#  BUSSARD –
Better Understanding Social Situations
for Autonomous Robot Decision-Making

Stefan Schiffer1 and Astrid Rosenthal-von der Pütten1 and Bastian Leibe2
1Stefan Schiffer and Astrid Rosenthal-von der Pütten are with the Chair
Individual and Technology, Department of Society, Technology, and Human
Factors, RWTH Aachen University, Aachen Germany {stefan.schiffer,
<EMAIL_ADDRESS>Leibe is with the Chair Computer Science 13
(Computer Vision), Department of Computer Science, RWTH Aachen University,
Aachen, Germany<EMAIL_ADDRESS>
###### Abstract

We report on our effort to create a corpus dataset of different social context
situations in an office setting for further disciplinary and interdisciplinary
research in computer vision, psychology, and human-robot-interaction. For
social robots to be able to behave appropriately, they need to be aware of the
social context they act in. Consider, for example, a robot with the task to
deliver a personal message to a person. If the person is arguing with an
office mate at the time of message delivery, it might be more appropriate to
delay playing the message as to respect the recipient’s privacy and not to
interfere with the current situation. This can only be done if the situation
is classified correctly and in a second step if an appropriate behavior is
chosen that fits the social situation. Our work aims to enable robots
accomplishing the task of classifying social situations by creating a dataset
composed of semantically annotated video scenes of office situations from
television soap operas. The dataset can then serve as a basis for conducting
research in both computer vision and human-robot interaction.

##  I Introduction

Social robots are envisioned to interact with humans in social scenarios, for
instance as assistants helping with daily tasks, reminding of appointments,
and delivering messages. To interact in a socially competent way, a robot
needs to consider the full social context – otherwise the acceptance of the
social robot is at stake.

In this paper we report on first findings in our effort to create a dataset of
social situations that can be used to help robots understand such situations
better and that is useful for research in computer vision, psychology, and
human-robot interaction alike.

The rest of the paper is organized as follows. First, we briefly review the
state-of-the-art both in social robotics and in computer vision. Then, we
sketch our project and report on the current state and lessons learnt so far.
Finally, we give an outlook on what work lies ahead of us.

##  II Background and Related Work

While context in human-robot interaction was initially merely a question of
user location, in recent years the term is increasingly understood as a more
complex concept: the system must draw conclusions from the perceived situation
in order to understand the current context in its entirety. This is crucial to
develop the sophisticated context models that are necessary to realize a smart
environment [10]. Just localizing users only provides one bit of information
necessary to understand what is going on. Two users in the same room can mean
many things: they can be friends or foes, they could be of equal or different
status, they could trust or distrust each other. [9] describe the challenges
and opportunities of socially intelligent agents, and break down social
context to the influencing factors situational context, social roles, social
norms and cultural conventions.

Overall, a large number of variables influences which behavior is socially
expected or desired in a particular situation, and which behavior is perceived
inappropriate. Human information processing manages to grasp this large amount
of information in the shortest time and thus makes it possible to judge social
situations at a glance. Social cognition usually takes place automatically,
unconsciously and with a minimum of cognitive effort [3]. Hence, the judgment
of a social situation comes naturally to humans, without considerable
cognitive effort, but they often cannot say explicitly why they decided one
way or the other or which factors determined the decision [4].

Computer vision scholars identified the lack of a social perspective as one of
the challenges in visual scene understanding: “The scarcity of a more social,
contextual perspective in the automated analysis of human-human interactions
is also reflected in computer vision literature, where interactions are
typically reduced to visually and temporally well-defined events.” [13, p. 1].
Research in computer vision has already addressed some aspects of social scene
understanding [13]. For instance, the presence of people in a scene can be
detected and tracked automatically [14, 7], their posture can be assessed [1,
12], and hand gestures can be determined [8]. One important factor limiting
research in this direction, which is also highlighted in recent surveys [13],
is the lack of large, well-curated and annotated datasets capturing complex
social interactions. They are too small (e.g., only 20 scenes, dataset “UT
interaction” by [11]), they are not embedded in real environments (e.g. only
in front of static background, with fixed perspective, or scripted/staged
interactions, such as the “SBU Kinetic Interaction” datasets (by [16]) and “UT
interaction”), and/or they only assess a very limited set of events (e.g. only
four classes of events like handshake, kiss, hug, high-five; such as the
datasets “TV Human Interaction” by [6] and “Hollywood2” by [5]).

##  III Approach and Lessons Learnt

Within BUSSARD, we aim to learn social context models from video sequences of
person-person interactions extracted from a video corpus of telenovela
episodes. The idea is to take advantage of repeated settings and known
personal relationships between the different characters in order to provide
rich annotations of the interaction context in a semi-automatic manner.
Automatic annotation of tv series episodes has already been explored by
several research groups, starting from the work by [2, 6]. We proceed in a
similar fashion, yet our focus is slightly different.

