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162
Polarization in Dynamic Networks: A Hopfield Mode! of Emergent Structure
Michael W. Macy, Comell University; James A. Kitts, University of Washington; Andreas
Flache, University of Gron~ngen; Steve Benard, Cornell University
ABSTRACT
Why do populations often self-organize into antagonistic groups even In the absence of
competition over scarce resources? Is there a tendency to demarcate groups of"us" and '`them"
that is inscribed In our cognitive architecture? We look for answers by exploring the dynamics of
influence and attraction between computational agents. Our mode} is an extension of Hopfield's
attractor network. Agents are attracted to others with similar states (the principle of homophiTy)
and are also influenced by others, as conditioned by the strength and valence of the social tie.
Negative valence implies xenophobia (instead of homophily) and differentiation (instead of
imitation). Consistent with earlier work on structural balance, we find that networks can self-
organ~e into two antagonistic factions, without the knowledge or intent ofthe agents. We mode!
this tendency as a Unction of network size, the number of potentially contentious issues, and
agents' openness and flexibility toward alternative positions. Although we find that polarization
into two antagonistic groups is a unique global attractor, we investigate the conditions under
which uniform and pluralistic alignments may also be equilibria. From a random start, agents can
self-organ~ze into pluralistic arrangements if the population size is large relative to the size of the
state space.
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2
INTRODUCTION: GROUP POLARIZATION
Why do populations often self-organize into antagonistic groups even in the absence of
competition over scarce resources? Is there a tendency to demarcate groups of"us" and '`them"
that is inscribed in our cognitive architecture? We look for answers by exploring the dynamics of
influence and attraction between computational agents.
Our model builds on recent work using agent-based models of cultural convergence and
differentiation (1,2,3). Scholars contend that the abundance of social groups containing highly
similar actors is due to dual processes of attraction and social influence. Formal models generally
assume that each agent chooses interaction partners that are similar to itself, while social
interaction also leads agents to adopt each other's traits and thus grow more similar. In this
positive feedback loop, a minimal initial similarity increases the probability of interaction which
then increases similarity.
This process of consolidation (4) will presumably continue until all agents engaged in
mutual interaction have converged to unanimity. While this self-reinforcing dynamic seems to
imply a "melting pot," or inexorable march toward homogeneity, scholars have shown that
global diversity can survive through impermeable barriers to interaction between distinct
subcultures. If bridge ties between groups are entirely absent, then this local convergence can
indeed lead to global differentiation, in spite of cultural conformity among the individual agents.
Stable minority subcultures persist because of the protection of structural holes created by
cultural differences that preclude interaction. Given such barriers, homogenizing tendencies
actually reinforce rather than eliminate diversity.
The models thus predict social stability only where a population is entirely uniform or
where agents cluster into mutually exclusive uniform subpopulations that are oblivious to one
another. This global pluriformity depends on the assumption that interaction across these
boundaries is impossible. Any allowance for interaction between dissimilar agents, no matter
how rare, leads diversity to collapse into global homogeneity. This also reflects earlier analytical
results obtained with models of opinion dynamics in influence networks (21,22,23). Abelson
proved that a very weak condition is sufficient to guarantee convergence of opinions to global
uniformity: The network needs to be "compact," such that there are no subgroups that are
entirely cut off from outside influences. This apparently ineluctable homogeneity in fully
connected networks led Abelson (21, p. 153) to wonder "... what on earth one must assume in
order to generate the bimodal outcome of community cleavage studies?"
We will pursue this interest in cleavages, or structural bifurcations, as an alternative
explanation for both the disproportionate homogeneity in social groups and the persistence of
diversity across groups. First let us revisit the generative processes, the psychological and
behavioral foundations of attraction and social influence.
These models of convergence implement one of the starkest regularities in the social
world: "homophily," or the tendency for each person to interact with similar others (5, 6, 7).
