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Discursive Voter Models on the Supercritical Scale-Free Network

John Fernley

TL;DR

This work analyzes discursive voter dynamics on supercritical scale-free networks by leveraging a duality to coalescing random walks and rigorous mixing-time control of the underlying variable-speed random walk on the Simplified Norros-Reittu graph. By embedding fast-mixing Erdős–Rényi cores into the giant component and extending them with bounded substructures, the authors obtain polylogarithmic bounds that yield polynomial consensus-time orders. In the small-world regime ($ au=1+1/ au$, $ au o[0,1/2]$) and ultrasmall regime ($ au o(1/2,1)$), they prove two-phase behavior depending on the temperature parameter $ heta$, with mean-field predictions from prior work (Moinet et al., 2018) verified up to polylog factors. The results extend mean-field accuracy to highly heterogeneous networks, including ultrasmall worlds with $ au o(2,3]$, and provide a rigorous framework for understanding opinion dynamics on complex graphs with heavy-tailed degree distributions.

Abstract

The voter model is a classical interacting particle system, modelling how global consensus is formed by local imitation. We analyse the time to consensus for a particular family of voter models when the underlying structure is a scale-free inhomogeneous random graph, in the high edge density regime where this graph features a giant component. In this regime, we verify that the polynomial orders of consensus agree with those of their mean-field approximation in [Moinet et al., 2018]. This "discursive" family of models has a symmetrised interaction to better model discussions, and is indexed by a temperature parameter which, for certain parameters of the power law tail of the network's degree distribution, is seen to produce two distinct phases of consensus speed. Our proofs rely on the well-known duality to coalescing random walks and a control on the mixing time of these walks, using the known fast mixing of the Erdős-Rényi giant subgraph. Unlike in the subcritical case [Fernley and Ortgiese, 2022] which requires tail exponent of the limiting degree distribution $τ=1+1/γ>3$ as well as low edge density, in the giant component case we also address the "ultrasmall world" power law exponents $τ\in (2,3]$.

Discursive Voter Models on the Supercritical Scale-Free Network

TL;DR

This work analyzes discursive voter dynamics on supercritical scale-free networks by leveraging a duality to coalescing random walks and rigorous mixing-time control of the underlying variable-speed random walk on the Simplified Norros-Reittu graph. By embedding fast-mixing Erdős–Rényi cores into the giant component and extending them with bounded substructures, the authors obtain polylogarithmic bounds that yield polynomial consensus-time orders. In the small-world regime (, ) and ultrasmall regime (), they prove two-phase behavior depending on the temperature parameter , with mean-field predictions from prior work (Moinet et al., 2018) verified up to polylog factors. The results extend mean-field accuracy to highly heterogeneous networks, including ultrasmall worlds with , and provide a rigorous framework for understanding opinion dynamics on complex graphs with heavy-tailed degree distributions.

Abstract

The voter model is a classical interacting particle system, modelling how global consensus is formed by local imitation. We analyse the time to consensus for a particular family of voter models when the underlying structure is a scale-free inhomogeneous random graph, in the high edge density regime where this graph features a giant component. In this regime, we verify that the polynomial orders of consensus agree with those of their mean-field approximation in [Moinet et al., 2018]. This "discursive" family of models has a symmetrised interaction to better model discussions, and is indexed by a temperature parameter which, for certain parameters of the power law tail of the network's degree distribution, is seen to produce two distinct phases of consensus speed. Our proofs rely on the well-known duality to coalescing random walks and a control on the mixing time of these walks, using the known fast mixing of the Erdős-Rényi giant subgraph. Unlike in the subcritical case [Fernley and Ortgiese, 2022] which requires tail exponent of the limiting degree distribution as well as low edge density, in the giant component case we also address the "ultrasmall world" power law exponents .
Paper Structure (7 sections, 21 theorems, 150 equations, 3 figures)

This paper contains 7 sections, 21 theorems, 150 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Paths in the Network
  4. Markov Chains
  5. Mixing Time for the Variable Speed Random Walk
  6. Discursive Voter Models
  7. Appendix

Key Result

Theorem 1.3

Take $\gamma\in[0,\tfrac{1}{2})$ and $0<\beta<1-2\gamma$. Then for the discursive voter model on $G_N$ from initial opinions distributed as $\mu_u$ of the Bernoulli process (see Definition def_bernoulli) with $u \in (0,1)$, we have

Figures (3)

  • Figure 1: We prove Theorems \ref{['theorem_disc_cons_small']} and \ref{['theorem_disc_cons_ultrasmall']} in the highlighted region (when $\gamma\geq1/2$ requiring Assumption \ref{['assumption_large_beta']}, the integral condition of which we cannot see in this plot), and conjecture the same exponents would be seen where $\{\gamma>1/2,\beta \leq 2\log 2\}$. In the other hole $\{\beta+2\gamma<1\}$, we see a very different subcritical graph structure and so the different consensus time exponents of Theorem \ref{['thm_subcrit']}.
  • Figure 2: We divide the degree mass and label by red or blue, for a typical rank one scale-free network with the step count $M=10$. Note there is some uncoloured mass.
  • Figure 3: We sketch the layers of the spanning subgraph that we construct in this proof. The first layer, $\mathscr{G}_1$, was necessary to have a fast-mixing core to build from; we aim to reach the point where the unexplored mass outside of the subgraph $\mathscr{G}_4$ has $\sum_{v \notin \mathscr{G}_4}\frac{\beta}{N}\left( \frac{N}{v} \right)^{2\gamma}<1$ so that the final growths will be subcritical. Further to keep these trees small, we must have bounded weights (in the sense of a $O^{\log N}(1)$ bound) which we achieve outside $\mathscr{G}_2$. $\mathscr{G}_3$ is a minor step to keep track of mixing, and then in $\mathscr{G}_4$ we use the large $\beta$ assumption to check that we have definitely passed the point of subcritical explorations. The graph $\mathscr{G}_5$ adds the promised subcritical trees which then spans the full giant $\mathscr{C}_{\rm max}$.

Theorems & Definitions (46)

  • Definition 1.1: Discursive voter model
  • Definition 1.2: Simplified Norros-Reittu graph $G_N$
  • Theorem 1.3: fernley2019voter
  • Definition 1.4: $\mu_u$
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7: The polylogarithmic factor
  • Definition 1.8: VSRW
  • Theorem 1.9
  • Remark 1.11: Smaller supercritical $\beta$
  • ...and 36 more