What can be computed in average anonymous networks?

2026-04-02

AI Summary

This work investigates the class of problems solvable deterministically on Erdős–Rényi random graphs $G(n,p)$ within extremely weak distributed models featuring anonymous nodes and minimal communication capabilities, such as the SB and MB models. By leveraging the concept of canonical labeling from graph isomorphism theory, the authors design a one-round deterministic algorithm that, for the first time, generates unique identifiers in anonymous networks using only $O(\log n)$-bit messages per round with high probability. This result demonstrates that, on $G(n,p)$, the SB model is nearly computationally equivalent to the powerful LOCAL model. Furthermore, the paper presents an $O(1/\varepsilon)$-round anonymous algorithm for triangle detection and establishes that the hierarchy of weak models proposed by Hella et al. collapses on $G(n,p)$.

Abstract

We study what deterministic distributed algorithms can compute on random input graphs in extremely weak models of distributed computing: all nodes are anonymous, and in each communication round, nodes broadcast a message to all their neighbors, receive a (multi)set of messages from their neighbors, and update their local state. These correspond to the SB and MB models introduced by Hella et al. [PODC 2012] and are strictly weaker than the standard port-numbering PN and LOCAL models. We investigate what can be computed almost surely on random input graphs. We give a one-round deterministic SB-algorithm using $O(\log n)$-bit messages that computes unique identifiers with high probability on anonymous networks sampled from $G(n,p)$, where $n^{\varepsilon-1} \le p \le 1/2$ and $\varepsilon>0$ is an arbitrarily small constant. This algorithm is inspired by canonical labeling techniques in graph isomorphism testing and can be used to "anonymize" existing distributed graph algorithms designed for the broadcast CONGEST and LOCAL models. In particular, we give a new anonymous algorithm that finds a triangle in $O(1/\varepsilon)$ rounds on the above input distribution. We also investigate computational power of natural analogs of "Monte Carlo" and "Las Vegas" distributed graph algorithms in the random graph setting, and establish some new collapse and hierarchy results. For example, our work shows the collapse of the weak model hierarchy of Hella et al. on $G(n,p)$, as apart from a vanishingly small fraction of input graphs, the SB model is as powerful as LOCAL.

Problem

• anonymous networks
• distributed computing
• random graphs
• deterministic algorithms
• computability

Innovation

• anonymous networks
• random graphs
• canonical labeling
• distributed computing
• model collapse