Several explanations for homophily have been proposed. Social psychologists posit a "Law of
Attraction" based on an affective bias toward similar others (8; see also 9,10,11,12). Even
disregarding such an emotional bias, structural sociologists (3,2,14) point to the greater
reliability and facility of communication between individuals who share vocabulary and syntax
as an inducement for homophilous relations.
Others counter that homophily is the spurious consequence of social influence, as in
classic research on 'pressures to uniformity" (16,17,18) and recent work in social networks
(19,20). Even if relations are held constant in an exogenously clustered social network, social
DYNAMIC SOCIAL NETWORK MODES AND ISIS
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3
In flu ence Tom network neighbors will lead to local homogeneity, without the need to assume
homophilous interaction.
Scholars have thus overwhelmingly attributed homophily In observed-social networks to
some combination of differential attraction and social influence, where agents choose similar
partners and partners grow more similar over time. Much less attention has been directed to an
alternative explanation - that homophily is largely a byproduct of its antipole. It is not so much
attraction to those who are similar that produces group homogeneity but repulsion Tom those
who are different. Xenophobia leads to the same emergent outcome as attraction between similar
actors: disproportionate homogeneity in relations. In fact, Rosenbaum (15) argues that many
experimental flndiIlgs of homophily in relations may have spuriously represented this effect of
repulsion Dom those who are different.
The most sustained treatment of positive and negative ties has appeared in the literature
on Balance Theory. Following Heider (24), scholars have assumed that actors are motivated to
mamta~n "balance" in their relations, such that two actors who are positively tied to one another
will fee! tension when they disagree about some third cognitive object. Formally, we can think of
this as a valued graph of agents A and B along with object X, where the graph will be balanced
only when the sign product A *Boxes positive. For example, if A and B are positively tied but A
positively values X and B negatively values X, then the graph is imbalanced. In order for this
dissonance to resolve and result In a stable alignment, there will either be a falling out between A
and B or one (but not both) of these actors will switch evaluations of X A n~rnile1 Hen
operates if the tie between A and B is negative.
- r~~~ r^~-_~V
A prolific line of research- Structural Balance Theory (32) - has examined a special case
of Heider's model, where the object X is actually a third actor, C. The mode! simply extends to
tnadic agreement among agents A, B. and C, where balance obtains when the sign product of the
tnad A-B-C is positive. This formalizes the adage that a Fiend of a friend or an enemy of an
enemy is a friend, while an enemy of a Fiend or a Fiend of an enemy is an enemy. Extended to a
larger network, it is well known that the elemental triadic case suggests a perfect senar~tion of
two mutually antagonistic subgroups.
Our mode} integrates the attraction-~nfluence feedback loop from formal models of social
convergence with the bivalent relations In Balance Theory. Following Nowak and ValIacher
(29), the model is an application of Hopfield's attractor network (25, 26) to social networks. Like
Heider's Balance Theory, an important property of attractor networks is that individual nodes
seek to minimize "energy,' (or dissonance) across all relations with other nodes. As we will see,
this suggests self-re~nforc~ng dynamics of attraction and influence as well as repulsion and
differentiation.
-look ~ rid row
More precisely, this class of models generally uses complete networks, with each node
characterized by one or more binary or continuous states and linked to other nodes through
endogenous weights. Like other neural networks, attractor networks leant stable configurations
by iteratively adjusting the weights between individual nodes, without any global coordination.
In this case, the weights change over time through a Hebbian learning rule (27~: the weight ~v'; is
a Unction of the correspondence of states for nodes i and j over time. To the extent that i and j
tend to occupy the same states at the same time, the tie between them will be increasingly
positive. To the extent that i and j occupy discrepant states, the tie will become increasingly
negative.
164
DYNAMIC SOCIAL N~TWORKMODELI?JG AND ANALYSIS

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4
Extending recent work (30), we apply the Hopfield model of dynamic attraction to the
study of polarization in social networks. In this application, observed sunilarity/difference
between states determines the strength and valence of the tie to a given referent.
