and the more integrated the network, the higher ego's marginality (with the foster child at the other extreme).

While Fig. 5 shows the moderate average tendency of female networks towards higher concentration and lower ego centrality, it also shows a great diversity. To explore further this diversity, let us consider our four case examples, each of which represents different combinations of differential centrality and concentration

**Fig. 5** Differential centrality (*x*-axis) and concentration (*y*-axis) of the 60 personal networks of the random sample, including the four example cases. Blue triangles and red circles (and the corresponding blue and red tendency lines) refer to male and female networks, respectively

**Table 4** Four case examples (individual attributes and personal network indicators)

(Table 4 and Box 1). As one can see from Betty's and Aloesso's networks, cases of female solitary long-distant migration (yielding ego-centered, dispersed networks), or of male dependent family migration (yielding alter-centered, concentrated networks), are not unusual.

**Betty**'s network represents the combination of high ego centrality and relatively low network integration typical for persons whose itineraries are shaped by high mobility outside the kinship network, such as teachers or soldiers, but also, as in the present case, female domestic workers. Besides the numerous isolated components she traversed in the course of her professional migrations, her social space also contains several more cohesive groups, one of which, organized around her husband, also channeled her professional itinerary.

**Omar**'s network likewise represents a relatively poorly integrated social network, but ego's relative centrality here is much less pronounced. We again meet the combination of several cohesive components—an initial uterine network centered around the mother, a transnational migrant trader, a component of friends, and a marital component—each of which has been important for professional connections. There are fewer isolated components, also due to ego's younger age. The mother's central position results from the fact that she constitutes the bridge between the two spaces (Togo and Benin) between which ego is oscillating.

**Sosime**'s network combines high network integration with a moderate decentralised position of ego. The high cohesion of the network results from the fact that it is largely composed of (active and passive) fosterage ties with close kin. After having been hosted by paternal relatives in childhood, she has subsequently hosted (and been accompanied by) a whole series of young female relatives, including the daughters of her former hosts, to be again hosted, in old age, by one of her former foster children. However, being childless herself, all these links are mediated through other members of her personal network, so that the high network concentration does not imply a high centrality of ego, but of the relative (her father) who serves as an intermediary. (Since fosterage typically implies a host-migrant relation, we did not include it into the relations of the personal network so as to avoid circularity. Otherwise, Sosime's centrality would have been considerably higher.)

**Aloesso**'s network represents the extreme case of a highly integrated network where ego holds a marginal position (expressed by highly negative differential centrality). In fact, the network's main component is actually not his but his elder brother's network, while the second, marital component has not developed into a larger network, since ego's marriage coincided with a definitive return to sedentary life in the natal village.

**Box 1** Personal network analysis of the four example cases

#### *3.2 Sequence Networks*

After having examined in a synchronic manner the way personal networks have been shaped by (and have shaped) people's itineraries, let us now turn to the question how these itineraries run through the various positions of these networks. We shall construct the corresponding relational sequence networks by classifying mobility events according to the type of relation or relational chain that links ego to his or her host. To facilitate comparison, we have used a very coarse classificatory grid, which, for example, lumps together all relatives other than the immediate parents into a single category,13 and generally does not distinguish between different instances of the same relational type (such as "landlord"), with the only exception of spouses. Even so, except for the three individuals who have never left their natal social

13The software Puck allows for any kind of precision for the classification of kinship chains.

**Fig. 6** Relational sequence networks of the four case examples. Nodes represent classes of events according to the type of relation between migrant and host; arcs indicate immediate succession of events of the corresponding type, numbered by order of occurrence in the itinerary. Numbers attached to nodes refer to loops (moves to new hosts within the same relational category). For the color code see Fig. 4

environment, the sequence networks of the 60 individuals of our random sample are all different from each other.

Figure 6 shows the relational sequence networks of our four example individuals, briefly discussed in Box 2. Each relational type is represented by a node. All itineraries start at the parental node (at the bottom), since the host of the first "mobility event" (birth) is by definition the birth mother.

The network representation of these relational sequences allows us to compare them by means of network-analytic indicators. Some of these indicators concern the local position of nodes—such as in- and outdegrees, which measure the diversity of relations that precede or succeed a given relation, or betweenness centralities, which **Aloesso**'s sequence network is the most simple: an oscillation between the parental home and a close maternal relative (his brother in Lomé), connected to an intermittent oscillation between the brother and a variety of hosts to whom Aloesso was only related by his brother's mediation. The simplicity of the sequence network is the corollary of Aloesso's marginality in the social network, by virtue of which the (orange) "relative" and "via relative" nodes absorb the quasi-totality of events.

**Omar**'s sequence network is more complex: it also starts with a move to non-parental relatives (first his maternal aunt in Benin, later his maternal grandmother and uncles in the village), but he uses these positions as springboards to engage with hosts of various types linked to his professional education and activity, such as masters, compatriots, strangers, and finally landlords.

A quite similar basic pattern can be observed in **Betty**'s case: again starting with a move to a maternal aunt, she soon begins staying with various employers and landlords. However, this professional network is completed by a series of social positions to which she accedes via marriage: her husband, the husband's employers, and the husband's relatives (i.e. her affines). Note that this marital network is not distinct from, but integrated into the professional network: just as was the case with her own relatives during her youth, her husband's relatives are at the same time her employers or mediate her relations to future employers.

The most complex sequence network is **Sosime**'s, who combines a long-term shift from the paternal home (where she is hosted first by her father then by her brother) to several successive husbands with oscillating movements to and from the landlords of the capital and the friends and market places of Nigeria. The brother's home serves as a constant haven of return in these movements, and also as the crossroad where she "picks up" the young female relatives who accompany her through the various stages of her career as a trader.

**Box 2** Sequence network analysis of the four example cases

indicate the importance of relations as transitions from or to other relations.14 Others concern the global network structure—such as the number of dyads or cycles, which correspond to the frequency of direct or indirect returns to relations that appeared earlier in the itinerary. In Betty's case, for example, her frequent but intermittent accommodation by her employers is evident in the fact that the employer relation has both the highest betweenness centrality and the highest outdegree and forms part of the two oriented cycles of her sequence network. (For a detailed discussion of network indicators such as density, centralization or homophily in sequence analysis, see Cornwell 2018).

To which extent do these sequence patterns follow a gendered logic? To answer this question, we can proceed in two alternative ways. The first consists in examining the aggregate male and female networks constructed by merging all sequence networks of people of the same gender. In these networks, line values represent the numbers of individual itineraries in which a given event sequence occurs (see Fig. 7, where line values are indicated by different line widths). Table 5 provides some basic indicators for the network position of the various types of relations.

14See above footnotes 11 and 12.

**Fig. 7** Aggregate sequence networks of men (top) and women (bottom) (initial random sample, *n* = 30 each). For the color code see Fig. 4

As our four examples have already suggested, both male and female networks are centered around a fundamental axis linking parents and non-parental relatives. Not only are relatives the most frequent successors of parents as hosts (which is partly accounted for by the widespread practice of fosterage), but a person who has been hosted by a relative is also likely to return to the parental home. For both men and women, a stay with relatives provides the springboard for engaging in monetarized relations with landlords, which then tend to replicate themselves. However, men enter two times as frequently into relations with landlords than

do women, and are also more than twice as likely to progress from a stay with relatives to a rented apartment. The most significant difference is, of course, the importance of marital homes for female itineraries, this having no equivalent in their male counterparts. Note, however, that these spouse-linked positions are far from representing endpoints: their outdegrees exceed or equal their indegrees; in other words, the positions they lead to are more diverse than those that lead to them, indicating continuing residential mobility. Also note that the parental home, followed by relatives' homes, is the most frequent base from which women (re)join their first and subsequent husband(s)—it is extremely rare for women to move directly from one husband to another.

To sum up, the aggregate female sequence network, rather than being totally distinct from the male network, represents a variant of it, rendered more complex by the integration of one or more additional focal nodes. While the typical male network is largely organized around a triangle formed by the parental home, relatives and the residential market, the female network contains in addition one or more spouse nodes, which may become the nuclei of marital subnetworks. As a consequence, the relative importance of the various focal nodes (as measured by their betweenness centrality) differs between male and female networks. In male networks, the parental home clearly constitutes the dominant central node—as might be expected in a rural environment with agnatic transmission of land rightswhile in female networks, it shares importance with the almost equally central first marital home, followed by subsequent marital homes.

