Fundamental Limits of Decentralized Self-Regulating Random Walks

TL;DR

This paper establishes fundamental limits for decentralized self-regulating random walks on graphs, providing universal bounds and stability conditions for their long-term behavior under various policies and trap scenarios.

Contribution

It introduces graph-dependent Laplace envelopes and viability-safety inequalities to characterize the stability and recurrence of SRRWs under general conditions.

Findings

Universal bounds on stationary fork intensity derived

Stability conditions for avoiding extinction and explosion

Positive recurrence established under certain policies

Abstract

Self-regulating random walks (SRRWs) are decentralized token-passing processes on a graph allowing nodes to locally \emph{fork}, \emph{terminate}, or \emph{pass} tokens based only on a return-time \emph{age} statistic. We study SRRWs on a finite connected graph under a lazy reversible walk, with exogenous \emph{trap} deletions summarized by the absorption pressure $\Lambda_{\mathrm{del}}=\sum_{u\in\mathcal P_{\mathrm{trap}}}\zeta(u)\pi(u)$ and a global per-visit fork cap $q$. Using exponential envelopes for return-time tails, we build graph-dependent Laplace envelopes that universally bound the stationary fork intensity of any age-based policy, leading to an effective triggering age $A_{\mathrm{eff}}$. A mixing-based block drift analysis then yields controller-agnostic stability limits: any policy that avoids extinction and explosion must satisfy a \emph{viability} inequality (births can overcome $\Lambda_{\mathrm{del}}$ at low population) and a \emph{safety} inequality (trap deletions plus deliberate terminations dominate births at high population). Under corridor-wise versions of these conditions, we obtain positive recurrence of the population to a finite corridor.