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 $$K_n$$ K n independently with probability $$p_n(e)$$ p ( e ) . Each vertex is assigned initial state $$+1$$ + 1 (with $$p_+$$ ) or $$-1$$ - $$1-p_+$$ ), updated at step following of its neighbors’ states. Under some regularity and density conditions sequence, if smaller than a threshold, then will display unanimous asymptotically almost surely, meaning that reaching consensus tends to one as $$n\rightarrow \infty $$ → ∞ The process has clear difference terms assignment probability: dense can be near half, while sparse vanishing. size dynamic monopoly also discussed.