Characterization of the critical values of branching random walks on weighted graphs through infinite-type branching processes

TL;DR

This paper analyzes the critical thresholds of branching random walks on weighted graphs, linking these thresholds to geometric graph parameters and exploring their implications for local and global survival/extinction.

Contribution

It introduces a characterization of strong and weak critical values of branching random walks on weighted graphs using geometric parameters, extending previous models.

Findings

Strong critical value corresponds to local extinction almost surely.

Weak critical value allows both global survival and extinction.

Critical values are characterized through geometrical graph parameters.

Abstract

We study the branching random walk on weighted graphs; site-breeding and edge-breeding branching random walks on graphs are seen as particular cases. We describe the strong critical value in terms of a geometrical parameter of the graph. We characterize the weak critical value and relate it to another geometrical parameter. We prove that, at the strong critical value, the process dies out locally almost surely; while, at the weak critical value, global survival and global extinction are both possible.