The problem with this kind of comparative macro-analysis is, of course, that it already presupposes that gender is a pertinent classification criterion for merging individual sequence networks, instead of deriving the classification criteria from a comparative analysis at the micro-level of individual sequence networks. This latter perspective characterizes the approach of optimal matching analysis (Abbott 1995) and its more recent network-analytic analogues, where sequence matching is replaced by graph similarity (Butts and Pixley 2004; Butts 2008; Fitzhugh et al. 2015). Since in our relational sequence networks each node type appears, by construction, at most once in each network (the only quasi-exception being first and subsequent spouses, which we have treated as distinct types), we can construct all sequence networks with identical node sets (absent nodes being left unconnected), so that the only edit operations necessary to transform one network into another consist in adding or removing arcs. Graph edit distance thus boils down to the (normalized) number of differently connected pairs of nodes (that is, the number of different cells in the adjacency matrices of the two graphs). Based on this graph distance matrix, hierarchical clustering techniques can be applied to represent the similarities and differences of the relational sequence networks. Figure 8 displays the agglomerative clustering tree of the 60 sequence networks of the initial random sample, constructed by the neighbor-joining algorithm (Saitou and Nei 1987),15 and spatialized by the Kamada-Kawai spring embedded algorithm implemented in Pajek. The graphs of the individual sequence networks have been plotted into the nodes of the tree, and border colors indicate gender (blue for male and red for female). Our four example cases are emphasized by the use of grey shades.

While the gender difference is neatly brought out by the graph (roughly speaking, female itineraries are located in the upper half, male itineraries in the lower half of Fig. 8), a closer look shows that, beyond the great divide introduced by virilocality (the presence or absence of "yellow" marital nodes in the sequence network), both male and female sequence networks are differentiated into several branches of the tree.

Most sequence networks, irrespective of gender, are based on the parent-relative axis, to which the simplest of them (such as Aloesso's) may reduce. In the "male" case, this basic axis frequently combines either with rent (as for the urban migrant workers), with hosting by the master or employer (as for the rural itinerary masons or woodcutters), or with both. This combination, eventually enriched by marital nodes, may also characterize the sequence networks of female domestic workers (such as Betty). A still more complex variant, characteristic of teachers' itineraries, combines rented accommodation with lodging by the state. Finally, the itinerary

15The neighbor-joining algorithm consists in sequentially linking the two nearest neighbors to a newly created "ancestor" node and recalculating the distances of all other nodes to the ancestor node from their distances to the joined neighbor nodes. Pairwise distances are adjusted by the average distance of both neighbors to the rest of the network. In Fig. 8 ancestor nodes are only represented as bifurcation points of outgoing arcs.

may involve lodging by friends or unrelated persons, frequently in a transnational context, as exemplified by Omar's case. Again, this pattern may be equally found among female traders (such as Sosime). A quite different type of itinerary is located at the bottom left of the tree, where the parental node is directly linked to the landlord node without passing through a relative's home. This is a minority pattern generally restricted to men who have migrated later in life for professional reasons, and to the few boys whose parents could afford to rent apartments for them while they attended high school. However, we also find this itinerary present in two women who chose not to marry.

While all the predominantly "male" patterns of sequence networks are accessible to women, the "female" upper half of the graph is significantly more homogenous. The upper left part in particular is made up of a large group of highly similar sequences completing the parent-relative-axis by one or more marital or affinal nodes, thus giving rise to several distinct clusters according to the complexity of the marital subnetwork. This pattern may further evolve by integrating stays with landlords, masters or employers (mainly for trade or domestic work), as can be seen on the upper right side (actually a female enclave encompassed by predominantly male patterns). By contrast, we almost never find a female pattern without the "relative" node in it. Women who leave their parental home directly for a rented apartment are even more rare then men (though, also here, the exception proves the rule).

#### **4 Conclusion**

Our analysis of the topology of both personal and sequence networks has produced convergent results with regard to the logics underlying gendered mobility patterns in South-east Togo. Male and female sequence networks both rest on a basic kinship axis (linking an "internal" parent pole and an "external" extended-kinship pole). For both genders (though more so for men), the parental home is a central place of (definitive or transitory) return, and for both genders, relatives outside the immediate family circle play an equal (for women even a stronger) role than parents, in particular before the age of marriage. The simplest migration pattern, and the starting point for most male and female itineraries, is an oscillation between parents in the village and relatives beyond.

The main difference between the genders consists in the way their respective social sequence networks develop from this double kinship core. Due to the sexual division of labor which renders wage-labor still an essentially male domain (women are either independent traders or work for relatives as domestic workers and market assistants), and to the virilocal residence rule, according to which women at least initially join their husbands' home, the networks of men tend to evolve through nonkinship links to friends, masters, employers, strangers or landlords, while women's networks grow through the emergence of affinal links to spouses and in-laws.

This difference is not only one of relational circuits but also of network morphology. Male non-kin relations, often radiating from the core in a star-like manner, are generally multiple but socially disconnected (except when they are linked to the kinship complex). By contrast, female affinal relations, though less numerous, are structurally productive: they develop into larger subnetworks that may succeed each other as disjoint spheres or merge into an integrated space of circulation. While male networks thus tend to evolve through a succession of structurally isolated non-kinship links, female networks develop into complex multifocal networks sewn together by marital and affinal ties—that is, the very ties that link together male and female itineraries. In other words, the central source of difference between the sequence networks of men and women is precisely their mutual relation.

Rather than just confirming the macro-tendencies for male and female mobility patterns (as stated in the demographic literature) at the micro-level of individual trajectories, sequence network analysis yields insight into the relational logics that bring these tendencies about. It serves not only to study the differences between gendered social networks, but also to understand gender itself as a relation between networks, that is, not just as an attribute of individuals, but as a structural trait of social space-time. In a more general perspective, the nascent integration of network and sequence analysis may be the first step towards a full-fledged social topology.

**Acknowledgements** The paper is part of a larger research project conducted with Karin Sohler since 2010. Preliminary results were the subject of joint presentations at the 1st European conference of Social Network Analysis in Barcelona in 2014 and at the 3rd meeting Réseaux-Histoire in Paris in 2015. Though Karin did not coauthor the present paper, its argument owes a fundamental debt to our collaborative work. I am also grateful to Véronique Hertrich, Ismaël Moya, Sorana Toma, Claire Lemercier and Karen Middleton, as well as to the anonymous reviewers of the LaCOSA II conference and for this volume, for their valuable and encouraging comments on earlier versions of this paper.

Fieldwork has been supported by the Laboratoire d'Anthropologie Sociale, the EHESS research fund and the French Cooperation in Togo. The 2010 fieldwork was carried out in cooperation with Ibitola Tchitou and Kokou Vignikin from the Unité de Recherche Démographique of Lomé University. This work would have been impossible without the competence and dedication of my field collaborators Komi Malou Kakanou and Toussaint Yakobi and the active cooperation and help of the interviewees, to whom I present my heartfelt thanks.

The sequence-network module of Puck has been developed with Christian Momon as part of the project "Kinsources" supported by the French National Research Agency ANR (research grant ANR-12-CORP-0008).

#### **References**

Abbott, A. (1995). Sequence analysis: New methods for old ideas. *Annual Review of Sociology, 21*(1), 93–113.

Abbott, A. (1997). Of time and space: The contemporary relevance of the Chicago School. *Social Forces, 75*(4), 1149–1182.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

## **Part IV Unfolding the Process**

## **Multiphase Sequence Analysis**

**Thomas Collas**

#### **1 Introduction**

Many common notions such as lifecycle, adulthood, turning point or ratchet effect are based on the idea that various sets of sequences—especially individual careers follow regular patterns defined by successive phases. Those phases that all cases encounter in the same order are usually defined by social moments—such as graduation, medical treatment, childbirth, job promotion, new research project, election as a congressman, Oscar nomination, etc.—or by calendar periods e.g., minutes, months, years or decades. Such a conception of sub-level temporal structures nested in sequences and linked to one another is a cornerstone of lifecourse studies (Levy and The Pavie Team 2005). Nonetheless, it has rarely been discussed as a methodological parameter of sequence analysis.

Implementing the notion of phase can in particular deepen our understanding of how institutionalised narratives shape social processes (Abbott 2001). Dividing into phases also helps to reduce data complexity. In this chapter, I elaborate on these ideas by presenting visual and metric tools of multiphase sequence analysis (MPSA). Throughout the text, I will develop an example of two-phase sequences drawn from a study of the careers of participants in professional *pâtissier* (pastry cook) competitions in France.