The general approach in BUSSARD then is on curating a dataset of annotated
social situations. On that data we first identify basic features (e.g.
visually perceptible features such as pose or gaze, temporal dynamics of
events) as well as ontological information (e.g., age, gender, relationship
between people, social rules on how to behave in a given situation). Then we
want to estimate which (combinations of these) features and ontological
information determines or allows for abstract judgements about the social
situations (e.g., whether there is a conflict or intimacy, a formal or
informal gathering, etc.). We want to explore both, how humans form such a
judgement as well as how robots can potentially classify social situations
like humans do.

The basic features mentioned above will first be labeled manually or, if
possible, they will be automatically computed with computer vision and
verified afterwards. On top of the basic features we will record a "scene
type" (e.g. business, leisure, …) and a "scene mood" (e.g. tense, joyful,
intimate). We are currently evaluating whether to use the adjectives from
PANAS [15] for the latter. Also, we register a one sentence description for
every scene (shown to clickworkers) in order to see whether our vocabulary is
appropriate or not.

So far, we have screened hundreds of hours of video data of more than ten TV
series. Only four series qualified to collect data from, and even with these
four series, only few scenes meet an extended set of requirements on the video
data. These requirements include sufficient realism, no overacting, no zooming
or moving camera effects and more. This is because the video material should
be as close as possible to what a robot will see in its future deployment
scenarios. We had to realize, that such requirements are rather exclusive and
only few scenes were usable. As an example, we took the German TV Series
"Sturm der Liebe".111https://daserste.de/unterhaltung/soaps-telenovelas/sturm-
der-liebe/index.html It is a telenovela and it is considered the most
successful daily TV series in Europe. From 100 episodes we could eventually
only collect 6:34 minutes of video data. Every clip has a length of 10 to
fifteen seconds. Scene types range from business settings to flirts and
romantic encounters.

We are now in the process of labeling the scenes with background knowledge and
automatic and manual annotations for the basic features before we give the
scene to clickworkers to classify the scenes according to higher-level
properties such as “intimate” or “tense” but also to confirm the manually
assigned basic features.

Once this data is being farmed, we plan to test both end-to-end machine
learning as well as a knowledge-based supervised learning method to
automatically classify scenes on a robot with these learnt models and to see
how they compare.

## ACKNOWLEDGMENT

We would like to thank Lena Plum, Luisa Escherich, Paula Winter, and Archit
Dharma for their support in screening, selecting, collecting, and annotating
the video data. This work presented in this paper was funded under grant
number OPSF728 in the Open Seed Fund 2022 pogram of the Exploratory Research
Space at RWTH Aachen University.

## References

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# Monotonicity in the averaging process

E. Ben-Naim  Theoretical Division and Center for Nonlinear Studies, Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA  P. L. Krapivsky
Department of Physics, Boston University, Boston, Massachusetts 02215, USA
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA  Skolkovo
Institute of Science and Technology, 143026 Moscow, Russia

###### Abstract

We investigate an averaging process that describes how interacting agents
approach consensus through binary interactions. In each elementary step, two
agents are selected at random and they reach compromise by adopting their
opinion average. We show that the fraction of agents with a monotonically
decreasing opinion decays as $e^{-\alpha t}$, and that the exponent
$\alpha=\tfrac{1}{2}-\tfrac{1+\ln\ln 2}{4\ln 2}$ is selected as the extremum
from a continuous spectrum of possible values. The opinion distribution of
monotonic agents is asymmetric, and it becomes self-similar at large times.
Furthermore, the tails of the opinion distribution are algebraic, and they are
characterized by two distinct and nontrivial exponents. We also explore
statistical properties of agents with an opinion strictly above average.

##  I Introduction

The averaging process models a system of agents who reach agreement via
compromise. The system consists of $N$ agents. In each step, two agents, say
$i$ and $j$, are chosen at random and their opinions $v_{i}$ and $v_{j}$ are
replaced by the average

$(v_{i},v_{j})\to\left(\frac{v_{i}+v_{j}}{2},\frac{v_{i}+v_{j}}{2}\right)\,.$
(1)

As this elementary step is repeated, the system moves closer and closer toward
consensus where all agents have the same opinion.

This averaging process has been studied by statistical physicists CFL ,
applied probabilists AL , computer scientists OT , and social scientists HK
with applications ranging from opinion dynamics flpr  to communication
algorithms for computer and sensor networks sdj ; pbb  to linguistics dcll .
For the pure averaging process (1), the system approaches perfect consensus.
However, when interactions are restricted to agents with sufficiently close
opinions, the system bifurcates into groups HK ; WDAN ; BKR ; JL ; OH ; BS ,
with all agents within the same group sharing the same opinion. Another
generalization of the averaging process (1) involves partial averaging where
the opinion difference is reduced by a fixed multiplicative factor in each
interaction. Partial averaging is equivalent to inelastic collisions, and it
has been used to model freely evolving and driven inelastic gases BK00 ; BMP02
; BK04 ; KB02 ; EB02 ; ADL ; PDSR ; GS ; NG ; book  that satisfy the Maxwell
model rules M67 .