MODEL DESIGN
In our application of the Hopfield model, each node has N-1 undirected ties to other
nodes. These ties include weights, which determine the strength and valence of influence
between agents. Formally, social pressure on agent i to adopt a binary state s (where s= +1) is
the sum of the states of all other agents j, conditioned by the weight (wit) of the dyadic tie
between i andj (-~.0 < wit ~ I.0):
N
~ WijSj
Pw=i=' , Jo (1)
Thus, social pressure (-0.5+%s, where % is a random number (-0.5<%<0.5) and ~ is an
exogenous error parameter (0~<~. At one extreme, - 0 produces deterministic behavior, such
that any social pressure above the trigger value always leads to s=l and pressure even slightly
below the trigger value always leads to sew. FoHow~ng Harsanyi (31), s>0 allows for a
"smoothed best reply" ~ which pressure levels near the trigger point leave the agent relatively
indifferent and thus likely to explore behaviors on either side ofthe threshold.
In the Hopfield model, the path weight wit changes as a function of similarity In the states
of node i and j. Weights begin with uniformly distributed random values, subject to the
constraints that weights are symmetric (wifwji). Across a vector of K distinct states so, (or the
position of agent i on issue k), agent i compares its OWD states to the observed states of another
agent j and adjusts the weight upward or downward corresponding to their aggregated level of
agreement or disagreement. Based on the correspondence of states for agents i end j, their weight
will change at each discrete hme point t ~ proportion to a parameter i, which defines the rate of
structural learning (0<~<~:
~ K
Wij,t+1 Wijt (1 2) +—~Sj~SiEt' j ~ i (3)
K k=1
DYN~MICSOCIALN~ TWORKMODE~G~D~^YSIS
165

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As correspondence of states can be positive (agreement) or negative (disagreement), ties
can grow positive or negative over time, with weights between any two agents always
symmetric. Note that both processes - adjustment of social ties and adjustment of behavior -
seek to maximize balance ~ relations between any two agents across their vectors of states.
Given an initially random configuration of states and weights, these agents will search for a
profile that ~ninim~zes dissonance across their relations.
METHODS
We tested the dynamics of network polanzation by manipulating three agent-level
behavioral rules:flexibili~y, broaci-mindedness, and open-mindedness:
I. Rigid vs. flexible (a.k.a. "wafflmg"~: the extent to which the state adoption decision is
stochastic near indifference, ranging Dom s=0 (agent i always picks the strictly preferred
position even when almost indifferent) to ~l (i adopts state s with propensity ~..s).
2. Narrow- vs. broad-minded: the multiplexity of the state space, that is, the number of
salient cross-cutt~ng dimensions that agents consider in evaluating each other.
3. Closed- vs. open-minded: the proportion ofthese salient dunensions that can be affected
by social pressure, with a minimum of at least one free state. A fixed state can be a
position on an issue about which an agent is strictly uncompromising. It can also be an
attribute that Is difficult to change, such as ethnicity or gender.
Thus, we consider a range of stylized character profiles for agents, ranging Dom
"extremists" who focus on ascriptive differences and are rigidly narrow- and closed-minded, to
more broad-minded "moderates" who tend to waffle on the issues, and are open to influence
(and/or focused on behavior rather than ascriptive traits).
We measure network polarization as the degree of segregation among mutually exclusive
cohesive subgroups of agents, measured after all weights and states have converged to
equilibrium. This measure is based on the graph theoretical ES-set (321. An LS-set is a subset of
agents who have stronger ties to members within the subset than they have to members outside
the subset. For the cohesive subgroups thus identified, we calculated their segregation in terms
of the normalized difference between the average internal tie strength and the average tie
strength between the subgroup and its complement. To obtain an overall polarization score, we
generated all partitionings of the graph into mutually exclusive cohesive subgroups and took
Tom these the average segregation of all subgroups In the most segregated partitioning.
