Fundamental Limits of Decentralized Self-Regulating Random Walks

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.

💡 Research Summary

This paper investigates the fundamental stability limits of decentralized self‑regulating random walks (SRRWs), a token‑passing mechanism in which each node makes local decisions—fork, terminate, or pass—based solely on the age of the token (the time elapsed since the node’s last visit). The setting is a finite, connected, undirected graph on which tokens move as independent lazy reversible random walks. A subset of vertices, the trap set, deletes visiting tokens with prescribed probabilities, giving rise to an average “absorption pressure” Λ_del that depends on the stationary distribution of the underlying walk.

The authors introduce a novel, controller‑agnostic analytical framework. First, they derive exponential tail bounds for return‑time distributions and use these to construct two graph‑dependent Laplace envelopes, L⁺_π(·) and L⁻_π(·). These envelopes bound the stationary per‑visit fork probability p_fork of any age‑based policy:
 q L⁺_π(A_eff) ≤ p_fork ≤ q L⁻_π(A_eff),
where q is the global per‑visit fork cap and A_eff is a single scalar—called the effective triggering age—that summarizes the steady‑state behavior of possibly non‑uniform, randomized policies. In the special case of a uniform policy with a common age threshold A, A_eff coincides with A.

With A_eff in hand, the paper derives two universal inequalities that any viable (non‑extinct) and safe (non‑explosive) SRRW must satisfy. The viability condition (V) requires that, when the population is low, the achievable fork intensity can overcome the absorption pressure:
 (V) q L⁺_π(A_eff) ≥ Λ_del.
The safety condition (S) demands that, when the population is high, trap deletions together with intentional terminations dominate births:
 (S) q L⁻_π(A_eff) − Λ_del − K_term ≤ 0,
where K_term denotes the mean intentional termination rate (the average probability that a token is deliberately killed upon visit).

The main theorem states that if a decentralized age‑based policy respects both (V) and (S) across appropriate population corridors, then the token count process {Z_t} is positive recurrent to a finite interval