Opinion and community formation in coevolving networks

In human societies, opinion formation is mediated by social interactions, consequently taking place on a network of relationships and at the same time influencing the structure of the network and its evolution. To investigate this coevolution of opinions and social interaction structure, we develop a dynamic agent-based network model by taking into account short range interactions like discussions between individuals, long range interactions like a sense for overall mood modulated by the attitudes of individuals, and external field corresponding to outside influence. Moreover, individual biases can be naturally taken into account. In addition, the model includes the opinion-dependent link-rewiring scheme to describe network topology coevolution with a slower time scale than that of the opinion formation. With this model, comprehensive numerical simulations and mean field calculations have been carried out and they show the importance of the separation between fast and slow time scales resulting in the network to organize as well-connected small communities of agents with the same opinion.

Figure 1

(Color online) Average degree $⟨k⟩$ (squares), average clustering coefficient $⟨C⟩$ (circles), average susceptibility $⟨s⟩$ (diamonds), average shortest path $⟨L⟩$ (up triangles), and average number of second neighbors $⟨n^{\left(2\right)}⟩$ (down triangles) for a network of size $N=200$ and averages taken over 100 runs. The arrows indicate the $g$ values where the susceptibility changes drastically. The inset shows such quantities as a function of system size $N$ for $g=1000$ and averaged over 1000 realizations. The graphs at the top are examples of final network configurations for $N=400$ and $g=5,{10}^3,{10}^5$, which show three different types of configurations emerging. White circles mean agents with $x=1$ and black those with $x=-1$.

Figure 2

(Color online) Communities found by the local fitness algorithm in the network configuration shown in the middle top of Fig. 1 for a system with $N=400$, $⟨k_0⟩=4$, and $g=1000$. The communities are distinguished by arbitrary coloring, regardless of the value of variable $x$ or opinion.

Figure 3

(Color online) The ratio $⟨k⟩/\left(N⟨C⟩\right)$ as a function of $g$, for various values of $N$. Each point corresponds to an average over 100 realizations. The inset shows the number of communities $X$ as a function of $N$ for $g=1000$ and averaged over 1000 realizations.

Figure 4

(Color online) Average fraction of undecided agents $⟨n_0⟩$ as a function of time, in units of $g$, for various networks of different size, where each point is an average over 1000 realizations. The mean field prediction fitted with Eq. (10) is shown as a black line.

Figure 5

The average final degree as a function of the average final clustering coefficient obtained from simulations in networks of size $N=400$ and $g=100$, and several initial degrees $⟨k_0⟩$ (open circles). The prediction by Eq. (11) for a random network is shown as a dashed line, and the mean field result of Eq. (20) is shown as a continuous line.