We allowed each experimental condition to repeat for 1000 iterations, which was
sufficient for almost all conditions to arrive at an equilibrium Only equilibrium outcomes
(converged solutions) were analyzed. For all the experiments, we set the structural learning rate
(~) at 0.5, which means that agents consider current states equally with previous impressions in
determ~n~g the tie weight for the next Arbitrary time ~nterval.t Each parameter combination
was repeated 20 times, y~eld~g a total of 44,000 observations.
More technically, for each of the subsets of an LS-set, average tie strength between members of We subset and
its complement within the LS set is higher than the average tie strength to members outside We LS set. We first
identified subgroups of agents that agree on those states that they are Bee to change. We then searched through
coalitions of these subgroups to find maximal LS-sets in the graph defined by agreement on Dee states. Finally,
we used these LS-sets to identify the maximally segregated partitioning of the network, based on the relational
weights (which reflect agreement on all states ~ fixed and Eee).
Qualitative patterns shown appear to be robust to a very broad range of assumptions about the learning rate,
though extremely low learning rates make waiting for convergence impractical.
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DYNAMIC SOCIAL NETWORK MODELING ED ISIS

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6
RESULTS
Experiment ~ tests the effects on network polarization of agent broad-mindedness, while
holding flexibility and openness constant (~0 and v=~. With zero flexibility and no fixed states,
network dynamics converged to equilibrium In every case. Network polarization is measured as
the degree of segregation In the most segregated equilibrium partitioning of the weight graph into
mutually exclusive cohesive subgroups. Maximum segregation (~.0) occurs in a perfectly
bifurcated network, composed of exactly two internally cohesive and mutually antagonistic
groups. Broad-mindedness is operational~zed as multiplexity, or the number of dimensions In the
state space. Intuitively, we would expect polarization to be most likely in a world where agents
focus single-mindedly on one highly salient issue, and indeed, Figure ~ confirms the intuition.
What is surprising, however, is that network polarization can also be caused by too many salient
issues. With 100 agents scattered over a state space larger than 2s, the dynamics quickly
approach the bifurcation we would expect in a single-issue population. As N declines, the non-
monotonic effect becomes less pronounced. For N>100, the U-shape function is little changed
but the critical value of K increases, Tom 5 dimensions with N<100 to I] dimensions with
N=300.
Polarization
N '~ °1 2
Multple~aty
0.8
0.4
-0.2
Figure I. Effect of network size (IO~N

7
expect less network polarization in a population that is less rigid and more polarization ~ a
population that is less open-minded. The surprising result is that the elect is quite the opposite.
Figure 2 displays the polarizing effects of flexibility and open mindedness in greater
detail. As in Experiment I, polarization is based on the relational partition that maximizes
segregation ofthe weight graph among coalitions of distinct subgroups in the matrix of Bee
states. Flexibility is simply the error level, ranging from s=0 (always pick the strictly preferred
position) to £=i (adopt the position with propensity At. Openness is the proportion of K states
that are affected by social pressure (or Ivs/K), where there must always be at least one bee state.
Results were averaged over 20 replications of each treatment condition, in which N ranged Mom
O to 100 and K ranged Tom 5 to ~ O (hence openness ranged Tom 0.2 to ~ .0~.
~ Polanzation
l
Flexibility ~.2
0.4
0.6
,S>~ ~ ~ I.
< ~~ ~ Ives l +~ s ~ ?.~
its a; ~~-~-~<,~t my' As- ·
~ ~~ 7 ';. i "I'm ~ ~ ~ Y it. ~—
a''.',',,,, \ ~ ~ ~ ~ ~ \ we;
,,~,~,i~,,,s~,,.,,.~~ ~~,&~ ~ y ~~ ~ A, ~^ 1
>~! ~> I,; ~ ~~ ~ ~t \ W~ 1
~ ~ ~ ~ S *
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8
differs from each oftwo opposing positions. As the number of dimensions increases, it is
possible for many more than two Unique combinations to persist at equilibrium, but for each,
there must always be some sunilarity with all other combinations except one. Surprisingly, as
these combinations explode, they lead not to pluralism but to polarization. The mechanism is
similar to that identified by Axeirod. The more opportunities for neighborhoods to be liniced via
overlapping positions, the higher the probability they will find a way to coalesce.