In the first section, I present key properties of multiphase sequences. In the second section, I exemplify and discuss the implications of the division into phases. In the third section, I present tools to render multiphase sequences, introducing sliced representations. In the fourth section, I introduce a dissimilarity measure of multiphase sequences called multiphase optimal matching (MPOM).

© The Author(s) 2018

T. Collas (-)

F.R.S.-FNRS – Université de Louvain, Louvain-la-Neuve, Belgium e-mail<EMAIL_ADDRESS>
G. Ritschard, M. Studer (eds.), *Sequence Analysis and Related Approaches*, Life Course Research and Social Policies 10, https://doi.org/10.1007/978-3-319-95420-2\_9

#### **2 Sequences as Multiphase Structures**

The pathbreaking proposition of sequence analysis was to study sequences as wholes (Abbott 1995), constructed as successions of atemporal timeslots.1 The notion of phase adds an intermediary level to this binary hierarchical structure (sequences and time slots) by introducing discontinuities within these successions.

#### *2.1 Characteristics of Multiphase Sequences*

Lesnard and Kan (2011) provide an example of multiphase sequences in their analysis of workweeks. The whole sequences they study are each made up of 672 15-min time slots. Lesnard and Kan represent them as successions of 7 days of 96 15-min time slots with a two-state alphabet (work/non-work). For each sequence, seven successions (days) are nested in a larger succession (a week). This example of multiphase sequences is extremely regular: all sequences are made up of seven phases which contain exactly 96 time slots. It is possible to conceive of more diverse types of multiphase sequences.

First, the phases may be of uneven length within sequences and from one sequence to another. For instance, in France, whenever a new government is formed, many recruits enter the ministerial *cabinets*. Their job careers can be divided into three phases: their positions before entering the *cabinets* (for example, by separating private and public sectors), their positions within the *cabinets* (by differentiating *cabinets* and hierarchical positions) and their positions after they leave these *cabinets* (with the same alphabet as in the first phase). The lengths of the different phases differ because they are not defined according to their duration but with respect to common events that may happen at different moments in a career. Take the first phase for example: some enter a *cabinet* after a 20-year long career whereas others have only worked for three or four years before being recruited. More broadly, the division into phases often implies length differences even for sequences of equal length.

Secondly, some phases may be empty, i.e., of length zero. For example, *pâtissier* competitions suppose a division of careers into two phases: a phase in which competitors are employed as a junior or apprentice *pâtissier*, followed by a phase in which they are a fully-fledged *pâtissier*. Nonetheless, some competitors may quit the trade just after their apprentice period or enter the trade directly as senior workers.

These characteristics lead to a formal definition. Multiphase sequences *S*1 and *S*2 are successions of *n* phases such as *S*1 = *(ζ S*1 1 *, ζ S*1 2 *,...,ζ S*1 *p ,...,ζ S*1 *n )* and *S*2 = *(ζ S*2 1 *, ζ S*2 2 *,...,ζ S*2 *p ,...,ζ S*2 *n )*, where *ζ S*1 *p* is phase *p* in sequence *S*1 and *ζ S*2 *p* is phase *p* in sequence *S*2. The length of each phase varies from 0 to the length of the sequence in which it is nested. If *n* = 1, we encounter the usual definition of sequences as continuous series of time slots.

1A time slot is atemporal since a single state is observed from the beginning to the end of it.

#### *2.2 Two Formal Properties of Phases and Two Methodological Assumptions*

Two formal properties of phases are captured by the "turning point—trajectories" model theorised by Abbott (2001, p. 253). In this model, turning points differentiate consistent episodes (trajectories) and link these episodes to the previous and following ones into a larger narrative. The first property is relative consistency: the division of a sequence into phases relies on the assumption that the succession of states within each phase is both consistent and different from successions within other phases.2 The second property is processual location: a phase is defined by its position within a sequence, hence by its position relative to other phases. Returning to the example of the workweek from Lesnard and Kan (2011), Friday is regarded as a consistent period for work scheduling and its position within the workweek immediately preceding Saturday distinguishes it from other days.

Two crucial methodological assumptions follow these formal properties.

First, as sequences, phases are approached as sites (or locations) of narratives: as successions of time slots each containing one state, phases and sequences are constructed as compartments for modelling narratives.3 To assert that a phase is relatively consistent and located in a sequence does not imply any assumption about the narrative it contains. "Stage" models in the traditions of Piaget or Parsons (for a synthesis, see Levy and The Pavie Team 2005) define the content of both typical sequences and typical phases within these sequences (the so-called stages). These models assume that each phase is the location of a single and specific kind of narrative (for instance, a certain behavioural development or a type of activity). The notion of phase discussed here helps one to appreciate the relevance of stage models but is not bound to these models. While comparing sequences, one assumes that there are patterns, i.e., types of narratives, to be discovered. While comparing multiphase sequences, one assumes that phase-structured patterns types of narratives including types of sub-narratives—are to be discovered. As for sequence analysis at large, the only assumption about the content of these subnarratives lies in the alphabets that are used and in the definition of the boundaries of the set of sequences. These elements limit the universe of possible narratives. Identifying these consistent sub-narratives is a different question from dividing into phases, just as identifying consistent narratives is a different question from delimiting a population of sequences.

Second, distinct phases are dissociated and should not be compared. In the workweek example, it would make little sense to compare one sequence's Tuesday to another's Sunday. Similarly, it does not seem relevant to directly compare contests

2As Abbott (2001) and Cornwell (2015, p. 94) indicate, this consistency can be conceived as a set of stable relationships between states as modelled by Markov chain models. A turning point is defined by Abbott (2001, p. 247) as a transition separating stable probability regimes.

3By methodological construction and to illustrate the notion of site, a narrative cannot be contained in a time slot, which can only contain one state.

in which apprentice *pâtissiers* compete with contests where senior *pâtissiers* compete. This incommensurability of phases has major implications for the comparison of multiphase sequences. I shall return to this point after discussing the practical operation of dividing into phases.

#### **3 Division into Phases: Reference Frame, Alphabet(s) and Phase-Structure**

This section starts with an example of division into phases before distinguishing three crucial aspects of this operation.

#### *3.1 A First Hint: The Extended Example*

Careers of participants in *pâtissier* competitions in France offer a case of two-phase sequences. The data are drawn from results of 2060 professional competitions. These competitions consist in making or presenting decorative sculptures (out of sugar, chocolate or ice), cakes or plated desserts before juries of peers.4

Here I present data from 777 *pâtissiers* who participated in two to 21 competitions and whose careers of participations in competitions began before 2002 and ended after 1990. Each time slot is a participation in a competition. Each participation is defined by an age category (apprentice, junior, senior), a rank (1st, 2nd, etc.) and a type of competition (preceded or not by screening contests).5

To be compared, these careers have been divided into two phases with respect to age categories since some contests are for apprentice and junior competitors only, others for senior competitors. This is a case of institutional division of careers into phases that is verified in the data. Indeed, returns to apprentice and junior competitions from senior competitions are unusual: amongst the competitors who participated in at least two competitions and at least one senior competition, only 7.6% participated in an apprentice or junior competition after competing in a senior competition. Interviews with *pâtissiers* within a larger research project made it clear

4For a detailed account, see Collas (2015).

5Data were gathered from archives and trade press collections. 2060 rankings covering the period 1933–2012 were coded. *Le Journal du pâtissier* (published since 1978), which is the main source, mentions competitions that are organised in different areas in France, while the other sources mention mainly competitions taking place in Paris. Beginning (before 2002) and end (after 1990) dates were chosen because of source heterogeneity and in order to limit right and left censoring. A first rank in the "*Un des meilleurs ouvriers de France*" competition was not kept since it was always gained at the end of a competitor's career. R software (R Core Team 2014) and the TraMineR package (Gabadinho et al. 2011) were used to visualise sequences, to compute OM-distances, to extract sets of representative sequences and to compute other sequence-related metrics.

that first participation in a senior competition is represented as an entry into a phase of evaluation which is not based on age or scholarship, but on the fact of being identified as a *pâtissier*. Thus, I postulate that participations in competitions preceding participation for the first time in a senior competition are not comparable with subsequent participations.

The dissociated phases are, first, the *ante*-senior phase—including only participations in apprentice and junior competitions and ending with the last participation before a participation in a senior competition or with the last participation if the competitor has never participated in a senior contest—and, second, the senior phase—which begins with the first participation in a senior competition and can include any type of participation. Each phase is defined by its relative position within the sequence and by its postulated internal consistency.

