Flache & Macy (2011) Small worlds and cultural polarization

Flache A, Macy MW (2011) Small worlds and cultural polarization. Journal of Mathematical Sociology 35:146-176. 

This paper looks at homophily, influence, and xenophobia in clustered small-world-type networks, specifically looking at the influence of long-range ties in facilitating polarization between clusters.

The model

• A Hopfield-type model.
• Each agent (node) has one or more continuous states, each relating to the agent’s position on some cultural dimension. The state of agent i on issue k  is s_ik, which can vary in [–1, 1].
• Each agent is linked to other agents through endogenous weights, which reflects the similarity between agents. The weight between agents i and j at time t is w_ijt. Weights can be positive or negative. If only positive weights are allowed, they represent the strength of homophily, or the absence of a connection. When they can extend into negative, they represent the range between homophily and xenophobia, with neutral in the middle.
• All agent states are initialized as random draws from the uniform distribution [–1,+1].
• Weights are dynamic and reflect the degree of similarity between agents. Technically, weights reflect the total similarity between agents, and are equal to 1 if the agents are identical in all states, and –1 if the agents are diametrically opposite in all states. For the version with only positive weights, the calculation is the same, but the weights are smooshed so they vary in [0,1].
• States are updated by taking adding all of one’s neighbors stated, multiplied by their weights, to one’s own state, and normalized. The distances moved are then again normalized as a proportion of their distance from the extremes of ±1.

The simulation experiments are conducted on a disconnected “caveman” graph with extra nodes added, resulting in a “small-world” style network with high local clustering but relatively small average path length between nodes. Used 100 agents, organized in 20 clusters of 5 fully connected agents (Figure 1).

Updating is asynchronous. Each time step, one agent is selected randomly. Either weights or states are selected for updating with a coin flip, but not both at the same time. Simulations always run to 100 time steps.

Outcome measure for polarization

To compute the population-level measure of polarization (P), they first calculate for every pair of agents (ij) in the population the opinion distance d_ij (averaged across all K cultural dimensions). This distance ranges from zero to 2. The level of polarization of the population is then the variance of the distribution of d_ij. This is kind of arbitrary, but it allows the polarization to vary between zero (total agreement) and 1 (complete polarization into two same-size factions who are completely opposed on every cultural dimension. With initial values drawn from uniform random distribution, P = 0.214 with K = 1 and P = 0.11 for K = 2.

Experiment 1

Compares polarization in disconnected caveman graph and same with a few random ties connecting clusters, both the valence of interaction (i.e., the weights) cannot or can take on negative (in addition to positive) values.

Without negatively valence interactions, long-range ties simply allow the whole population to converge more closely, in line with previous results (Figures 2 and 3).

With negatively valence interactions, long-range ties significantly contributed to polarization. Without the ties, most caveman groups still converged, but there was polarization in many groups. However, polarization was much greater with long-range ties, because the individuals with more extreme views were more likely to be influential, pulling everyone along with them toward an extreme (Figures 4 and 5).

Robustness: These effects were robust, up to a point, to differences in “cave” size and differences in the number of cultural dimensions, K.  When cave size was very large (and negative valence permitted), only very large caves, size 50 (and number of clusters 2) did it not matter if there were long-range ties, since the cluster was large enough that extreme individuals could exert influence. As K increased, the effect sizes decreased, as did the overall levels of polarization. This is largely because, with many aspects to potentially agree on, the distance between individuals is likely to be low, and so weights are less likely to be negative (see Figs 6 and 7 in the paper).

Experiment 2

Is the effect really because random long-range ties create connections between previously disconnected caves, or simply because random ties increase network density? Here, they repeat the experiment with a connected caveman graph, such that each cluster has one connection with an adjacent cluster, in a ring. They then add either random short-range ties (which linked a random node in each cave to a random node on the right side of it in a circular arrangement), or long-range ties (same number of ties added, but between random pairs of nodes from the entire graph). In both cases, density is the same, but long-range ties significantly decrease average path length.

For runs without negatively valenced weights, polarization decreased, and long- vs. short-range ties had no effect. In contrast, random ties significantly increased polarization relative to short-range ties (Figures 9 and 10).

 These effects were robust to increases in cave size and K, as in experiment 1.

From the Discussion:

• An important scope condition for our argument is that actors who are connected with a long-range tie are more likely to differ sharply from each other than are actors connected with a short-range tie. Under that condition, interactions in long-range ties are more likely to be driven by the negative mechanisms of social interaction, xenophobia and differentiation, than are interactions within short-range ties. In our model, this condition arises because from a random start, there is on average more assimilation between agents than there is differentiation. This implies that agents within close proximity in the network become more similar in their opinions than agents within large distance from each other.
• In the online appendix of this article, we have explored alternative assumptions about the initial distribution of weights that render positive ties initially equally likely than negative ties. The results show that then additional long-range ties in the network no longer have a stronger effect on polarization than additional short-range ties.