Stabilization Bounds for Influence Propagation from a Random Initial State

TL;DR

This paper analyzes the stabilization time of influence propagation processes on graphs, showing bounds for both majority and minority dynamics under various conditions, with implications for understanding how long such processes can last.

Contribution

It provides new bounds on stabilization times for influence processes starting from random states, including proportional switching scenarios with different parameters.

Findings

Stabilization can last up to Θ(n^2) steps with small improvements.

For λ < 1/3, stabilization can last Ω(n^{1+c}) steps.

For λ > 1/2, stabilization time is bounded by O(n log n).

Abstract

We study the stabilization time of two common types of influence propagation. In majority processes, nodes in a graph want to switch to the most frequent state in their neighborhood, while in minority processes, nodes want to switch to the least frequent state in their neighborhood. We consider the sequential model of these processes, and assume that every node starts out from a uniform random state. We first show that if nodes change their state for any small improvement in the process, then stabilization can last for up to $\Theta(n^2)$ steps in both cases. Furthermore, we also study the proportional switching case, when nodes only decide to change their state if they are in conflict with a $\frac{1+\lambda}{2}$ fraction of their neighbors, for some parameter $\lambda \in (0,1)$. In this case, we show that if $\lambda < \frac{1}{3}$, then there is a construction where stabilization can indeed last for $\Omega(n^{1+c})$ steps for some constant $c>0$. On the other hand, if $\lambda > \frac{1}{2}$, we prove that the stabilization time of the processes is upper-bounded by $O(n \cdot \log{n})$.