Polarization in Dynamic Networks: A Hopfield Model ...

1 Polarization in Dynamic Networks: A Hopfield Model of Emergent Structure Michael W. Macy, Cornell University; James A. Kitts, University of Washington; and Andreas Flache, University of Groningen 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 model is an extension of Hopfield’s attractor network. Agents are attracted to others with similar states (the principle of homophily) 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 selforganize into two antagonistic factions, without the knowledge or intent of the agents. We model this tendency as a function 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 global uniformity or pluralistic alignments may also be equilibria. From a random start, agents can self-organize into pluralistic arrangements if the population size is large relative to the size of the state space.

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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. Others counter that homophily is the spurious consequence of social influence, as argued 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

3 influence from 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 from 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 findings of homophily in relations may have spuriously represented this effect of repulsion from 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 maintain “balance” in their relations, such that two actors who are positively tied to one another will feel 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*B*X is 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 parallel process operates if the tie between A and B is negative. 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 model simply extends to triadic agreement among agents A, B, and C, where balance obtains when the sign product of the triad A-B-C is positive. This formalizes the adage that a friend of a friend or an enemy of an enemy is a friend, while an enemy of a friend or a friend of an enemy is an enemy. Extended to a larger network, it is well known that the elemental triadic case suggests a perfect separation of two mutually antagonistic subgroups. Our model integrates the attraction-influence feedback loop from formal models of social convergence with the bivalent relations in Balance Theory. Following Nowak and Vallacher (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-reinforcing dynamics of attraction and influence as well as repulsion and differentiation. 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 learn 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 wij is a function 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.

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 similarity/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 (wij) of the dyadic tie between i and j (-1.0 < wij < 1.0): N

Pis =

∑w s j =1

ij j

N −1

, j≠i

(1)

Thus, social pressure (-1< Pis 0.5±χε, where χ is a random number (0