In this study, we analyze the pure averaging process (1). The system
approaches perfect consensus and the difference between the typical opinion
and the consensus opinion decays exponentially with time BK00 . While the
typical opinion follows a unidirectional path en route to consensus, an
individual opinion may increase or decrease due to fluctuations. We focus on
agents with a monotonically decreasing opinion (similar behavior is exhibited
by agents with a monotonically increasing opinion).

We find that the fraction $M(t)$ of monotonic agents decays exponentially with
time,

$M(t)\simeq A\,e^{-\alpha t}\,,$ (2)

in the long-time limit (see Fig. 1). Our main result is that the exponent
$\alpha$ is nontrivial

$\alpha=\frac{1}{2}-\frac{1+\ln\ln 2}{4\ln 2}=0.271517\ldots\,.$ (3)

The opinion distribution of monotonic agents becomes self-similar at
sufficiently large times. Further, this distribution has two algebraic tails
that are characterized by two different exponents. These features of the
opinion distribution enable us to determine the exponent $\alpha$ which is
selected as the extremal value from a spectrum of possible values.

The rest of this paper is organized as follows. We begin with a brief overview
of the averaging process in Sec. II. In Sec. III, we consider monotonic agents
with an opinion that only decreases with time. We obtain the exponent (3), and
also study the opinion distribution of monotonic agents. In Sec. IV, we
investigate the behavior of agents with strictly positive opinion. In Sec. V,
we summarize our findings and discuss possible avenues for future work. The
generalization of the pure averaging process (1) to partial averaging is
outlined in Appendix A, and corrections to the leading asymptotic behavior (2)
are discussed in appendix B.

##  II The averaging Process

In the averaging process, there are $N$ interacting agents. In each averaging
event, two agents are selected at random, and their opinions evolve according
to (1). This pairwise interaction is repeated indefinitely, and time is
augmented by $2/N$ after each interaction, so that agents experience one
averaging event per unit time. The distribution $F(v,t)$ of agents with
opinion $v$ at time $t$ satisfies the nonlinear rate equation BK00

$\frac{\partial F(v,t)}{\partial
t}=2\int_{-\infty}^{\infty}du\,F(u,t)F(2v-u,t)-F(v,t)\,,$ (4)

in the limit $N\to\infty$. The convolution term reflects the binary nature of
(1), and the linear loss rate reflects that agents participate in one
interaction per unit time. The averaging process (1) conserves the number of
particles and the total opinion, and consequently, the rate equation (4)
conserves the two lowest moments of the opinion distribution: the normalized
distribution, $\int dv\,F(v,t)=1$, and the average opinion, $\langle
v(t)\rangle=\int dv\,v\,F(v,t)={\rm const}$.

The averaging process (1) is invariant under translation $v\to v+{\rm
const.}$, as well as dilation, $v\to{\rm const.}\times v$. Hence, without loss
of generality, we may consider initial distributions with zero average,
$\langle v(0)\rangle=0$, and unit variance, $\langle v^{2}(0)\rangle=1$ finite
. Furthermore, we restrict our attention to symmetric distributions as Eq. (4)
implies $F(v,t)=F(-v,t)$ when $F(v,0)=F(-v,0)$.

The system approaches consensus, $F(v,t)\to\delta(v)$, as all agents acquire
the average initial opinion in the long-time limit. The second moment $\langle
v^{2}(t)\rangle=\int_{-\infty}^{\infty}dv\,v^{2}F(v,t)$ quantifies the
distance between the typical opinion and the consensus opinion. This quantity
decays exponentially with time BK00

$\langle v^{2}(t)\rangle=e^{-t/2}\,,$ (5)

as follows from Eq. (4). Moreover, the distribution $F(v,t)$ becomes self-
similar in the long-time limit, and the second moment (5) sets the scale for
the typical opinion. In particular, the distribution $F(v,t)$ adheres to the
scaling form BMP02

$F(v,t)=e^{t/4}\mathcal{F}(V),\quad{\rm
with}\quad\mathcal{F}(V)=\frac{2}{\pi}\,\frac{1}{(1+V^{2})^{2}}\,,$ (6)

and the scaling variable $V=v\,e^{t/4}$. This scaling behavior holds in the
limits $t\to\infty$ and $v\to 0$. Also, the scaling function is normalized
$\int_{-\infty}^{\infty}dV\mathcal{F}(V)=1$.