Since many competitors participate only in one type of competition (senior ones on the one hand, junior and apprentice ones on the other), a large fraction of sequences include a phase that does not contain any participation (an empty phase, i.e., of null length): these sequences are made up of participations in competitions during only one phase. The senior phase includes participations in 84.4% of the sequences, the *ante*-senior phase in 44.7% of the sequences.

Division into phases impacts the sequences' states alphabet(s). In that example, division leads to a shorter alphabet and, as a consequence, reduced data complexity: age categories are not taken into account in the definition of each participation since phases already bear this age dimension.

Two dimensions are used to define the alphabet. First, competitions preceded by screening contests are regarded as distinct. Amongst these competitions, two are singular and thus isolated as distinct states in the alphabet: a national plated dessert competition named *Championnat de France du dessert* (CFD) (labelled "C") and the oldest competition preceded by screening contests still organised today named *Un des meilleurs ouvriers de France* (One of the best craftsmen in France exam, labelled "M"). This competition is for senior competitors only. Several other competitions are gathered under the "S" label (for Screening contests).6 Other types of competitions are labelled "W" (for Without screening contests). Second, each participation is defined by the rank awarded, in three categories: 1st rank (labelled "L", for laureate), 2nd or 3rd rank ("P", for podium) and 4th rank or below ("O", for off-the-podium). For example, a state "LC" indicates a first rank in the *Championnat de France du dessert*.

In order to include long pauses between two participations in the analysis, a state named "4Y" (for four years) was created to indicate every period lasting more than four years and less than eight years between two successive participations. The alphabet contains eleven possible states (Table 1). One, related to a senior competition (OM), is observed only during the senior phase.

6*Meilleur Apprenti de France* (Best Apprentice in France), *Meilleur Apprenti du Monde* (Best Apprentice in the World), *Coupe du Monde de la Pâtisserie*, *Grand prix international de la chocolaterie*, World Chocolate Master.

#### **Table 1** States alphabet

#### *3.2 Three Aspects of Division into Phases*

This case sheds light on three aspects of division into phases.

The first one is the reference frame of the division. Here, the reference frame is endogenous: two phases of participations in competitions are dissociated according to a characteristic of participations in competitions (the first participation in a senior competition). In other cases, the reference frame is exogenous: an external dimension is introduced in relation to a research question. For example, careers of participants in competitions could be divided according to job positions. Taking another example, in many systems, academic careers are structured by a limited number of phases (lecturer, assistant professor, associate professor, etc.). The dissociation of phases of academic activities (e.g., publications) according to these successive academic ranks helps to explore how institutionalised episodes impact scientific outputs. Such a division into phases is also a way to reduce data complexity: what can be regarded as two distinct channels (Pollock 2007; Gauthier et al. 2010)—academic ranks and publications—are reduced to one channel cut into successive phases.

The second aspect is the definition of the alphabet(s). Here, the phase division reduces by a half the number of possible states with a very limited loss of information regarding the age categories of each participation (only the above mentioned 7.6% of senior competitors are affected). But the division into phases can also accompany a definition of several alphabets. Since phases are regarded as consistent episodes within sequences, the number of relevant states for each phase may be quite low, especially when the division is endogenous. Such is the case for the careers in ministerial *cabinets* mentioned above. The types of possible positions during the *cabinet* phase are both more limited and more specific than before and after this phase. As a consequence, two alphabets may be defined. By delimiting a relevant universe of possible states for each phase, the plurality of alphabets reduces data complexity.

Determining the number and level of phases is a third key aspect to phase division. The case developed here is simple: only two successive phases are dissociated. As illustrated by several examples mentioned, this number can be higher, but the number of assumptions about the structure of the sequences rises accordingly. We can also envision sub-phases nested within phases. A tennis match is such a two-level nested structure: points are clustered within games and games within sets. Careers including gradations within ranks (2nd class, 1st class, etc.) such as academic careers in France present such a nested structure.

#### **4 Rendering Multiphase Sequences**

Two types of graphical representations help to render multiphase sequences. The event-aligned variety of simple alignment representations is suited for two-phase sequences. Sliced representations, introduced here, generalise the logic of eventaligned representations to *n*-phase sequences.

#### *4.1 Simple Alignment on a Specific Event*

Drawing on previous studies (Blanchard 2010; Giudici and Gauthier 2009), Colombi and Paye (2014) discuss a visualisation method that transforms the usual left- or right-aligned representation suited for one-phase sequences into a representation aligned on a specific event. This event is regarded as a turning point between two phases: "After synchronisation, each sequence (e.g., series of job positions) is positioned according to an event that takes place in a particular moment for each individual (e.g., childbirth)" (Colombi and Paye 2014, p. 250).

Two steps are followed. First, a relative time axis aligned on the specific event under study is introduced: the temporal scale is negative for states observed before this event, positive for states observed after. The format of each time slot is preserved but the time axis is distorted so as to preserve a social timing according to a supposed turning point. Second, blank time slots are inserted at the beginning and end of each sequence. As a result, the length of every sequence is equal to the length of the longest observed period preceding the studied event summed with the length of the longest observed period following this event.

Taking a toy example, if *A* marries at age 25 and *B* at age 28, their job sequences from 22 to 31 are left-aligned and event-aligned (on marriage date) as shown in Table 2 (E stands for employment, U for unemployment, m on the axis stands for marriage).

**Table 2** Different alignments of sequences

#### *4.2 Multiple Alignment by Sliced Representation*

Event-alignment is convenient for keeping the continuous representation of leftaligned sequences but such a technique is limited to two-phase sequences. Sliced representations help to render *n*-phase sequences. In the previous example, if one wants to include divorce in the reference frame of division and if *A* gets divorced at 29, the representation implies three dissociated phases (see bottom subtable in Table 2).

As for event-aligned representations, the temporal scale is relative, but the time axis is indexed on phases, not on a single event. In this example, the axis is leftaligned for each phase. The origin point is the first time slot of each phase. In case of a right-alignment, the origin point is the highest possible length for a given phase. A right-alignment for certain phases and a left-alignment for others can be envisioned. In that sense, the event-aligned representation is a special case of sliced representations.

Figure 1 shows three representations of the same sample of five sequences of participations in competitions. Division into phases underlines the proximity of the last two sequences during the senior phase. While the event-aligned representation highlights the sequences' continuity as the usual left-aligned or right-aligned representations of sequences, sliced representation focuses on the different successions within each phase, not on the sequels and aftermath of a supposed turning point.

**Fig. 1** Three representations of the same careers of participations in competitions (see explanation of state labels in Table 1)

#### **5 Measure and Interpretation of Pairwise Distances Between Multiphase Sequences: Multiphase Optimal Matching**

Besides visualisation, dissimilarity measures are commonly used tools for comparing sequences. When measuring dissimilarities between multiphase sequences, phases are regarded both as dissociated incommensurable episodes and as sites of narratives. These are the basic principles of multiphase optimal matching (MPOM) introduced here. This section focuses on optimal matching (OM) for three reasons: it is a seminal and widespread dissimilarity measure in the social sciences; the cost definition operations that OM implies have consequences for MPSA; the principles of OM are adapted to the example under study (this point is discussed below). Nevertheless, MPOM's analytical logic can be extended to other dissimilarity measures when they are applied to multiphase sequences.7

#### *5.1 Analytical Logic*

MPOM's analytical logic is twofold. First, pairwise distances between sequences are measured with respect to equivalent phases. Time slots belonging to Phase *P*1 in Sequence *S*1 are only compared with time slots belonging to Phase *P*1 in Sequence *S*2. Second, each phase is regarded as an ordered set of time slots. Equivalent phases

7For a review of dissimilarity measures between sequences, see Studer and Ritschard (2016).

are compared with the three basic operations of OM (Abbott and Forrest 1986): substitution, insertion and deletion. Costs assigned to these operations are defined for each phase.8

For each pair of sequences, MPOM measures a distance per phase (distance between *S*1 and *S*2 on Phase *P*1, distance between *S*1 and *S*2 on Phase *P*2, etc.) through OM operations and then a distance between sequences by summing distances between equivalent phases. The matrix of pairwise distances between sequences is the sum of the matrices of pairwise distances per phase. Thus, the contribution of each phase to the distance between two sequences depends on the differences in state composition of this phase from one sequence to another and on its maximal length (the longer a phase is, the heavier its impact on pairwise distance can be, due to the number of insertions and/or deletions necessary to edit one phase into another).

MPOM-measure is a fractal generalisation of OM measure. In the case of a onelevel phase division, each phase is approached as OM approaches a sequence. In the case of a higher-level phase division (with multiple levels of phases, phases nested within phases), the same operation is reproduced at each level. Thus, an *n*-level MPOM-distance implies nested sums. For instance, a two-level MPOM-distance (see the tennis match example above) is a sum of one-level MPOM-distances.

