Stabilization of stochastic networks in Markovian environment

TL;DR

This paper provides criteria for the stabilization of stochastic networks in Markovian environments, confirming a conjecture by analyzing eigenvalues of toppling matrices and using a toppling random walk approach.

Contribution

It introduces new eigenvalue-based criteria for network stabilization and confirms a previously conjectured condition in stochastic network theory.

Findings

Stabilization characterized by the largest eigenvalue of a modified toppling matrix.

Confirmation of Conjecture 7.2 from Levine-Greco [GL23].

Use of toppling random walk in the proof.

Abstract

We establish criteria under which stochastic networks in a Markovian environment stabilize, thus confirming Conjecture 7.2 from Levine-Greco [GL23]. The networks evolve on finite connected graphs $G=(V,E)$, and their dynamics are encoded by $V \times V$ toppling matrices $M$, whose columns record the expected number of topplings when the environment is in stationarity. Stabilization and non-stabilization are characterized by a parameter $\rho$ which depends on the largest eigenvalue of the matrix $M+\alpha I$, with $\alpha=1+\max\{-M(v,v):v\in V\}$. The proofs rely on the toppling random walk, in which toppled vertices are sampled according to the eigenvector associated with the largest eigenvalue of $M$.