##  III Monotonicity

The ultimate opinion of every agent vanishes, $v_{i}\to 0$ as $t\to\infty$.
Further, according to Eqs. (5)–(6), the typical magnitude of the opinion $|v|$
decreases monotonically, $|v|\sim e^{-t/4}$. In this study, we focus on agents
with a monotonically decreasing opinion. At time $t$, we refer to agents with
opinion that satisfies the inequality $v_{i}(t_{1})\geq v_{i}(t_{2})$ for all
$t_{1}\leq t_{2}\leq t$ as monotonic agents. In the context of opinion
dynamics, monotonic agents change their opinion only in one direction, say
toward the left of the political spectrum only. According to (1), monotonic
agents interact only with agents who have a smaller opinion. We stress that
monotonic agents have an opinion that strictly decreases with time, but the
sign of the opinion is not constrained. In particular, monotonic agents may
start with a positive opinion and end up with a negative one.

Figure 1: The fraction of monotonic agents $M(t)$ versus time $t$. A dashed
line with slope given by Eq. (2) is displayed for reference. The inset shows
that the local slope $\alpha(t)\equiv-d\ln M/\ln t$ varies linearly with
$1/t$. The dotted line shows a linear fit that yields the estimate
$\alpha=0.272\pm 0.003$.

We denote by $M(t)$ the fraction of monotonic agents at time $t$, and by
$M(v,t)$ the density of such agents with opinion $v$. Of course, $M(t)=\int
dvM(v,t)$. By symmetry, the dual fraction of agents with monotonically
increasing opinion equals $M(t)$, and their density is given by $M(-v,t)$. The
density $M(v,t)$ is coupled to the total density $F(v,t)$, and it satisfies
the linear rate equation

$\frac{\partial M(v,t)}{\partial
t}=2\int_{v}^{\infty}du\,M(u,t)F(2v-u,t)-M(v,t)\,.$ (7)

The loss term reflects that on average, each agent experiences one interaction
per unit time. The gain term in Eq. (7) resembles the gain term in Eq. (4),
but the lower limit of integration ensures that the opinion of a monotonic
agent may only decrease.

By integrating the master equation (7) over all opinions, we find that the
fraction $M(t)$ decreases with time according to the rate equation

$\frac{dM(t)}{dt}=-\int_{-\infty}^{\infty}du\,M(u,t)\int_{u}^{\infty}dv\,F(v,t)\,.$
(8)

This evolution equation reflects that monotonic agents may only interact with
agents having smaller opinions. The fraction $M(t)$ has two bounds,
$e^{-t}\leq M(t)\leq 1$. The upper bound follows from the inequality
$M(v,t)\leq F(v,t)$. The lower bound reflects that agents experiencing zero
interactions are necessarily monotonic. Since agents interact once per unit
time, the overall density $N(t)$ of noninteracting agents decays exponentially
with time, $N(t)=e^{-t}$. The bounds $\alpha=0$ and $\alpha=1$ are realized in
limiting cases of the partial averaging process, as discussed in Appendix A.

When the opinion $v$ is sufficiently large, the dominant contribution to the
integral in (7) comes from the vicinity of $u=2v$. Using the normalization
$\int_{-\infty}^{\infty}dv\,F(v,t)=1$ we arrive at the linear equation

$\frac{\partial M(v,t)}{\partial t}=2M(2v,t)-M(v,t)\,,$ (9)

that holds for sufficiently large $v$. The derivation of Eq. (9) relies on the
fact that the distribution $F(v,t)$ is normalized, but remarkably, the precise
form of that distribution is not utilized. Indeed, as follows directly from
(1), the outcome of interactions involving an agent with a very large opinion
$v$ is not affected by the opinion of the interaction counterpart.
Consequently, large opinions are reduced by a factor $2$ with each
interaction, thereby leading to a simple multiplicative process bm

$v\to v/2\to v/4\cdots.$ (10)

Equation (9) merely reflects this multiplicative process.

Figure 2: The normalized scaling function $A^{-1}\mathcal{M}(V)$ versus the
scaling variable $V$. This quantity is the probability distribution function
of the scaled opinion $V$, as follows from Eqs. (2) and (11).

In the long-time limit, the density $M(v,t)$ of monotonic agents adheres to
the scaling form

$M(v,t)\simeq e^{-\alpha t}e^{t/4}\,\mathcal{M}(V)$ (11)

with the same scaling variable $V=ve^{t/4}$. By integrating (11), we obtain
the exponential decay (2) with $A=\int_{-\infty}^{\infty}dV\,\mathcal{M}(V)$.
Figure 2 convincingly shows that the scale $v\sim t^{-1/4}$ also characterizes
the opinion distribution of monotonic agents.