Regarding empty phases, if Phase *P*1 is of length *l P* 1 = 0 in *S*1 and of length *l P* 2 ≥ 1 in *S*2, the impact of emptiness on the distance between *S*1 and *S*2 is equal to the cost of insertion and deletion (*indel*) multiplied by the length of the longest version of Phase *P*1 (here *l*2). More broadly, the impact of a length difference between two sequences is equal to this difference multiplied by the indel (|*l*2 − *l*1| × *indel*).9

Thus, MPOM rests on two methodological principles that can be applied to other dissimilarity measures: dissociation of phases and combination of phase pairwisedistances into sequence pairwise-distances.

#### *5.2 MPOM Applied to Careers of Participants in 'Pâtissier' Competitions*

Turning to careers of participants in *pâtissier* competitions, optimal matching measure offers an appropriate tool for searching for regular patterns in these data for two reasons. First, these sequences are characterised by a high level of instability from one participation to another regarding ranking (one may rank first, then ninth,

8This involves a multiplication of cost-setting operations which may seem dubious since many criticisms of OM have focused on cost-setting operations (Abbott and Tsay 2000). Constant or data-driven substitution costs may be relevant in some cases.

9Since division into phases often implies length differences, pairwise distances between phases can be standardised with respect to the maximal possible distance for each phase, which I do not do here precisely in order to take length differences into account.

**Table 3** Substitution costs

then third, etc.); insertion/deletion operations reduce distance due to lags. The second reason is relations between states. A first rank is closer to a second rank than to a tenth rank. Multiple substitution costs lie in the postulate of distance instead of strict difference between states.

So as to limit the number of assumptions, the same substitution costs between states are used for both phases. These costs have been set manually with respect to states' formal closeness: competitions with screening contests are closer to one another than to competitions without screening contests; a first rank is closer to a second rank than to a fourth rank; a second or third rank is closer to a fourth rank than to a first rank (due to the singular situation of ranking first). Minimum substitution cost between two distinct states is 1, maximum cost is 2. Minimum cost is used for a substitution between two states indicating the same position in two competitions preceded by screening contests (e.g., CFD and World Cup). Other substitution costs are set between 1.7 and 2 (Table 3). Several trials intended to take into account the unequal length of sequences without making it the first criterion of distance lead me to set insertion-deletion cost (*indel*) at 1.35 for both phases.10

10This indel is higher than the minimum substitution cost for distinct states (1) and lower than the minimum substitution cost for states differing in terms of both ranking and screening (1.7). For example, it is less costly to edit one sequence (ABC) into another sequence which is one time slot shorter and otherwise identical (AB) than to edit one sequence into another same-length sequence which is different only for the last time slot (ABC into ABD) only if states C and D are similar with respect to ranking or anterior screening contests.

**Fig. 2** MDS Sequence Index Plot for each cluster (sorted according to the first factor of multidimensional scaling following Piccarreta and Lior 2010). SC = competitions preceded by screening contests. (See explanation of state labels in Table 1)

**Table 4** Sets of representative sequences for each cluster

*N.B.*: In each cluster, the distance of at least one sequence out of two from one of the representative sequences is inferior to 30% of the maximum distance in the cluster (for details on the centrality criterion, see Gabadinho and Ritschard 2013). Sequences are sorted by representativeness. Each state is followed by its number of successive occurrences

A comparison between several clustering methods invited me to opt for a ninecluster Ward's (1963) partition (see Fig. 2 for sequence index plots and Table 4 for representative sequences).11

11Any distance-based clustering method could be used including the property-based and fuzzy methods addressed by Studer (2018) in this bundle. Here, using the R package WeightedCluster (Studer 2013), several algorithms have been compared for a division into two to ten classes: hierarchical cluster solutions named Ward, single, complete, average (UPGMA), McQuitty (WPGMA) and beta-flexible (flexible-UPGMA) (for a presentation, see Müllner 2013; Belbin et al. 1992) and non-hierarchical partition around medoids algorithm (PAM) (Rousseeuw and Kaufman 1990). PBC, HC, HG and ASW measures have been used to compare the quality of the different clustering solutions (Hennig and Liao 2010). Ward's nine-cluster solution was the most relevant regarding both quality measures and readability. Except for PAM, other algorithms tend to produce a partition between a very heterogenous group containing more than 80% of the cases and two to

Three key elements of interpretation arise: participation in senior competitions, length and tonality (the most frequent state or family of states within the sequence).

Cluster 1 (4% of sequences) gathers sequences mainly characterised by an empty *ante*-senior phase (only two sequences out of 31 do not start with a senior competition) and a long senior phase (mean length is 6.87 time slots) including mainly first ranks and participations in competitions preceded by screening contests (64.7% of participations). Cluster 2 (3.5% of sequences) gathers medium to long senior careers (mean length is 8.63) in which first ranks are rare (less than 11% of participations). Cluster 3 (11.7% of sequences) gathers short to medium length senior careers (mean length is 4.99), mainly characterised by second and third ranks (70% of participations). Cluster 4 (28.8% of sequences) gathers short length senior sequences with no shared tonality. Clusters 5 (16.1% of sequences) and 6 (10.3% of sequences) are defined by at least one four-year pause, respectively during the senior phase and during the *ante*-senior phase. Cluster 7 (17.8% of sequences) gathers sequences defined by a short *ante*-senior phase and an empty senior phase with no shared tonality. Cluster 8 (1.8% of sequences) is made up of very long senior careers (mean length is 13) including mainly first to third ranks (76% of participations). Sequences in Cluster 9 (6% of sequences) share a symmetrical intensity regarding participations in *ante*-senior and senior phases and a relatively low rate of 4th ranks or below.

How far does this clustering take phases into account? First, the Ward twocluster solution separates Clusters 6 and 7 from the seven other clusters, that is to say careers first defined by *ante*-senior participations from careers first defined by senior participations. Second, Clusters 6 and 7, both characterised by *ante*-senior participations, are distinct from one another with respect to participation in senior competitions. Third, a quarter of the sequences counting one or more ante-senior participations are not clustered in Clusters 6 and 7. In other words, closeness does not only rest on the (non-)emptiness of phases, but also on phases' tonality. Fourth, when, as here, the reference frame of the division is endogenous, that division greatly simplifies the interpretation: once the phases mainly portrayed by each cluster have been identified, interpretation is primarily based on ranking.

#### *5.3 MPOM Compared*

MPOM takes cues from two other families of dissimilarity measures that assume a division of sequences into dissociated and incommensurable episodes.

First, MPOM generalises Hamming Distance (Hamming 1950) and Dynamic Hamming Distance (DHD) (see Lesnard 2008, 2014), which measure dissimilarities

nine easy to analyse but very small groups. Quality measures are higher for Ward compared to PAM. A nine-cluster solution is associated with the highest value of HG and HC indexes, the second highest value of PBC and the fourth highest value of ASW.

position-wise. DHD can be approached as a specification of MPOM-distance in which each time slot is a phase (of length 1) and in which substitution costs are derived from transition rates before and after each time slot. This specification is suited for sequences defined by a limited number of states, observed in each sequence and spanning long spells.12

Second, compared with Qualitative Harmonic Analysis (QHA) (Deville and Saporta 1983; Robette and Thibault 2008; Robette and Bry 2012), in which sequences are divided into periods that are modelled as bundles of states varying from one another in proportions of time spent, MPOM pays attention to the order of the states within phases within phases.

Thus, the main advantage of MPOM is to take into account a unit nested in a sequence that is more malleable than time slots—its length varies from 0 to the whole sequence's length—and that is studied as a time-ordered structure.

#### **6 Conclusion**

This chapter has introduced the idea of multiphase sequences and several tools to study them, especially sliced representations and a multiphase dissimilarity measure (called MPOM) the logic of which can be extended to other dissimilarity measures. Two general issues have been raised and invite further investigation in the development of MPSA.

