A Note on the Majority Dynamics in Inhomogeneous Random Graphs

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 $$n\rightarrow \infty $$ . 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.