The voter model on the hyperbolic graph

Abstract

We consider the voter model on the giant component of a hyperbolic random graph, which is a spatial scale-free network, in the sparse and linear-giant regime $α\in(1/2,1)$. We find that the quenched expected consensus time has order $n^{2-1/α}$, as the number of vertices $n\to\infty$, with probability arbitrarily close to one. This is generalised to the voter model where each vertex changes its opinion at rates $q(v)={\rm d}(v)^φ$, where we also establish the consensus time orders for all $φ\geq 0$. These orders have 3 regimes, with a phase transition at $φ=2-2α$. For the upper bounds, our main proof idea is to connect the meeting set to some fixed target vertex of appropriate height in the product chain electrical network, to make rigorous an argument due to Durrett.