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Fundamental Limits of Decentralized Self-Regulating Random Walks

Ali Khalesi, Rawad Bitar

TL;DR

This work establishes controller-agnostic stability limits for decentralized self-regulating random walks (SRRWs) on finite graphs with trap-induced absorption. It introduces graph-dependent Laplace envelopes L_{\pi}^{\pm}(A) and an effective triggering age A_{\mathrm{eff}}, which bound the stationary per-visit fork intensity under any age-based policy. The main result provides universal viability and safety inequalities, q L_{\pi}^{+}(A_{\mathrm{eff}}) \ge \Lambda_{\mathrm{del}} and q L_{\pi}^{-}(A_{\mathrm{eff}}) - \Lambda_{\mathrm{del}} - K_{\mathrm{term}} \le 0, ensuring positive recurrence to a finite corridor when satisfied. By coupling to stationarity, analyzing block-wise drift, and applying Foster–Lyapunov arguments, the paper connects network geometry, mixing, and trap structure to intrinsic limits on token population growth, guiding controller design for robust decentralized SRRWs. These results unify various failure models (e.g., universal trap deletions and Pac-Man-like adversaries) under a single framework and provide principled constraints for ensuring resilient SRRW-based protocols in distributed systems and decentralized learning contexts.

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 $Λ_{\mathrm{del}}=\sum_{u\in\mathcal P_{\mathrm{trap}}}ζ(u)π(u)$ 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 $Λ_{\mathrm{del}}$ 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.

Fundamental Limits of Decentralized Self-Regulating Random Walks

TL;DR

This work establishes controller-agnostic stability limits for decentralized self-regulating random walks (SRRWs) on finite graphs with trap-induced absorption. It introduces graph-dependent Laplace envelopes L_{\pi}^{\pm}(A) and an effective triggering age A_{\mathrm{eff}}, which bound the stationary per-visit fork intensity under any age-based policy. The main result provides universal viability and safety inequalities, q L_{\pi}^{+}(A_{\mathrm{eff}}) \ge \Lambda_{\mathrm{del}} and q L_{\pi}^{-}(A_{\mathrm{eff}}) - \Lambda_{\mathrm{del}} - K_{\mathrm{term}} \le 0, ensuring positive recurrence to a finite corridor when satisfied. By coupling to stationarity, analyzing block-wise drift, and applying Foster–Lyapunov arguments, the paper connects network geometry, mixing, and trap structure to intrinsic limits on token population growth, guiding controller design for robust decentralized SRRWs. These results unify various failure models (e.g., universal trap deletions and Pac-Man-like adversaries) under a single framework and provide principled constraints for ensuring resilient SRRW-based protocols in distributed systems and decentralized learning contexts.

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 and a global per-visit fork cap . 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 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 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.
Paper Structure (11 sections, 7 theorems, 94 equations)

This paper contains 11 sections, 7 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. System Model and Problem Formulation
  3. Notation and Model
  4. Problem Formulation
  5. Main Result
  6. Token Distribution on the Graph
  7. Effective Triggering Age
  8. Proof sketch of Theorem 1
  9. Conclusion
  10. Lemma 1
  11. Proof of Theorem 1

Key Result

Theorem 1

For any decentralized age-based policy obeying the W--AC rule that avoids extinction and explosion, the following conditions must hold: where $A_{\text{eff}}$ is the effective triggering age, $K_\mathrm{term}$ is the mean intentional termination rate, and $q\in(0,1]$ denotes the global per-visit fork bound, i.e. Here, $q_u^\mathrm{fork}\xspace(A)$ is the probability that node $u$ forks a visitin

Theorems & Definitions (12)

  • Theorem 1
  • Lemma 1: Stationary distribution of an undirected RW
  • Lemma 2: Exponential envelopes and Laplace envelopes
  • proof : Proof sketch for \ref{['Theorem~1']}
  • Lemma 1: Stationary distribution of an undirected RW
  • proof
  • Theorem 1
  • proof : Proof of Theorem 1
  • Lemma 2: Exponential envelopes and Laplace envelopes
  • proof
  • ...and 2 more