At the beginning of this chapter, sequences were defined as hierarchical structures, as narratives including sub-narratives and nested into larger narratives. This definition is partly consistent with Dumont's (1980) perspective of hierarchies as nested entities. Any sequence (marital biographies, job careers, dances, etc.) can be approached as a fragment of larger social processes (Abbott 2016) including other fragments of social processes. The approach to sequences as hierarchical structures could be further developed by investigating the variety of relations between nested temporal structures. Hybridisations of network analysis and sequence analysis (Cornwell 2015) may be a possible way to study these relations as a multilevel issue (Lazega and Snijders 2016).13

12To preserve a division into phases in the comparison of workweeks, Lesnard and Kan (2011) wrap each phase into an atomic time slot, thus describing each phase by a single state. Their two-step method identifies types of narratives through DHD and clustering procedure and then assembles these types in week-sequences analysed with DHD and clustering. Compared to MPOM, this wrapping solution is suited for periods of identical time spread (such as hours or days). Its main limitation is that its second step is based on the heterogenous inputs of a clustering procedure.

13Regarding MPOM, the analysis of the articulation of distances between phases and distances between sequences could be further developed since two sequences can be identical along some phases and clearly different along others. A related question is the importance of specific phases in the definition of a whole sequence. Various theoretical perspectives assume that certain phases are more crucial than others (childhood in a whole life-course for example). Such assumptions can orientate the parameters of MPOM by differentiating phases' weights.

A methodological assumption of MPSA is to approach phases as sites of narratives nested within sequences, which are sites of larger narratives. This assumption differentiates phases and the narrative patterns that the division into phases makes it possible to unveil. That echoes other notions centred on temporal substructures, such as subsequence (Elzinga et al. 2008) or motif (Han 2014). There is an open field for research on sub-narratives, their typical position(s) within sequences and their overlaps.14 MPOM assumes a division into phases prior to the identification of narrative patterns. Different alphabets are defined, thus determining what patterns can be identified. A comprehensive method would manage three different steps: identifying types of narratives, identifying phases, identifying patterns of relation between phases and narratives. In other words, the division into phases could be dynamically revised and preceded by a moment of identification of patterns for different fragments of sequences under study.15

These elements invite renewed exploration of the various interrelations and continuities between temporal structures, a major question that sequence analysts have already extensively explored.

**Acknowledgements** I thank the participants to the LaCoSA 2 conference and the editors and anonymous reviewers of this book for valuable suggestions. I thank Claire Lemercier, Laurent Lesnard, Etienne Ollion and Loretta Platts from whom I received insightful comments on earlier versions of this chapter. I am thankful to Richard Nice for a meticulous proofreading of my English.

#### **References**

14Regarding overlaps, my purpose was focused on clear-cut phases, divided from the single viewpoint of contractual events (a divorce, a recruitment, etc.), which is simplistic. For instance, the idea that a divorce begins with a contract in couples' histories is debatable. In most cases, the identification of phases as relevant locations of narratives is a challenging issue since many narratives overlap. Nevertheless, the notion of overlap is premised upon the idea of typical locations of narratives. Identifying those typical locations is thus a preliminary step to the study of overlaps. 15This identification could rely on tools for detecting consistency within segments, such as the

stationarity test proposed by Bakeman and Gottman (1997).

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## **Unpacking Configurational Dynamics: Sequence Analysis and Qualitative Comparative Analysis as a Mixed-Method Design**

**Camilla Borgna and Emanuela Struffolino**

#### **1 Introduction**

Sequence analysis (SA) was introduced in the social sciences in the 1980s: Andrew Abbott, inspired by the treatment of DNA strings in biology, developed it as a technique to study social processes that unfold over time as sequences of events (Abbott and Forrest 1986). In contrast to the variable-oriented approach of techniques like event-history analysis, which focus on the timing of specific transitions, SA's approach is holistic and case-oriented (Billari 2005). This implies looking at the whole picture of individuals' longitudinal realizations rather than at the effects of variables on single-event outcomes. Against this scenario, processes of change over the life course are regarded as successions of states/actions located within constraining or enabling structures. This reflects (Abbott 1992)'s idea that holistic and analytical approaches can be combined into a multicase narrative methodology, based on which researchers can compare the unfolding of processes to pursue categorization and generalization.

SA has been extensively applied in demography and life-course research to study changes and continuities in individual pathways over time (e.g. Aassve et al. 2007;

C. Borgna (-) Collegio Carlo Alberto, Turin, Italy e-mail<EMAIL_ADDRESS>
E. Struffolino (-)

The original version of this chapter was revised. A correction to this chapter is available at https:// doi.org/10.1007/978-3-319-95420-2\_17

WZB Berlin Social Science Center – Research Group 'Demography and Inequality' and Humboldt University of Berlin, Berlin, Germany e-mail<EMAIL_ADDRESS>
Brzinsky-Fay and Solga 2016; Fasang and Raab 2014; Raitano and Struffolino 2013). Other fields of application include social policy analysis (Frank et al. 2000), democratization research (Wilson 2014), electoral participation studies (Buton et al. 2014), historical sociology (Abbott and Forrest 1986; Mercklé and Zalc 2014), and developmental psychology (Dietrich et al. 2014). The mainstream application of SA relies on optimal matching techniques to measure the distance between sequences that represent the individual realization of a certain process and on the clustering of such sequences to identify typical trajectories. Regression analysis is often applied as a second step to investigate why (and not only how) sequences resemble or differ from each other (Aisenbrey and Fasang 2010). Yet, multivariate regression analysis, with its focus on the net effects of single variables, seems to be at odds with the case-oriented nature of SA. In this chapter, we propose to use qualitative comparative analysis (QCA) as a second, explanatory step for SA. Compared to regression analysis, QCA is more coherent with SA's epistemological framework, because it shares the notion of analytically approaching social phenomena without disregarding their complexity. QCA was introduced in the 1980s (Ragin 1987) and relies on methodological tools that were uncommon in social science until then: in this case, logic and in particular Boolean and fuzzy-set algebra. Its developer Charles Ragin, like Abbott, aimed at bridging the divide between variable-oriented and case-oriented research. However, the joint holistic and analytical perspective of QCA does not concern over-time trajectories, but rather configurations of factors at given time points.

We contribute to the growing literature on mixed-methods research (e.g. Creswell 2009; Teddlie and Tashakkori 2006) by putting forward a novel "sequential mixedmethod design" (Hollstein 2014; Teddlie and Tashakkori 2006) that consists of applying a recent innovation in SA—the so-called discrepancy analysis of state sequences (Studer et al. 2011; Struffolino et al. 2016)—as a first step, and crisp-set (Ragin 1987) or fuzzy-set (Ragin 2000, 2008) QCA as a second step. Our proposed framework allows researchers to analyze individual trajectories as a whole and to identify combinations of factors that are systematically linked to variations in the unfolding of such trajectories at potentially critical turning points. This can be read as an answer to the call for a "processual sociology" (Abbott 2016) that can take on the challenge of investigating both the dynamics of social phenomena as lineages of successive events and the complexity of contextual characteristics of "present" moments.

#### **2 Sequence Analysis and Qualitative Comparative as a Sequential Mixed-Methods Design**

Qualitative comparative analysis (QCA) is a method that permits systematic comparisons of cases through the highly-formalized tools of logic and set theory. Its perspective on the study of social phenomena is inherently analytical, but at the same time holistic and case-oriented (Berg-Schlosser et al. 2009; Ragin 1987; Schneider and Wagemann 2012).

In the QCA framework, cases are understood as complex entities of interrelated attributes. Empirically, this means that cases are classified as members or nonmembers of multiple and possibly overlapping sets. For instance, when investigating countries as cases, researchers can label them as "democracies" but also as "industrial economies," "corporatist welfare states" and so on. Each of these labels represents a set to which the country case can either belong or not. Similarly, if cases are represented by individuals rather than countries, researchers could envisage sets such as "working population," "women," "mothers," etc. While belonging to one set always implies not belonging to its negation, different sets are not mutually exclusive, as all mothers are women (subset relation) and some women are part of the working population (set intersection), for instance. In its original formulation later known as crisp-set QCA—set membership was dichotomous, while the later development of fuzzy-set QCA (Ragin 2000, 2008) permits a more fine-grained assessment: Set membership can vary in a continuum from 0 to 1 (as, for instance, individuals who work few hours a week could be considered as partial members of the set "working population").

QCA can be used as a classification tool, for example for typology building (Berg-Schlosser et al. 2009). However, most of the applications so far have taken an explanatory perspective in assessing the empirical regularities that exist between some factors and an outcome (Marx et al. 2014). Typical fields of application include comparative politics (Schneider 2009), welfare studies (Emmenegger 2011; Vis 2009), policy and administration (Sager and Thomann 2016), sociology of work and education (Borgna 2016; Glaesser and Cooper 2010), and organization research (Fiss 2011). The technique consists of a first step dedicated to identifying (combinations of) conditions in presence of which the cases systematically display a given outcome (a procedure known as truth-table construction). In a second step, redundant elements are removed from these configurations by applying Boolean or fuzzy-set algebra (i.e. truth-table minimization). The resulting Boolean expression represents the logically-minimal combinations of factors that are sufficient for the occurrence of the outcome and are sometimes defined as "outcome-enabling context" (Schneider and Wagemann 2012).

