Vertex-pursuit in random directed acyclic graphs

TL;DR

This paper studies a vertex-pursuit game called Seepage in directed acyclic graphs (DAGs), analyzing the minimum number of agents needed to block an intruder in stochastic models with different degree sequences.

Contribution

It introduces a generalized stochastic model for DAGs and provides asymptotic bounds and precise values for the green number in regular and power law degree sequences.

Findings

Asymptotic bounds for the green number in stochastic DAGs.

Precise green number values in certain degree sequences.

Analysis of Seepage in both regular and power law DAGs.

Abstract

We examine a dynamic model for the disruption of information flow in hierarchical social networks by considering the vertex-pursuit game Seepage played in directed acyclic graphs (DAGs). In Seepage, agents attempt to block the movement of an intruder who moves downward from the source node to a sink. The minimum number of such agents required to block the intruder is called the green number. We propose a generalized stochastic model for DAGs with given expected total degree sequence. Seepage and the green number is analyzed in stochastic DAGs in both the cases of a regular and power law degree sequence. For each such sequence, we give asymptotic bounds (and in certain instances, precise values) for the green number.