Different topologies for a herding model of opinion

Understanding how opinions spread through a community or how consensus emerges in noisy environments can have a significant impact on our comprehension of social relations among individuals. In this work a model for the dynamics of opinion formation is introduced. The model is based on a nonlinear interaction between opinion vectors of agents plus a stochastic variable to account for the effect of noise in the way the agents communicate. The dynamics presented is able to generate rich dynamical patterns of interacting groups or clusters of agents with the same opinion without a leader or centralized control. Our results show that by increasing the intensity of noise, the system goes from consensus to a disordered state. Depending on the number of competing opinions and the details of the network of interactions, the system displays a first- or a second-order transition. We compare the behavior of different topologies of interactions: one-dimensional chains, and annealed and complex networks.

Figure 1

Finite-size scaling of the transition in a system with a two-dimensional opinion vector with random interactions. The figure compares the frequency of agents with the same dominant opinion versus the noise $\eta$ for different population sizes $N$. (a) Original data. (b) Finite-size scaling. Each point corresponds to an average over 10–20 runs with different random seeds.

Figure 2

The transition of the system for different numbers of opinions, $O$, and different initial conditions. The population size is 500. (a) The outcome of a system of two opinions (circles) is independent of its initialization. (b) In the case of three opinions (dashed line), the curves present a hysteresis and the results are different if the field is initialized randomly (squares) or with the system being already in the consensus state (stars). (c) A system with an opinion vector containing ten opinions also exhibits hysteresis (dash-dotted line). (d) Average value of the dominant opinion $a_{o_{\mathit{max}}}$ vs $\eta$ for the same simulation as in (c).

Figure 3

Ten opinions on a one-dimensional chain. These results illustrate simulations without noise $(\eta =0)$ and a system of 1000 agents. (a) The figure zooms in on the first 200 agents of the population, where each line corresponds to a different element of the opinion vectors. Only the first five opinions are displayed. The agents form local clusters of different dominant opinions. (b) The cluster sizes distribute following an exponential decay.

Figure 4

The size of the largest cluster increases in time until it reaches the population size $N=1000$. Here, we see the results of different random initializations of the system on a chain. The number of opinions is 10, and the noise $\eta =0.2$.

Figure 5

(Color online) Graphical illustration of the temporal behavior of the system on a chain. The color (gray tone) corresponds to the value of the first of the two opinions of the system. The simulations run over 100 000 time steps, drawing each 1000 iterations a new point on the vertical axis beginning at the bottom. The horizontal axis depicts the location of each agent on the chain, altogether consisting of 1000 agents. The noise $\eta$ is 0.05 in (a), 0.2 in (b), 0.35 in (c), and 0.45 in (d).

Figure 6

Histogram of the time steps needed to reach the consensus state. Each curve corresponds to simulations with the same parameters: 100 agents, 2 opinions, random initialization. For each value of the noise parameter we carried out 200 000 runs with different random seeds.

Figure 7

(a) The fraction of agents with the same dominant opinion versus $\eta$ are compared for different population sizes $N$ in a system of two opinions, where only the nearest neighbors on a chain interact. (b) Finite-size scaling of the transition.

Figure 8

(Color online) (a) Influence of the topology of networks on the transition to consensus $(D=1)$ as a function of noise $\left(\eta \right)$. The transition on two different scale-free networks, the Apollonian (solid line) and the Barabási-Albert network (BA, triangles), is similar to the one observed for annealed interactions (AI, plus signs), and differs from the transition on a regular lattice (circles). In the three insets we plot the value of the dominant opinion, $a_{o_{\max }}$ vs time. (b) Comparison of the behavior of $a_{o_{\max }}\left(t\right)$ on the four networks: Apollonian (solid line), Barabási-Albert (BA, dotted line), annealed (AI, dashed line) and regular (long dashed line) for a fixed noise $(\eta =0.2)$. One observes that for this noise, which is below the critical noise, in the regular network the emergence of consensus takes longer than in scale-free and annealed interactions, which have similar behavior (three upper curves). (c) Near below the transition, for $\eta =0.4$, we compare the response of the regular and the Apollonian network. It is observed that for the former there is an intermittency among consensus $D=1$ and $a_{o_{\max }}=0.79$ and not consensus $D=0.5$ and $a_{o_{\max }}=0.5$. This behavior is not observed in the complex networks. (d) Above the transition $(\eta =0.6)$ the consensus is broken and the dynamics of the opinion $a_{o_{\max }}$ vs time behaves similarly in regular and complex networks. All simulation runs are with systems of 124 agents.