In a nutshell, QCA is a powerful method for systematically comparing cases without ruling out their potentially configurational nature. This complements the approach of SA, which underlines the complexity of cases in terms of sequencing of states and duration of events.

In this contribution, we bring together the strengths of SA and QCA in a sequential mixed-methods design. We propose applying SA in a first research stage, in order to describe both qualitatively and quantitatively the temporal complexity of a given social process. By applying QCA in a second stage, it is possible to shed light on the configurational complexity at given phases along the process and to reduce such complexity to synthetic combinations of explanatory factors. This combination is not a simple juxtaposition of methods but rather a genuine example of crossfertilization between two methodological traditions that arose from the same desire to "bring cases back" in quantitative analysis (Ragin and Becker 1992). By adopting our proposed research design, researchers can approach complexity from different angles and thus explore its temporal and configurational dimensions within the same analytical framework. This exploration is a first step towards the identification of what (Abbott 1992, 2009) has called "turning points" along the unfolding of social processes.

Practically, the steps of our proposed framework are the following: (1) sequence construction, where each time point is designated as a categorical state identifying the outcome of interest; (2) discrepancy analysis of state sequences, where we identify the phases when a given factor is mostly or increasingly relevant to explain inter-sequence differences; (3) truth-table construction, where we assess for each of these phases which combinations of factors are systematically associated with the outcome; (4) truth-table analysis, where the logically-sufficient factors for the occurrence of the outcome are identified.

#### **3 Empirical Illustration**

To illustrate the added value of our approach, we apply it to the study of women's employment trajectories in divided Germany (1955–1990). The leading research question for this exercise is: what conditions enabled women in East and West Germany to continue their education or employment over early- and mid-adulthood? This question is suitable for our illustrative purposes because there is extensive literature that accounts for the factors shaping women's labor-market participation within the two contexts (Diewald et al. 2006; Rosenfeld et al. 2004). This enables us to evaluate our results against a wide range of established, substantive evidence.

#### *3.1 Background*

During the period of division (1955–1990), the German Democratic Republic (henceforth, East Germany) and the Federal Republic of Germany (henceforth, West Germany) differed to a great extent concerning their economic and welfare systems.

The social market economy in West Germany was coupled with the promotion of a pro-traditional male breadwinner model in a corporative welfare state (Engelhardt et al. 2002; Rosenfeld et al. 2004). A highly gendered division of labor was indirectly supported by social policies that did not promote the compatibility of work and family and by a taxation system that penalized working wives (Brückner 2004; Cooke 2011; Sainsbury 1999). Female labor-market participation was around 50%—mostly part-time—and it was discouraged especially for women with children, while for male employment rates were around 80% (Diewald et al. 2006).

In the egalitarian centrally planned economy of East Germany, women's employment was supported by an infrastructure and an ideology that affirmed the right and duty to work. Pro-natalist family policies aimed at improving compatibility of work and family: The normative pressure to have children in one's early twenties was combined with widespread childcare, and state-controlled resources (e.g. housing or loans) were available to those who started a family (Kreyenfeld 2004). As a result, women's employment rates paralleled men's, reaching 90% also for women with children (Huinink et al. 1995).

When considering enabling conditions for employment, we expect gender to play a major role in the explanation of inter-individual differences along the whole life course, and especially so when specific life transitions occurred. Moreover, given the institutional differences outlined above, we expect gender to be a prominent explanatory factor in the West more than in the East. Finally, within each context, we expect some combinations of factors to be consequential for women's employment. We focus on three drivers that the literature indicates to be relevant to female labor participation (Engelhardt et al. 2002; Kreyenfeld 2004; Rosenfeld et al. 2004): partnership status, number of children, and parental education.

To develop our empirical illustration, we analyze longitudinal-retrospective data from the Starting Cohort Six of the *National Educational Panel Study* (NEPS) (Blossfeld et al. 2011). We selected a subsample of individuals born in East and West Germany (N = 374 and N = 1,695 respectively) between 1944 and 1955, i.e. individuals who experienced most of their early- and mid-adulthood in divided Germany.

#### *3.2 Empirical Analysis*

In a first step, we construct individual sequences representing longitudinal employment trajectories of men and women in West and East Germany. We then apply discrepancy analysis of state sequences to estimate the proportion of the variation in employment trajectories explained by gender at each time point in the sequences. This allows us to identify the phases when gender is mostly or increasingly relevant.

In a second step, we focus on these time points and apply fuzzy-set QCA to the sample of women to identify the configurations of factors sufficient for them to be employed or pursuing education in West and East Germany.

#### **3.2.1 Step 1: Sequence Analysis**

Sequences' Construction

We construct individual employment trajectories from ages 15 to 40 as sequences of monthly intervals1: each of the 300 months is encoded according to a binary definition that distinguishes being in education or employment from being out of education and out of employment (unemployment, inactivity, military service, maternity leave). We adopt this binary definition because both crisp- and fuzzyset QCA require a two-dimensional outcome. This does not rule out the possibility

1All analyses are performed in R.3.2.5 by using the packages TraMineR, TraMineRextras (Gabadinho et al. 2011), and WeightedCluster (Studer 2013). For a discussion on the choice of the distance measure see Studer and Ritschard (2016).

**Fig. 1** State distribution plot by gender and context: Percentage of individuals in a certain state at each point in time. (Source: authors' elaboration on NEPS data)

of adopting more elaborate definitions of states, as SA could be combined with multi-value QCA (Cronqvist and Berg-Schlosser 2009). From a substantive point of view, being in employment or in education is often considered a "positive" outcome during early adulthood and one that is associated with greater chances of societal integration later on. We censor our observational window at age 40 because we assume that the most important steps for the establishment of one's career have already occurred during this phase of the life course.

Figure 1 shows the state distribution plot by gender in East and West Germany. For men, periods spent out of employment or education were concentrated in the years around military service in both contexts. In contrast, women's employment trajectories differed according to context: while women in the East exhibited very limited amounts of time spent out of employment over the whole timespan considered, in West Germany women's likelihood of being in employment or education decreases over time.

#### Discrepancy Analysis

We apply discrepancy analysis of state sequences (DA) (Studer et al. 2011), which translates the ANOVA framework into sequence analysis. The resulting pseudo-*R*2 values can be interpreted as the share of the total variability between sequences explained by a specific factor of interest at each point in time (for a detailed description see Struffolino et al. 2016). The input of DA is a pairwise distance matrix based on a dissimilarity measure computed by assigning a cost to each transformation needed to align the couples of sequences (insertion—deletion and substitution costs). We used the optimal matching dissimilarity measure with an insertion—deletion cost of 1 and a substitution cost of 2. Notice that in this case, the sequences are composed of only two states, and therefore any more complex specification of costs would have been pointless (Studer and Ritschard 2016). Our explanatory factor of interest is gender. Hence, the pseudo-*R*2 value for betweengroup differences at each time point can be interpreted as the share of the total variability in the pool of sequences that is explained by gender.

Figure 2 displays the pseudo-*R*2 values for gender in East and West Germany. In the East, gender explains 12% of the variation between employment trajectories around age 20: this is most probably due to men being in military service. After age 23, gender explains very little of the variation between trajectories, because employment rates were very high for both men and women. In contrast to this, in the West gender explains an increasing share of the variability between employment trajectories starting from age 22; it reaches 15% between age 30 and 36.

The peaks in the pseudo-*R*2 represent phases of the life course where gender differences reach their relative maximum: These phases are of particular substantive interest because they might represent 'turning points' along the unfolding of employment trajectories (Abbott 2009). However, within the SA framework, change is understood as lasting over a period of time, rather than as happening at specific time points (Shanahan 2000). Divergence between sequences may then result from an extended succession of critical moments and not necessarily concentrate at single peaks. Therefore, both the peaks and the phases where gender differences display sharp increase can be considered as potential turning points. We randomly select one specific time point (month) for phases of peaks and sharp increases. To be able to compare between the two contexts, we additionally selected time points in the West and applied them to the East, and vice versa. This results in five time points that constitute the starting point for our second step of analysis: 19 years, 20 years and 9 months, 23 years, 31 years, and 35 years.