By substituting the scaling form (11) into (9), we find that the scaling
function $\mathcal{M}(V)$ satisfies

$\frac{V}{4}\,\frac{d\mathcal{M}(V)}{dV}+\left(\frac{5}{4}-\alpha\right)\mathcal{M}(V)=2\mathcal{M}(2V)$
(12)

when $V\gg 1$. This difference-differential equation admits an algebraic
solution, $\mathcal{M}(V)\sim V^{-\nu}$, with

$\alpha=\frac{5-\nu}{4}-2^{1-\nu}\,.$ (13)

The right-hand side of (13) is bounded from above, see Fig. 3. The maximal
value quoted in (3) occurs at

$\nu=3+\frac{\ln\ln 2}{\ln 2}=2.471233\ldots\,.$ (14)

Thus, the algebraic tail for $V\gg 1$ yields the upper bound
$\alpha\leq\alpha_{*}$. We postulate that this extremal value is realized,
$\alpha=\alpha_{*}$, for the averaging process. Our numerical simulations give
the estimate $\alpha=0.272\pm 0.003$ and hence support this theoretical
prediction (see the inset to Fig. 1 and also, Appendix B). We note that our
assumption that the behavior in the neighborhood of $u=2v$ dominates the
integral in Eq. (7) is consistent with the fact that the tail $M(v)\sim
v^{-\nu_{*}}$ is shallower than the tail $F(v)\sim v^{-4}$. The selection of
the extremal value emerging from the dispersion-like relation (13) very much
resembles the selection of the extremal propagation velocity in traveling
waves that are governed by partial differential equations in deterministic KPP
; Fisher ; Bramson ; Saarloos ; Saarloos-Rev  and stochastic systems BD97 ;
KM00 ; KM02 ; BK03 ; BD06 .

Figure 3: The exponent $\alpha$ in (13) versus $\nu$. The maximal value
$\alpha=0.271517$ occurs at $\nu=2.47123$.

Next, we substitute the scaling form (11) into the master equation (7) and
arrive at the linear integro-differential equation

$\frac{V}{4}\,\frac{d\mathcal{M}}{dV}+\left(\frac{5}{4}-\alpha\right)\mathcal{M}=2\int_{V}^{\infty}dU\,\mathcal{M}(U)\,\mathcal{F}(2V-U)$
(15)

with $\mathcal{F}(V)$ given in Eq. (6). We stress that this equation governs
the scaling equation $\mathcal{M}(V)$ for all values of $V$, in contrast with
Eq. (12) that applies only when $V\gg 1$. The parameter $\alpha$ is an
eigenvalue of this equation, and in principle, a solution for the eigenvalue
$\alpha$ requires a solution for the entire eigenfunction $\mathcal{M}(V)$.
However, in our particular problem, extreme-value analysis suffices.

To understand the behavior when $V\ll-1$, we introduce the change of variables
$U=2V-W$, and thereby recast the right-hand side of Eq. (15) into
$2\int_{-\infty}^{V}\\!dW\,\mathcal{M}(2V-W)\,\mathcal{F}(W)$ . In the limit
$V\to-\infty$, this integral is negligible and Eq. (15) simplifies to
$V\mathcal{M}^{\prime}+(5-4\alpha)\mathcal{M}=0$. Therefore, there is a second
algebraic tail, $M(V)\sim(-V)^{-(5-4\alpha)}$ when $V\ll-1$.

Thus, the scaling function $\mathcal{M}(V)$ has two distinct algebraic tails

$\mathcal{M}(V)\sim\begin{cases}(-V)^{-(5-4\alpha)}&V\ll-1\,,\\\ V^{-\nu}&V\gg
1\,.\end{cases}$ (16)

The asymmetry of $\mathcal{M}(V)$ is reflected by the inequality
$\nu<5-4\alpha$ as $5-4\alpha=3.913932$. Interestingly, both tails of the
scaling function $\mathcal{M}(V)$ are shallower than the tails of the scaling
function $\mathcal{F}(V)$ as both $\nu<4$ and $5-4\alpha<4$. Of course, the
inequality $M(v,t)\leq F(v,t)$ holds for all $v$. By comparing the two
densities $e^{-\alpha t}\mathcal{M}(V_{\pm})\sim\mathcal{F}(V_{\pm})$, we find
that (6) and (11) hold simultaneously in the scaling region $V_{-}\ll V\ll
V_{+}$. Both scales $V_{+}$ and $V_{-}$ grow exponentially with time, and
hence, the two scaling forms (6) and (11) hold over an exponentially growing
range of scaled opinions.

Another consequence of the asymmetry of $\mathcal{M}(V)$ is that the majority
of monotonic agents have a positive opinion. This fraction saturates at a
finite value

$M_{+}=\int_{0}^{\infty}dV\,\mathcal{M}(V)\,.$ (17)

Using numerical simulations we find $M_{+}=0.74\pm 0.01$. The complementary
fraction $M_{-}$ of monotonic agents with negative opinion, $M_{-}=1-M_{+}$,
is roughly three times smaller than the fraction of monotonic agents with a
positive opinion.

