Discursive Voter Models on the Supercritical Scale-Free Network

SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1285-1314, June 2024.
Abstract. The voter model is a classical interacting particle system, modeling how global consensus is formed by local imitation. We analyze 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 [A. Moinet, A. Barrat, and R. Pastor-Satorras, Phys. Rev. E, 98 (2018), 022303]. This “discursive” family of models has a symmetrized interaction to better model discussions and is indexed by a temperature parameter that, 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 novel bound on the mixing time of these walks using the known fast mixing of the Erdős–Rényi giant subgraph. Unlike in the subcritical case [J. Fernley and M. Ortgiese, Random Structures Algorithms, 62 (2023), pp. 376–429], which requires tail exponent of the limiting degree distribution [math] as well as low edge density, in the giant component case, we also address the “ultrasmall world” power law exponents [math].