The key variable here is the density ofthe population distribution across the state space.
When there are many agents but few dimensions, then, from a random start, every possible
combination of states (corresponding to "ideologies" if the states are variable and to ~`identities''
if the states are fixed) will find multiple incumbents. The presence of identical neighbors (as well
as dissimilar antipodes who serve as negative referents) creates strong pressures to remain loyal
to a shared set of states. This resistance to change can then support a pluriform equilibrium.
Increasing the number of dimensions expands the set of possible combinations, allowing groups
to mobilize around a variety of configurations of states. Accordingly, polarization initially
decImes with increasing multiplexity. However, if population density In the state space fats
below a critical level (either due to low population or high multiplexity), it becomes impossible
for every distinct vector of states to find an incumbent. The sparseness of the population relative
to the number of unique positions ~ state space requires that some agents begin with neither
allies (who overlap perfectly3 nor enemies (who overlap not at all). These agents lack sufficient
pressure to stick to their guns and will thus be pulled into coalitions with agents who overlap
partially with their positions. As the multiplexity ofthe state space increases, equilibria
ncreas~gly depend on a delicate balance, where a distnbution of agents on ideologies and
identities needs to be found that corresponds to the distances between positions, such that they
are not so close to each other that they are pulled together and not so far apart that they are
pushed toward the opposite pole. In this higher range, increasing multiplexity makes it more
difficult for pluriform configurations to persist, such that collapse into a simple bifurcation
becomes the most likely alternative.
Flexibility promotes polarization by the same process. Network self-organization can
become trapped in high-energy pluriform equilibria that are nevertheless a local minimum ofthe
energy landscape, as depicted in Figure 3.
issues can disturb these local solutions, allowing the system to continue searching for the global
attractors. Thus, increased flexibility washes out the non-monotonic effect of multiplexity
evident ~ Figure I. Flexibility leads to polarization even when agents attend to multiple
dimensions of differentiation that would otherwise produce the cross-cutt~ng cleavages
characteristic of a pluriform society.
Agents who express ambivalence by "waffling" on the
I'
\ ~
DYNAMIC SOCIAL NETWORK MODELING AND ANALYSIS
169
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Figure 3. How agent volatility allows the network to find a lower energy minimum- hence
greater polarization.
Homogeneity is also a stable equilibrium of the model, but this equilibrium cannot obtain
sinless we preclude negative weights and negative influence (as In the earlier models of cultural
convergence and differentiation discussed above). If we initialize the mode} at homogeneity, it
will remain there indefinitely. However, from a random start, the system cannot find the s~ngle-
group solution without falling into an inescapable trap of is-group/out-group hostility. Further, a
sufficient exogenous perturbation will disturb the unitary equilibrium and yield a two-group
solution, but the reverse shin is much more difficult to obtain and is extremely unlikely.
To explore this interpretation further, we measured the energy level of equilibria by
summing over the product of agreement and tie strength across all dyads in the weight graph. As
expected, we found that networks with perfect polarization in two opposing camps had an energy
level of zero, while energy levels increased with the number of different positions in the state
space that co-existed at equilibrium. That analysis examined tension in dyads for the variety of
converged configurations. A continuous index of graph balance (35), assessing balance in triads
and cycles of any higher length, yielded the same conclusion. Equilibria with either one or two
internally cohesive groups represent maximum balance, while convergence at any higher number
of groups implies imbalances relations in the social network.
CONCLUSIONS AND IMPLICATIONS
This study has explored polarizing tendencies of a self-organizing network, using a
Hopfield mode} of dynamic attraction. The project integrates Axeirod's positive feedback mode}
of ~nfluence-~nteraction with a bivalent mode! of social relations and cognitive balance. The
Hopfield mode! emulates well-known processes of homophily, xenophobia, and influence Tom
positive and negative referents. Thus, we may account for emergent social phenomena with a
rigorous and parsimonious model, mcludina onlY basic behavioral orincinles that have horn
broadly supported in experimental work.
