The Offended Voter Model

TL;DR

The paper analyzes the offended voter model, a coevolving-network variant of the voter model where discordant interactions either lead to agreement with probability $q$ or cause the edge to be deleted with probability $1-q$. It establishes that segregation and a weaker form of consensus occur with positive probability for all $q o(0,1)$, and provides asymptotic bounds via a beta-function $eta$ on segregation probabilities; segregation becomes dominant as $q o0$. It further shows that if $1-q_N$ decays sufficiently fast with $N$, consensus occurs with high probability while the network remains densely connected, and for certain regimes consensus is also likely for fixed $q$ with simulations supporting sharper bounds. The paper introduces two constructions—the delayed OV-model and the dynamical deletion graph—and uses Erdős–Rényi comparisons and level-based analyses to obtain rigorous results, complemented by simulations and extensions to multiple opinions.

Abstract

We study a variant of the voter model on a coevolving network in which interactions of two individuals with differing opinions only lead to an agreement on one of these opinions with a fixed probability $q$. Otherwise, with probability $1-q$, both individuals become offended in the sense that they never interact again, i.e. the corresponding edge is removed from the underlying network. Eventually, these dynamics reach an absorbing state at which there is only one opinion present in each connected component of the network. If globally both opinions are present at absorption we speak of "segregation'', otherwise of "consensus''. We rigorously show that segregation and a weaker form of consensus both occur with positive probability for every $q \in (0,1)$ and that the segregation probability tends to $1$ as $q \to 0$. Furthermore, we establish that, if $q \to 1$ fast enough, with high probability the population reaches consensus while the underlying network is still densely connected. We provide results from simulations to assess the obtained bounds and to discuss further properties of the limiting state.

Figures (5)

• Figure 1: Left: Empirical segregation probabilities observed in simulations with $q\in\{0,0.05,0.1,\ldots,1\}$ and varying $N$. For each combination $(q,N)$, $1000$ simulations were carried out. Right: Comparison of the simulated empirical segregation probabilities with $N=1024$ individuals to the lower bound from equation \ref{['eq:split']}.
• Figure 2: Histogram of $\tau_{\textnormal{abs}}$ with $N=1024$ and $q=0.3$.
• Figure 3: Segregation cases of 1000 simulations for each combination of $q=0.3$ and $N\in\{512,1024\}$. (a) Size $C_1$ of largest versus size $C_2$ of second largest connected component. (b) Number of remaining edges in final graph. (c) Histogram of $C_1$. (d) Number of connected components in final graph.
• Figure 4: Simulation of three opinions. Depiction of 633 segregation states reached in 1000 runs of $N=1026$, $q=0.3$, initial complete graph with $342$ nodes of each opinion. (a) Sizes of largest versus second largest connected component. (b) Largest versus third largest connected component. (c) Ternary plot of opinion frequencies at absorption. (d) Comparison of empirical probabilities for $q\in\{0,0.05,0.1,\ldots1\}$ and $K=3$. Teal: Any segregation. Black: Segregation with 3 opinions present. Gray: Segregation with $K=2$ initial opinions (cf. Figure \ref{['fig:bound_split']} for $N=1024$).
• Figure 5: Left: Subdivision of the domain of the random walk into finitely many "levels". Right: Starting from level $k$, the level $k+1$ is hit w.h.p. before returning to $r_N^{-k+\eta}N$, $\eta\in(0,1)$, (or $N- r_N^{-k+\eta}N$) and hence before returning to level $k-1$.

Theorems & Definitions (39)

• Definition 2.1: Limit classification
• Remark 2.2
• Theorem 3.1
• Theorem 3.2
• Proposition 3.3
• Theorem 3.4
• Conjecture 4.1
• Conjecture 4.2
• Conjecture 4.3
• Conjecture 4.4