Opinion dynamics on dense dynamic random graphs

Abstract

We consider two-opinion voter models on dense dynamic random graphs. Our goal is to understand and describe the occurrence of consensus versus polarisation over long periods of time. The former means that all vertices have the same opinion, the latter means that the vertices split into two communities with different opinions and few disagreeing edges. We consider three models for the joint dynamics of opinions and graphs: one with a one-way feedback and two which are co-evolutionary, i.e., with a two-way feedback. In the first model only coexistence is attainable, meaning that both opinions survive, but with the presence of many disagreeing edges. In the second model only consensus prevails, while in the third model polarisation is possible. Our main results are functional laws of large numbers for the densities of the two opinions, functional laws of large numbers for the dynamic random graphs in the space of graphons, and a characterisation of the limiting densities in terms of Beta-distributions. Our results are supported by simulations. To prove our results we develop a novel method that involves coupling the co-evolutionary process to a mimicking process with one-way feedback. We expect that this method can be extended to other dense co-evolutionary models.

Figures (10)

• Figure 2.1: The top row displays the empirical graphon when $n=100$ and $T=0.5,1,1.5$, the bottom row displays the corresponding functional law of large numbers. Simulations are based on a single run. A dot represents an edge. The labels of the vertices are updated dynamically so that they are ordered lexicographically, i.e., the vertices with opinion $+$ have lower labels than the vertices with opinion $-$, and then by increasing type.
• Figure 2.2: Empirical distribution of $(y_i(T))_{i\in [n]}$ when $n=6000$ and $T=700$ for varying $\gamma_{+-}$,$\gamma_{-+}$. For any $i\in[n]$, $y_i(0)$ is chosen uniformly at random in $[0,1]$, independently of the other vertices. The red line indicates the corresponding $\hbox{Beta}(\gamma_{-+},\gamma_{+-})$ density function. Simulations are based on a single run.
• Figure 3.1: The top row displays the empirical graphon when $n=100$ and $T=1,2,3$, the bottom row displays the corresponding functional law of large numbers. Simulations are based on a single run. A dot represents an edge. The labels of the vertices are updated dynamically so that they are ordered lexicographically, i.e., the vertices with opinion $+$ have lower labels than the vertices with opinion $-$, and then by increasing type.
• Figure 3.2: Eventually all the vertices end up holding opinion $+$ in model 2.
• Figure 3.3: Empirical distribution of $(y_i(T))_{i\in [n]}$ when $n=600$ and $T=3,6,9$, for $\beta=0.66$, $\pi_+=\pi_-=0.6$. For any $i\in[n]$, $y_i(0)$ is chosen uniformly at random in $[0,1]$, independently of the other vertices. The red line indicates the corresponding $\hbox{Beta}(\beta p_+,\beta(1-p_+))$ density function, where $p_+$ denotes the proportion of vertices holding opinion $+$. Simulations are based on a single run.

Theorems & Definitions (32)

• Remark 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Remark 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8
• Lemma 2.9
• Theorem 3.1