_^ I,^ .,._,^~.,_ - i,.. ~~ ~ _ TV
The results have interesting implications for the notion of"structural balance" In social
~ .. . . ~ . .
relations. Although the agents in this model are clearly designed to maintain balance in their
behaviors with both positive and negative referents, this assumption is not '`~red m" to the
relations themselves, as it is In Structural Balance Theory. That is, two agents A and B fee] no
direct need for consistency ~ their relations with a third agent C. ~deed, A has no knowledge of
the B-C relationship and thus no ability to adjust the ~4-B or A-C relations so as to balance the
triad. NotablY. the results show that triads for cycle of TV lengthy Hm tend tn horning hulas
~ V A ~ ~ ~ _~) ~~ —_~ ~~ V_~v~~—~ v
. . ~ . . . . .
over time as agents seek balance in Shelf dvadic relatinn~ However there in no relaxant-- in this
~ 3
model that they will achieve a globally optimal state In structural balance' and we also observe
equilibrium outcomes where more than two subgroups persist indefinitely. This outcome cannot
be reconciled with Structural Balance Theory, which predicts that system-level stability can only
occur when the group either has become uniform or has nc~larizer1 into two intern~liv unhip
and mutually antipathetic cliques.
--~ r ~ _~,,,_-,, _
The model also has interesting implications for Social Identity Theory (36), which posits
an ~n-group bias toward those who share a salient trait, prejudice against the out-group, and a
tendency to ignore or change discrepant traits. The Hopfield model produces dynamic networks
that self-organize into a similar pattern, but without a higher-order cognitive Damework of social
categories. In fact, these agents are not even aware that they belong to "groups" at all. This
170
9
DYNAMIC SOCIAL NETWO=MODEL~G^D ISIS
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10
demonstrates the possibility that ~n-group/out-group differentiation and antagonism are emergent
properties of network self-organ~zation and are not inscribed In agents' cognitive architectures as
assumed by identity theonsts. In effect, the model demonstrates a distributed representation of
the formation of social identities, which are not reducible to local representations In the minds of
m~ividual agents.
Fmally, the mode! offers an important methodological demonstration. The results show
how analytical methods focusing on equilibria may be misleading for prediction of opinion
distributions that are likely to obtain. An analysis of possible equilibria clearly shows that more
multiplexity and larger population size increase the number of possible pluriform equilibria, i.e.
Opinion configurations in which for all agents and all issues the aggregated pressures to stick to
one's opinion exceed the aggregated pressures to change. However, our computational model
showed that despite a larger number of theoretically possible equilibria, these conditions
decreased the number of equilibria that were actually reached by the opinion dynamics. The
reason is that equilibrium analysis fails to talce into account the ~tnlc.tilr~l 1erlrnino Her
~ ~ ~ I__ ~~= r~ IF _ —~7
~ ~ ~ t ~
Inrougn wmcn outcomes are selected. 1I1 structural reaming, equilibria become increasingly
unlikely as the complexity of the coordination process increases. As a consequence, the
computational model identifies polarization as the global attractor in a multiplex opinion space,
an important substantive result that is overlooked by static equilibrium analysis.
Previous theoretical work has emphasized the global stability of social homogeneity,
where convergence to unanimity is an almost irresistible force In closely interacting populations.
If it is possible for some ties to be negative, our model suggests instead that a social structure is
most stable when the network self-organ~zes along a single dimension of differentiation which
thus determines a clear "right" and '~wrong" choice on all behavioral dimensions. In comparison,
social homogeneity is highly brittle.
ACKNOWLEDGEMENT
The research was supported by a grant to the first author Dom the U.S. National Science
Foundation (SES-0079381~. The research ofthe third author was made possible by a fellowship
of the Royal Netherlands Academy of Arts and Sciences.
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