A General Stabilization Bound for Influence Propagation in Graphs

Abstract

We study the stabilization time of a wide class of processes on graphs, in which each node can only switch its state if it is motivated to do so by at least a $\frac{1+\lambda }{2}$ fraction of its neighbors, for some $0<\lambda <1$. Two examples of such processes are well-studied dynamically changing colorings in graphs: in majority processes, nodes switch to the most frequent color in their neighborhood, while in minority processes, nodes switch to the least frequent color in their neighborhood. We describe a non-elementary function $f(\lambda )$, and we show that in the sequential model, the worst-case stabilization time of these processes can completely be characterized by $f(\lambda )$. More precisely, we prove that for any $\epsilon >0$, $O(n^{1+f(\lambda )+\epsilon })$ is an upper bound on the stabilization time of any proportional majority/minority process, and we also show that there are graph constructions where stabilization indeed takes $\Omega (n^{1+f(\lambda )-\epsilon })$ steps.