Randomized Zero Forcing

TL;DR

Randomized zero forcing (RZF) is a stochastic process on directed graphs that models color propagation based on in-neighborhoods, with applications to network centrality and graph analysis.

Contribution

The paper introduces RZF, extending probabilistic zero forcing to directed and weighted graphs, and analyzes its propagation time and extremal properties.

Findings

Expected propagation time is monotonic with initial blue set and edge weights.

Exact values and asymptotics are derived for specific graph families.

Application demonstrates RZF as a dynamic centrality measure in networks.

Abstract

We introduce randomized zero forcing (RZF), a stochastic color-change process on directed graphs in which a white vertex turns blue with probability equal to the fraction of its incoming neighbors that are blue. Unlike probabilistic zero forcing, RZF is governed by in-neighborhood structure and can fail to propagate globally due to directionality. The model extends naturally to weighted directed graphs by replacing neighbor counts with incoming weight proportions. We study the expected propagation time of RZF, establishing monotonicity properties with respect to enlarging the initial blue set and increasing weights on edges out of initially blue vertices, as well as invariances that relate weighted and unweighted dynamics. Exact values and sharp asymptotics are obtained for several families of directed graphs, including arborescences, stars, paths, cycles, and spiders, and we derive tight extremal bounds for unweighted directed graphs in terms of basic parameters such as order, degree, and radius. We conclude with an application to an empirical input-output network, illustrating how expected propagation time under RZF yields a dynamic, process-based notion of centrality in directed weighted systems.