Our Monte Carlo simulations utilize a straightforward implementation of the
averaging process. Initially, there are $N$ agents whose opinions are drawn
from a uniform distribution: $F(v,0)=1$ for $|v|<1/2$ and $F(v,0)=0$
otherwise. We stress that our main results are independent of the shape of the
initial distribution: the same scaling functions $\mathcal{F}(V)$ and
$\mathcal{M}(V)$ are realized for compact distributions as well as non-compact
distributions finite . In each simulation step, two agents are selected at
random, and their opinions are updated according to the averaging rule (1).
Time is augmented by $2/N$ subsequently. We maintain a counter for the number
of monotonic agents, and whenever an agent interacts for the first time with
an agent having a larger opinion, the counter decreases by one. The simulation
results shown throughout this paper represent an average over $10^{3}$
independent Monte Carlo runs in a system of size $N=10^{8}$.

We made the following choices to optimize the simulations: (i) we adjust the
average initial velocity to zero, (ii) we keep track of agents with
monotonically increasing opinions as well as agents with monotonically
increasing opinions, and (iii) we rescale the opinion $v\to ve^{1/4}$ once per
unit time. The latter rescaling enables direct measurement of $V=ve^{t/4}$,
and additionally, it prevents simulation of exponentially small opinions.

##  IV Positivity

Figure 4: The normalized fraction of positive agents $2P(t)$ versus time $t$
(since only one half of all agents start with a positive opinion, $2P(0)=1$).
The inset shows $\beta(t)\equiv-d\ln P/\ln t$ versus $\exp(-3t/8)$. A linear
fit, shown as the dashed line, yields the estimate $\beta=0.188\pm 0.001$.

We also studied a related subset of agents with an opinion that remains
strictly above average. Since the average opinion is zero, these are the
agents with a strictly positive opinion. Let $P(t)$ be the fraction of such
positive agents at time $t$. Our numerical simulations show that $P(t)$ decays
exponentially with time (Fig. 4)

$P(t)\simeq B\,e^{-\beta t}\quad{\rm with}\quad\beta=0.188\pm 0.01\,.$ (18)

We analyze the density $P(v,t)$ of agents with opinion $v>0$, from which the
overall density follows, $P(t)=\int_{0}^{\infty}dv\,P(v,t)$. Of course, there
is a dual set of agents who maintain a strictly negative opinion, with an
overall fraction $P(t)$ and density $P(-v,t)$. In the context of opinion
dynamics, positive and negative agents can be viewed as agents that remain
consistently on the left side or the right side of the political spectrum. The
density $P(v,t)$ evolves according to the linear equation

$\frac{\partial P(v,t)}{\partial
t}=2\int_{0}^{\infty}du\,P(u,t)F(2v-u,t)-P(v,t)\,,$ (19)

for $v>0$. This equation reflects that the density $P(v,t)$ is coupled to the
overall density $F(v,t)$, and it differs from (7) only in the lower limit of
integration. Using (19), we deduce the rate equation

$\frac{dP(t)}{dt}=-\int_{0}^{\infty}du\,P(u,t)\int_{u}^{\infty}dvF(v,t)$ (20)

for the fraction $P(t)$. Again, this evolution equation differs from (8) only
in the lower limit of integration.

In the long-time limit, the distribution $P(v,t)$ acquires the scaling form

$P(v,t)\simeq e^{-\beta t}e^{t/4}\,\mathcal{P}(ve^{t/4})\,.$ (21)

This form is consistent with the exponential decay (18) with
$B=\int_{0}^{\infty}dV\,\mathcal{P}(V)$ . Our numerical simulations confirm
that this scaling behavior applies at sufficiently large times (Fig. 5). The
scaling function $\mathcal{P}(V)$ is qualitatively similar to $\mathcal{M}(V)$
(see Fig. 2): both functions are non-monotonic and are maximal at a nonzero
value of $V$. For positive agents, the “depletion” region near $V=0$ reflects
that positive agents with a sufficiently small opinion are less likely to
remain positive.

By substituting the scaling forms (6) and (21) into the evolution equation
(19), we find that the scaling function $\mathcal{P}(V)$ satisfies the linear
integro-differential equation

$\frac{V}{4}\,\frac{d\mathcal{P}}{dV}+\left(\frac{5}{4}-\beta\right)\mathcal{P}=2\int_{0}^{\infty}dU\,\mathcal{P}(U)\,\mathcal{F}(2V-U).$
(22)

This equation poses an eigenvalue problem with the scaling function
$\mathcal{P}(V)$ being the eigenfunction, and the exponent $\beta$ being the
eigenvalue. The eigenfunction is subject to the constraint $\mathcal{P}(V)>0$
for all $V>0$.

Figure 5: The normalized scaling function $B^{-1}{\cal P}(V)$ versus the
scaled opinion $V$.

