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Stabilization of stochastic networks in Markovian environment

Robin Kaiser, Martin Klötzer, Ecaterina Sava-Huss

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 $ρ$ which depends on the largest eigenvalue of the matrix $M+αI$, with $α=1+\max\{-M(v,v):v\in V\}$. The proofs rely on the toppling random walk, in which toppled vertices are sampled according to the eigenvector associated with the largest eigenvalue of $M$.

Stabilization of stochastic networks in Markovian environment

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 , and their dynamics are encoded by toppling matrices , whose columns record the expected number of topplings when the environment is in stationarity. Stabilization and non-stabilization are characterized by a parameter which depends on the largest eigenvalue of the matrix , with . The proofs rely on the toppling random walk, in which toppled vertices are sampled according to the eigenvector associated with the largest eigenvalue of .

Paper Structure

This paper contains 5 sections, 16 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stackwise representation
  4. Toppling random walk
  5. Proof of the Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $G=(V,E)$ be a finite, connected, directed graph and let $(\eta_n)_{n\in \mathbb{N}}$ be a stochastic network in a Markovian environment on $G$ which satisfies (eq:only-local-reduction), (eq:bfb) and (eq:irr). Let $M$ be the expected offspring matrix as in eq:expected-toppling-matrix and where $r(M+\alpha I)$ is the Perron-Frobenius eigenvalue of $M+\alpha I$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Example 2.1: Multitype branching process in a Markovian environment
  • Example 2.2: Stochastic sandpiles
  • Lemma 3.1: Abelian property
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.1: Least-action principle
  • proof
  • Proposition 3.2
  • ...and 23 more