Fundamental Limits of Decentralized Self-Regulating Random Walks

2026-01-29

AI Summary

This study investigates the stability and fundamental limits of population control in decentralized self-regulating random walks (SRRW) on finite connected graphs, where agents locally decide to fork, terminate, or pass a token based solely on age statistics of return times. By introducing absorption pressure and a global forking upper bound—combined with an exponential envelope on return-time tails, a graph-dependent Laplacian envelope, and mixing time analysis—the work proposes the notion of “effective triggering age” and establishes a steady-state upper bound on forking intensity that holds for arbitrary age-based policies. Employing block drift techniques and Markov chain ergodic theory, the paper proves positive recurrence of the population process under corridor-like conditions and derives universal inequalities guaranteeing both survival and safety, thereby providing controller-agnostic stability guarantees and theoretical limits for decentralized token systems.

Technology Category

• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Sequential Decision Making
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• decentralized
• self-regulating random walks
• stability limits
• trap deletions
• population control

Innovation

• self-regulating random walks
• decentralized token passing
• return-time age
• stability limits
• positive recurrence