A Note on the Majority Dynamics in Inhomogeneous Random Graphs

Abstract

In this note, we study discrete time majority dynamics over an inhomogeneous random graph G obtained by including each edge e in the complete graph K_n independently with probability p_n(e). Each vertex is independently assigned an initial state +1 (with probability p_+) or -1 (with probability 1-p_+), updated at each time step following the majority of its neighbors’ states. Under some regularity and density conditions of the edge probability sequence, if p_+ is smaller than a threshold, then G will display a unanimous state -1 asymptotically almost surely, meaning that the probability of reaching consensus tends to one as nrightarrow infty . The consensus reaching process has a clear difference in terms of the initial state assignment probability: In a dense random graph p_+ can be near a half, while in a sparse random graph p_+ has to be vanishing. The size of a dynamic monopoly in G is also discussed.