[Paper Review] The Evolving Voter Model on Thick Graphs

Anirban Basak, Rick Durrett+1 more|arXiv (Cornell University)|Dec 24, 2015

Opinion Dynamics and Social Influence14 references3 citations

TL;DR

This paper rigorously analyzes the evolving voter model on Erdős-Rényi random graphs with intermediate average degree $L = N^a$ ($0 < a < 1$), focusing on how rewiring mechanisms—rewire-to-same or rewire-to-random—affect consensus formation. Using an approximate master equation and large-deviation techniques, it proves that for rewire-to-same, a phase transition occurs at a critical $\alpha_c$ independent of initial opinion density $p$, while for rewire-to-random, $\alpha_c(p)$ depends on $p$, with a universal curve for $\alpha > \alpha_c(p)$. The key result is that the system reaches consensus in $O(NL)$ steps with high probability under these dynamics.

ABSTRACT

In the evolving voter model, when an individual interacts with a neighbor having an opinion different from theirs, they will with probability $1-α$ imitate the neighbor but with probability $ α$ will sever the connection and choose a new neighbor at random (i) from the graph or (ii) from those with the same opinion. Durrett et al. used simulation and heuristics to study these dynamics on sparse graphs. Recently Basu and Sly have studied this system with $1-α= ν/N$ on a dense Erdős-Rényi graph $G(N,1/2)$ and rigorously proved that there is a phase transition from rapid disconnection into components with a single opinion to prolonged persistence of discordant edges as $ν$ increases. In this paper, we consider the intermediate situation of Erdős-Rényi random graphs with average degree $L=N^a$ where $0 &lt; a &lt; 1$. Most of the paper is devoted to a rigorous analysis of an approximation of the dynamics called the approximate master equation. Using ideas of \cite{LMR} and \cite{Silk} we are able to analyze these dynamics in great detail.

Motivation & Objective

• To rigorously analyze the evolving voter model on random graphs with intermediate average degree $L = N^a$ ($0 < a < 1$), bridging the gap between sparse and dense graph regimes.
• To investigate how rewiring mechanisms—rewire-to-same or rewire-to-random—affect the persistence of discordant edges and consensus formation.
• To establish the existence of a phase transition in the system's behavior depending on the rewiring probability $\alpha$, particularly in relation to initial opinion density $p$.
• To validate the heuristic and simulation-based findings of prior work [7] through rigorous probabilistic analysis using large-deviation bounds and approximate master equations.

Proposed method

• The authors employ an approximate master equation to model the dynamics of opinion and network structure coevolution, simplifying the full stochastic process.
• They use large-deviation bounds and concentration inequalities to control the number of rewiring and imitation events over time, particularly focusing on rare but impactful events.
• Key lemmas track the number of times edges are picked, rewired, or lead to opinion changes, using bounds on vertex degrees and edge counts.
• The analysis distinguishes between edges within the minority set $S$, between $S$ and $T$, and within $T$, to bound the total number of discordant edge updates.
• The proof relies on high-probability bounds: for instance, showing that the number of rewirings to any vertex is bounded by $20L$ with high probability.
• The system is analyzed in discrete time with each oriented edge updated at rate 1, and the dynamics are shown to terminate in $O(NL)$ steps with high probability.

Experimental results

Research questions

• RQ1 Does a phase transition occur in the evolving voter model on thick graphs with $L = N^a$ ($0 < a < 1$), and if so, how does it depend on the rewiring probability $\alpha$ and initial opinion density $p$?
• RQ2 How does the rewiring mechanism—rewire-to-same versus rewire-to-random—affect the persistence of discordant edges and the final consensus state?
• RQ3 Can the heuristic and simulation-based results from prior work [7] be rigorously proven for intermediate-degree random graphs using large-deviation techniques?
• RQ4 What is the typical time scale for consensus to emerge in the evolving voter model on such graphs, and how does it scale with $N$ and $L$?

Key findings

• For the rewire-to-same mechanism, a phase transition occurs at a critical $\alpha_c$ independent of $p$, with $\pi \approx p$ for $\alpha > \alpha_c$ and $\pi \approx 0$ for $\alpha < \alpha_c$, confirming the existence of a universal curve.
• For the rewire-to-random mechanism, the critical $\alpha_c(p)$ depends on $p$, and for $\alpha > \alpha_c(p)$, $\pi \approx p$, while for $\alpha < \alpha_c(p)$, $\pi(\alpha, p) = \pi(\alpha, 1/2)$, indicating a universal behavior for $\alpha < \alpha_c(p)$.
• The system reaches consensus in $O(NL)$ steps with high probability, as shown by bounding the total number of updates to discordant edges across $R_{SS}$, $R_{ST}$, and $R_{TT}$.
• With high probability, the number of rewirings to any vertex is bounded by $20L$, ensuring that once a vertex becomes 'stubborn' (i.e., has high degree), it remains in its state for a long time.
• The total number of updates to discordant edges is bounded by $6.9NL$ with high probability, implying that the system terminates in $O(NL)$ steps.
• The analysis confirms the separation of time scales: the system remains close to a quasi-stationary distribution indexed by the minority opinion density, with slow changes governed by a diffusion-like process.