Latent Voter Model on Locally Tree-Like Random Graphs

In the latent voter model, which models the spread of a technology through a social network, individuals who have just changed their choice have a latent period, which is exponential with rate λ, during which they will not buy a new device. We study site and edge versions of this model on random graphs generated by a configuration model in which the degrees d(x) have 3 ≤ d(x) ≤M . We show that if the number of vertices n → ∞ and log n λn n then the fraction of 1’s at time λnt converges to the solution of du/dt = cpu(1− u)(1− 2u). Using this we show the latent voter model has a quasi-stationary state in which each opinion has probability ≈ 1/2 and, with high probability, persists in this state for a time that is ≥ nm for any m <∞. Thus, even a very small latent period drastically changes the behavior of the voter model, which has a one parameter family of stationary distributions and reaches fixation in time O(n).