Max-linear models in random environment

TL;DR

This paper extends max-linear models to infinite and random graphs, exploring their relation to percolation theory and phase transitions, with applications in communication networks and influence propagation.

Contribution

It introduces max-linear models on infinite and random graphs and analyzes their phase transition behavior related to percolation theory.

Findings

Percolation critical probability induces phase transition in max-linear models.

Dependence structure is influenced by the underlying graph topology.

Applications include modeling influence propagation in communication networks.

Abstract

We extend previous work of max-linear models on finite directed acyclic graphs to infinite graphs as well as random graphs, and investigate their relations to classical percolation theory, more particularly the impact of Bernoulli bond percolation on such models. We show that the critical probability of percolation on the oriented square lattice graph $\mathbb{Z}^2$ describes a phase transition in the obtained model. Focus is on the dependence introduced by this graph into the max-linear model. We discuss natural applications in communication networks, in particular, concerning the propagation of influences.