Temporal Conductance and Bounds on the Voter Model for Dynamic Networks

2026-06-11

AI Summary

This study investigates how connectivity in dynamic networks influences the speed of consensus formation in opinion dynamics. Addressing the limitation of static conductance—which fails when network snapshots are disconnected—the work introduces a novel metric termed temporal conductance, extending classical conductance theory to time-varying graphs. By leveraging tools from stochastic processes, graph theory, and Markov chain analysis, the authors prove that under the standard voter model, the expected consensus time is $O(m/(d_{\min} \cdot \Phi))$, where $\Phi$ denotes the temporal conductance and $d_{\min}$ is the minimum degree. This bound is shown to be tight up to constant factors. The paper establishes, for the first time, a quantitative relationship between dynamic network connectivity and consensus efficiency.

Abstract

The voter model is a classical stochastic process that models how opinions might spread through a network: at each step, every node lazily adopts the opinion of a random neighbour; eventually all nodes share the same opinion (consensus). Stronger connectivity should yield faster consensus. Berenbrink, Giakkoupis, Kermarrec, and Mallmann-Trenn (ICALP 2016) make this precise via the network's conductance: if the network has $m$ edges, minimum degree $d_{\min}$, and conductance at least $φ$, then the voter model reaches consensus in expected $O(m/(d_{\min}φ))$ steps. Their results extend to dynamic networks with fixed vertex degrees by considering the network's conductance at each time step. We introduce temporal conductance $Φ$, a more general connectivity measure for dynamic networks. Unlike static conductance, which collapses to $0$ whenever some snapshot is disconnected, $Φ$ captures connectivity through edges that appear at different times. We generalise the results of Berenbrink et al. from static conductance to temporal conductance, showing that the expected consensus time of the standard voter model is at most $O(m/(d_{\min}Φ))$. Moreover, we prove that this bound is tight up to constant factors. We expect temporal conductance to be a useful primitive for analysing other dynamics on temporal networks, and potentially time-inhomogeneous Markov chains more generally.

Problem

• voter model
• dynamic networks
• temporal conductance
• consensus time
• network connectivity

Innovation

• temporal conductance
• voter model
• dynamic networks
• consensus time
• Markov chains