Randomized Zero Forcing

2026-02-18

AI Summary

This study investigates a stochastic propagation process on directed graphs governed by in-neighborhood structure and its inherent limitations on global spread. The authors propose the Random Zero Forcing (RZF) model, wherein white nodes turn blue with probability proportional to the fraction of blue in-neighbors, and extend it to weighted directed graphs. This work is the first to integrate zero forcing dynamics with the in-neighborhood topology of directed graphs, revealing the critical role of edge directionality in propagation dynamics, and establishes an equivalence between RZF processes on weighted and unweighted graphs. Leveraging probabilistic graph theory, expected propagation time analysis, and extremal graph theory, the paper derives exact and asymptotic propagation times for canonical directed structures—including trees, stars, paths, cycles, and spider graphs—and introduces a dynamic centrality measure based on RZF, whose efficacy is validated on empirical networks.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Web Mining and Content Analysis: Content-based information diffusion

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.

Problem

• randomized zero forcing
• directed graphs
• propagation time
• stochastic process
• weighted graphs

Innovation

• Randomized Zero Forcing
• directed graphs
• propagation time
• weighted networks
• dynamic centrality