#### **3.2.2 Step 2: Qualitative Comparative Analysis**

In the second research stage, we shift the focus to women, because it is their patterns that diverge from those of men, at least in the West, and investigate which configurations of factors are systematically associated with being in employment or in education at each of the five time points identified in the first research stage and within the two contexts of East and West Germany. Our explanatory conditions refer

**Fig. 2** Share of the variability in employment trajectories over time explained by gender in East and West Germany: Pseudo-*R*2 values. (Source: authors' elaboration on NEPS data)

to women's partnership status, number of children, and family background. While many other factors could be listed as potentially explanatory, for the sake of simplicity of this empirical exercise we focus on the three identified as the most relevant according to previous literature (see Sect. 3.1). Moreover, parsimony in identifying potentially explanatory conditions is required when applying QCA, as the number of possible combinations grows exponentially with the inclusion of new conditions.

#### Calibration

As mentioned in Sect. 2, to apply QCA, the cases under observation must be classified as members or non-members of the sets corresponding to the outcome and the explanatory conditions. While in principle researchers can attribute set membership based on their own substantive knowledge of the cases, set membership is often assigned following the more systematic procedure of calibration, which consists of applying external criteria to transform an interval-scale variable into a crisp or fuzzy set.2

Two calibration techniques can be distinguished: Following the direct method (Ragin 2008, 94–97), researchers need to specify three values of the interval scale (also known as "source variable") that will identify three qualitative breakpoints in the set membership: full inclusion, full exclusion, and maximum ambiguity or crossover. For fuzzy sets, full inclusion usually corresponds to a set membership of 0.99, full exclusion to 0.01, and the crossover to 0.5, while intermediate values are assigned according to a logistic function. For crisp sets, only the value corresponding to the crossover must be specified, as set membership can only assume the value of 0 or 1. Instead, when following the indirect method of calibration, one should use substantive knowledge to determine the correspondence between values of the source variable and set membership scores. The recommended set membership scores are: 1 for full inclusion, 0.8 for cases "mostly but not fully in the target set," 0.6 for cases "more in than out of the target set," 0.4 for cases "more out than in of the target set," 0.2 for cases "mostly but not fully out of the target set," and 0 for full exclusion (Ragin 2008, 95–96). For both the direct and indirect calibration methods, the threshold criteria need to be justifiable and transparent. In our empirical application, cases are constituted by individuals—or more specifically, women—in East and West Germany observed at each of the five time points selected in the SA analytical step. The criteria for the calibration of the sets corresponding to the outcome and to the explanatory conditions are detailed in the following.

*Outcome*: Crisp set of "women in employment *or* in education". In this case no calibration is necessary because the source variable, corresponding to the two states analyzed in the first research stage, is already dichotomous (see above).

#### *Explanatory Conditions*:

2Remember that for crisp sets membership is dichotomous, while for fuzzy sets it can vary in a continuum between 0 and 1.

3For the sake of simplicity, in what follows we shall talk of "highly educated parents" vs. "nonhighly educated parents". However, cases belonging to this set should be understood as women originating from households where at least one parent has at least an intermediate level of education—hence, a proxy for a relatively advantaged social background.

**Table 1** Descriptive statistics for the outcome and the conditions, by context and time point

Source: authors' elaboration on NEPS data *m* months

to account for the differences in the degrees of educational expansion and in the educational systems. East Germany: 15.5 years (university education): 0.99; 12.5 years (above the 12 threshold corresponding to upper-secondary education): 0.5; 11 years (less than upper-secondary education): 0.01. West Germany: 15.5 years (university education): 0.99; 11 years (above the 10 threshold corresponding to the intermediate school certificate): 0.5; 9.5 (no or low school certificate only): 0.01.

Table 1 reports descriptive information on the source variables for the outcome and the conditions for each time point and context.

**Table 2** Truth tables for the presence of the outcome, by context and time points

Source: authors' elaboration on NEPS data. Logical remainders (N *<* 20) omitted for parsimony. Consistency threshold: 0.75. In bold: configurations associated with the presence of the outcome.

#### Truth-Table Construction

Before proceeding to the analysis, we explore the empirical configurations displayed by the cases and their relation with the outcome for each time point and context. To this scope, we construct a so-called "truth table," whose rows correspond to possible combinations of conditions. Based on the empirical distribution of cases, each combination is connected to either the presence or the absence of the outcome. The 10 resulting truth tables are displayed in Table 2.

The intersection of our three conditions (CHILDREN, HIGH-PAREDU, PART-NERSHIP) produces eight possible configurations. However, not all configurations display empirical instances and some rows are therefore empty (the so-called "logical remainders"). To minimize random noise, we set a frequency threshold of 20 cases, meaning that configurations with less than 20 cases are classified as logical remainders.4 In Table 2, logical remainders are omitted for parsimony. Each row represents a combination of conditions (with 1 indicating the presence, and 0 the absence of the condition) to which a certain number of cases (N) belongs. For instance, the first row in the right part of the table indicates that, in the sample of West Germany, when they were 19 years old, 635 women did not have any children, came from highly educated families, and were not in a partnership (CHILDREN = 0, HIGH-PAREDU = 1, PARTNERSHIP = 0). Among them, the vast majority was in employment or in education (OUT = 1), as indicated by the consistency parameter (Cons. = 0.942). This parameter, roughly speaking, returns the proportion of women belonging to the 0-1-0 configuration who were in employment or in education.5

Table 2 reveals that in each of the five time points, for both the East and the West, multiple configurations were associated with the outcome. In East Germany, this applies to all existing configurations (i.e. displaying at least 20 empirical instances), in line with the fact that in this context the great majority of women remained in employment (or education) throughout their whole life course (see Fig. 1).

Looking more closely at Table 2, we note that, with the exception of the second time point, there are only two configurations for East Germany. Moreover, they are constant for the last three time points: women in stable partnerships and with children, with (1-1-1) and without (1-0-1) highly educated parents. This finding is consistent with previous research underlining the standardization of individual life courses in the GDR (Fasang and Raab 2014; Struffolino et al. 2016). More diversity existed in West Germany. This is visible first of all in the number of rows that exhibit empirical instances: Women are generally grouped in four or five configurations. Second, such configurations vary considerably over time, as can be seen from the comparison of the rows across time points. Finally, some configurations are associated with the presence and some with the absence of the outcome.

#### Truth-Table Analysis

In order to remove the redundant elements from the configurations associated with the outcome and thus identify the "outcome-enabling" conditions, we perform the fuzzy-set minimization of the 10 truth tables.6 Different strategies exist in the literature for dealing with the issue of limited diversity, which emerges in QCA as non-existent configurations or logical remainders (Schneider and Wagemann 2012, 160–177). Given the descriptive purpose of our analysis, we refrain from

4Alternatively, one could set a relative threshold corresponding to a given share of cases, e.g. 5%. 5More precisely, the consistency parameter indicates the extent to which the fuzzy set corresponding to the 0-1-0 configuration is a subset of the fuzzy set corresponding to the outcome.

6The analyses were performed using the QCA package (Thiem and Du¸sa 2013) in R.3.2.5 using the Enhanced Quine-McCluskey Algorithm without row dominance rule.

using logical remainders, meaning that configurations with less than 20 cases are not considered in the minimization process. Therefore, our solutions are to be intended as conservative (*ivi*: 182). Additionally, before proceeding to the analysis one must set a consistency threshold in order to exclude configurations that are not sufficient subsets of the outcome. We apply a consistency threshold of 0.75, which is generally considered acceptable in the QCA literature (*ibidem*), but results are generally robust even with stricter consistency levels.7

The results are summarized in Table 3, whose columns represent the five time points and whose rows display the "prime implicants." The prime implicants are the essential components of a configuration identified by the truth-table analysis that, read in combination with each other, constitute the logically-minimal Boolean expression sufficient for the presence of the outcome (Ragin 1987, 95–98). For example, for 19-year-old West-German women, in order to be in employment or in education, it was sufficient not to have children *and* not to be in a partnership, *or*, as an alternative, not to have children *and* to have highly educated parents. The resulting logically-minimal Boolean expression is therefore: "child\*partnership + child\*HIGH-PAREDU".8

Table 3 thus shows the dynamic development of the combinations of factors logically sufficient for being in employment or in education. Indeed, these combinations changed over time during the phases of early- and mid-adulthood in both contexts. In both East and West Germany, not having children *and* not being in a partnership emerges a prime implicant for the first two time points considered, although in combination with different prime implicants (high-paredu\*PARTNERSHIP in the East and child\*HIGH-PAREDU in the West), but later disappears. As women reached the stage of their life course when most entered partnerships and had children, the