To determine the asymptotic behavior of $\mathcal{P}(V)$ when $V\gg 1$, we
repeat the approach used in Section III and arrive at an equation that is
entirely analogous to (15),

$\frac{V}{4}\,\frac{d\mathcal{P}}{dV}+\left(\frac{5}{4}-\beta\right)\mathcal{P}=2\mathcal{P}(2V)\,.$
(23)

Hence, the tail of the scaling function is algebraic, $\mathcal{P}(V)\sim
V^{-\mu}$, and the dispersion relation reads

$\beta=\frac{5-\mu}{4}-2^{1-\mu}\,.$ (24)

By substituting the Monte Carlo simulation result (18) into the dispersion
relation (24), we expect $\mu=3.58\pm 0.01$. The numerical simulation results
(Fig. 6) give $\mu=3.6\pm 0.2$, consistent with (24). The eigenvalue $\beta$
does not correspond to an extremum in the dispersion equation (24), so it must
be determined as the eigenvalue of the full integro-differential equation
(22). As was the case for monotonic agents, the large-$v$ tail of the opinion
density $P(v,t)$ is algebraic. Furthermore, the algebraic tail of the scaling
function $\mathcal{P}(V)$ is shallower than the tail of the scaling function
$\mathcal{F}(V)$ as $\mu<4$.

Figure 6: The scaling function $\mathcal{F}(V)$ and $\mathcal{P}(V)$ for $V\gg
1$. Also shown as a reference are the theoretical results (6) and (24).

##  V Discussion

In summary, we considered the averaging process and studied sub-classes of
agents with an opinion that maintains a certain property throughout the
evolution. In particular, we probed the fraction of agents with a
monotonically decreasing opinion and the fraction of agents with a positive
opinion. These fractions decrease exponentially with time, and the exponents
characterizing these decays are eigenvalues of linear integro-differential
equations. In the case of monotonic agents, we were able to find the
eigenvalue analytically using an extremum selection principle, analogous to
velocity selection in traveling waves KPP ; Fisher ; Bramson ; Saarloos ;
Saarloos-Rev ; BD97 ; KM00 ; KM02 ; BK03 ; BD06 . Both monotonicity and
positivity can be viewed as types of persistence, and as is typically the
case, nontrivial persistence exponents characterize the time evolution dhp ;
dbg ; Bray13 .

For the averaging process, the opinion distribution becomes self-similar, and
in particular, the scaling form (6) is characterized by the second moment (5).
However, moments of the opinion distribution exhibit multiscaling and are not
characterized by the second moment. The moments decay exponentially with time
$\langle v^{n}\rangle\sim\exp(-\sigma_{n}t)$ with a nonlinear spectrum of
exponents, $\sigma_{n}/n\neq\sigma_{2}/2$ when $n>2$. We anticipate that
moments of the opinion distribution of monotonic agents also exhibit
multiscaling. Finding the corresponding spectrum of exponents is challenging
because the evolution equations that govern the moments of $M(v,t)$ are not
closed.

We also studied numerically the ultimate fraction $\rho$ of agents with an
opinion that obeys $|v(t)|\leq|v(0)|$ during the entire evolution history,
$0<t<\infty$. Interestingly, the fraction of such agents is finite, and for a
uniform distribution, the simulations yield $\rho=0.6488\pm 0.0001$. In
contrast with the universal exponents $\alpha$ and $\beta$, the fraction
$\rho$ does depend on the initial distribution.

A natural generalization of the averaging process is to multi-component
opinions. When the opinion of each agent is a vector, rather than a scalar,
one may study monotonicity properties of the magnitude of the opinion vector.
Additionally, one can investigate agents with one component of the opinion
vector being always larger than all other components.

Monotonicity can also be studied in systems that reach a steady state, and in
particular, averaging processes that are forced into a steady state bm .
Monotonicity can be probed in an even broader class of stochastic processes
since the trajectory of any fluctuating quantity may include segments where
all changes in the value of the fluctuating quantity occur in the same
direction.

###### Acknowledgements.

We dedicate this paper to Robert Ziff, whose singular style and influential
work continue to guide, inspire, and challenge an entire generation of
statistical physicists.

##  Appendix A Partial averaging

In the partial averaging process, agents make a partial compromise by moving
part-way toward each other. The post interaction opinions are linear
combinations of the pre-interaction opinions

$(v_{i},v_{j})\to\left(pv_{i}+qv_{j},qv_{i}+pv_{j}\right)\,,$ (25)

with $0\leq p\leq 1$ and $p+q=1$ so that the average opinion is conserved.
Each interaction reduces the opinion difference by factor $|p-q|$, as in an
inelastic collision BK00 .

Treatment of the partial averaging process is a straightforward generalization
of analysis above. For the random process (25), the rate equation governing
the density $F(v,t)$ of agents with opinion $v$ at time $t$ becomes

$\frac{\partial F(v,t)}{\partial
t}=\tfrac{1}{q}\int_{-\infty}^{\infty}\\!\\!\\!du\,F(u,t)F\left(\tfrac{v-pu}{q},t\right)-F(v,t)\,.$
(26)

We can verify that the distribution remains normalized, $\int dv\,F(v,t)=1$,
and that the average opinion is conserved, $\int dv\,v\,F(v,t)=0$. We restrict
our attention to symmetric distributions, and from (26), it follows that the
second moment decays exponentially with time, $\langle v^{2}(t)\rangle=\langle
v^{2}(0)\rangle\,e^{-2pqt}$. In the long-time limit, the opinion distribution
follows the scaling form

$F(v,t)=e^{pqt}\mathcal{F}(V)$ (27)

with the scaling variable $V=ve^{pqt}$. Independent of $p$, the scaling
function $\mathcal{F}(V)$ is given by (6).

