Phase transitions in social networks inspired by the Schelling model

We propose two models of social segregation inspired by the Schelling model. Agents in our models are nodes of evolving social networks. The total number of social connections of each node remains constant in time, though may vary from one node to the other. The first model describes a “polychromatic” society, in which colors designate different social categories of agents. The parameter $\mu$ favors/disfavors connected “monochromatic triads,” i.e., connected groups of three individuals within the same social category, while the parameter $\nu$ controls the preference of interactions between two individuals from different social categories. The polychromatic model has several distinct regimes in $(\mu ,\nu )$-parameter space. In $\nu$-dominated region, the phase diagram is characterized by the plateau in the number of the intercolor connections, where the network is bipartite, while in $\mu$-dominated region, the network looks as two weakly connected unicolor clusters. At $\mu >{\mu }_{\text{crit}}$ and $\nu >{\nu }_{\text{crit}}$ two phases are separated by a critical line, while at small values of $\mu$ and $\nu$, a gradual crossover between the two phases occurs. The second “colorless” model describes a society in which the advantage or disadvantage of forming small fully connected communities (short cycles or cliques in a graph) is controlled by a parameter $\gamma$. We analyze the topological structure of a social network in this model and demonstrate that above a critical threshold, ${\gamma }^{+}>0$, the entire network splits into a set of weakly connected clusters, while below another threshold, ${\gamma }^{-}<0$, the network acquires a bipartite graph structure. Our results propose mechanisms of formation of self-organized communities in international communication between countries, as well as in crime clans and prehistoric societies.

Figure 1

(a) Switching of links preserving degrees of all vertices; (b) example of a local network updating process which increases the number of red and green triads by one. Thick lines (right) highlight edges connecting new unicolor (red and green) triads that were absent before the switch (left). To distinguish nodes in a color-less mode, we painted the core of green vertices in black and the core of red vertices in white.

Figure 2

Phase portrait of the network in the parameter $(\mu ,\nu )$–plane: (a) Density plot of the fraction of cross-color links, ${\rho }_{GR}(\mu ,\nu )$; (b) density plot of the third moment of the spectral density, which measures bipartitness of the network. In the simulations there are 128 nodes of each color, $p=0.15$, averaging is performed over 500 realizations.

Figure 3

(a) Fraction of cross-color bonds, ${\rho }_{RG}$, as the function of the chemical potential of unicolor triads, $\mu$ at fixed chemical potential of cross-color bonds, $\nu =0.1$ – upper panel, and the behavior of the second eigenvalue of Laplacian matrix, ${\lambda }_2\left(\mu \right)$ – lower panel; (b) The behavior ${\rho }_{RG}\left(\nu \right)$ at fixed $\mu =0.1$ – upper panel, and the behavior ${\lambda }_2\left(\nu \right)$ – lower panel.

Figure 4

Typical sample of $M=4$-color network in the “intra-community” phase near the transition boundary. The “ambassadors” provide communication gates between clusters. Each cluster has 64 nodes.

Figure 5

The phase diagrams (density plots of ${\rho }_{RG}$) in the $(\mu ,\nu )$-plane for three different values of $N$: (a) $N=128$, (b) $N=256$, (c) $N=512$.

Figure 6

The density plots of phase diagram for the three-color network having 85 nodes of each color (compare to the two-color network shown in Fig. 2): (a) the cross-color density of links; (b) the averaged third moment of spectral density.

Figure 7

(a) Switching of network links preserving degrees of all vertices; (b) example of a local network updating move which increases the number of triangles.

Figure 8

Few typical samples of intermediate stages of network evolution: (a) Networks with strictly conserved vertex degree (our model); (b) networks with nonconserved vertex degree (“Strauss model”) (recalculated on the basis of Ref. [28]).

Figure 9

Critical behavior at negative $\gamma$.