The density of monotonic agents $M(v,t)$ satisfies

$\frac{\partial M(v,t)}{\partial
t}\\!=\\!\tfrac{1}{q}\int_{v}^{\infty}\\!\\!\\!du\,M(u,t)F\left(\tfrac{v-pu}{q},t\right)-M(v,t)\,.$
(28)

In the long-time limit, the overall density of monotonic agents decays as in
(2), and the density $M(v,t)$ of monotonic agents approaches the scaling form
$M(v,t)\simeq e^{-\alpha t}e^{pqt}\,\mathcal{M}(V)$ with the scaling variable
$V=ve^{pqt}$. The tail of the scaling function is algebraic,
$\mathcal{M}(V)\sim V^{-\nu}$ when $V\gg 1$, and equation (13) that relates
the exponents $\alpha$ and $\mu$ becomes

$\alpha=1+pq(1-\nu)-p^{\nu-1}\,.$ (29)

The extremum occurs at (figure 7)

$\alpha=1-pq\,\frac{1+\ln\left(\tfrac{1}{pq}\ln\tfrac{1}{p}\right)}{\ln\tfrac{1}{p}}\,.$
(30)

Our numerical simulations confirm that this extremum is indeed selected by the
dynamics. The exponent $0\leq\alpha\leq 1$ decreases monotonically as $p$
increases. The maximal value $\alpha=1$ is achieved when the interaction is
strongest ($p=0$), and the minimal value $\alpha=0$ is achieved when the
interaction is weakest ($p=1$).

Figure 7: The exponent $\alpha$, given by (30), versus $p$.

The exponent $\nu$ which characterizes the tail of the opinion distribution is
given by

$\nu=2+\frac{\ln\left(\tfrac{1}{q}\ln\tfrac{1}{p}\right)}{\ln\tfrac{1}{p}}\,.$
(31)

The exponent $\nu$ increases monotonically with $p$. Since $\nu<4$, the tail
of the scaling distribution $\mathcal{M}(V)$ remains less steep than the tail
of the scaling distribution $\mathcal{P}(V)$.

##  Appendix B Correction to the leading asymptotic behavior (2)

In problems admitting traveling wave solutions, the speed $w$ and the
wavenumber $k$ of the traveling wave are related via the dispersion relation
$w=\Phi(k)$, and the propagation velocity $w=w_{*}=\Phi(k_{*})$ is selected at
an extremum of the dispersion curve. This happens in a broad set of problems.
Moreover, the asymptotic approach to the speed $w_{*}$ is remarkably universal

$w(t)=w_{*}+\frac{3}{2k_{*}}\,t^{-1}+B_{2}t^{-3/2}+\cdots\,.$ (32)

The leading $t^{-1}$ correction was derived by Bramson Bramson  in the context
of the Fisher-Kolmogorov equation KPP ; Fisher , and then confirmed for many
other deterministic Saarloos ; Saarloos-Rev  and stochastic BD97 ; KM00 ; KM02
; BK03 ; BD06  systems. The second $t^{-3/2}$ correction was established in
Ref. Saarloos .

In our problem, the dispersion relation is given by Eq. (13). By assuming (32)
is valid, the correction to the leading asymptotic behavior (3) is given by

$\alpha(t)=\alpha_{*}+\frac{3}{2\nu_{*}}\,t^{-1}+B_{2}t^{-3/2}+\cdots\,.$ (33)

This form implies an algebraic correction to the leading asymptotic behavior
(2) as $M(t)\simeq A\,t^{-a}\exp(-\alpha t)$ with $a=3/(2\nu_{*})$. The inset
in Fig. 1 shows that $\alpha(t)\equiv-d\ln M/\ln t$ varies linearly with the
inverse time $1/t$. A two-parameter linear fit, accounting only for the
leading correction to the asymptotic behavior, yields the value
$\alpha=0.272\pm 0.003$. However, equation (33) which accounts for the two
leading corrections also involves only two parameters, $\alpha$ and $B_{2}$
because $\alpha$ and $\nu$ are related by (13). By using (33) and (13), we
obtain the improved estimate $\alpha=0.2715\pm 0.0005$ from the very same
simulation results. Such an approach is in the spirit of Bob Ziff’s work on
numerous problems including percolation Ziff-perc , planar dimer tilings Ziff-
dimers , and random sequential adsorption Ziff-RSA , where he perfected the
art of using finite-time and finite-size corrections for producing high-
precision measurements from Monte Carlo simulations.

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