What can be computed in average anonymous networks?

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.

• The paper introduces a deterministic, one-round canonical labeling algorithm that assigns unique O(log n) identifiers to nearly all nodes in random graphs.
• It demonstrates that several distributed tasks, including Hamiltonian cycle detection and triangle finding, can be performed without traditional identifiers.
• The study shows the collapse of weak anonymous models to the power of ID-based algorithms in random graphs while outlining soundness limitations.

Deterministic Computability in Average-Case Anonymous Networks

Introduction and Problem Formulation

This paper investigates the deterministic computational power of extremely weak models of distributed computing on random (Erdős-Rényi) graphs, specifically considering situations where all nodes are anonymous and possess minimal symmetry-breaking information. In the considered models—the set-broadcast $\mathcal{S}$ and multiset-broadcast $\mathcal{M}$ models—nodes, in each synchronous communication round, broadcast a (possibly short) message to all neighbors and receive either the set or multiset of all such messages, updating their local state accordingly. These settings are strictly weaker than classical port-numbering or ID-based models and have been previously shown incapable of solving standard symmetry-breaking tasks in the worst-case due to inherent inability to break structural graph symmetries [Angluin80, yamashita1988computing].

Departing from worst-case inputs, the work instead adopts an average-case (random graph) lens, probing whether typical instances permit significantly more computational power in these highly symmetric, restricted models. Specifically, the study formalizes what distributed tasks can be solved asymptotically almost surely (a.a.s.) (i.e., with probability $1-o(1)$ as $n\to\infty$) by deterministic, anonymous algorithms, as well as the power of analogues to "Monte Carlo" (correct with high probability) and "Las Vegas" (always correct or explicit failure) paradigms in the random graph regime.

Principal Technical Results

One-Round Generation of Unique Identifiers via Canonical Labeling

The central technical contribution is a deterministic, single-round $S^*$ (set-broadcast with $O(\log n)$ message bound) algorithm that assigns unique $O(\log n)$-bit identifiers to all nodes in $G(n,p)$ a.a.s., provided the edge probability $p$ satisfies $n^{\varepsilon-1}\leq p \leq 1/2$ for any constant $\mathcal{M}$0. The key insight is that canonical labeling techniques from graph isomorphism testing—judiciously adapted for distributed implementation—allow local degree patterns (collected in one round) to act as distinguishing short identifiers in almost all random graphs. This result enables simulation of any broadcast-LINIAL-style algorithm in these extremely weak and anonymous models with a cost of only one extra round and no increase in message length, as IDs are locally computable with high probability. Compared to prior canonical labeling schemes [BabaiES80, GaudioRS25, MosselR19], this protocol achieves both asymptotic optimality in communication and extends the success range of $\mathcal{M}$1 dramatically.

Algorithmic Consequences: Triangles, Hamiltonian Cycles, and Subgraph Detection

This one-round node distinction enables anonymization of a broad class of deterministic distributed algorithms robustly designed for models with IDs. For instance, the paper shows that the state-of-the-art deterministic Hamiltonian cycle algorithm for random graphs, originally needing unique identifiers and $\mathcal{M}$2 rounds for $\mathcal{M}$3 [Turau20], can be directly implemented in the $\mathcal{M}$4 model with at most one extra round, removing the dependency on input IDs.

The authors introduce a new, asymptotically almost surely correct, $\mathcal{M}$5-round triangle-finding algorithm for $\mathcal{M}$6, robust to the anonymity of the nodes, and achievable in expected $\mathcal{M}$7 rounds if $\mathcal{M}$8. This exploits a new structural lemma on the eccentricity profile of triangles in random graphs, highlighting that, on almost all instances, local neighborhoods suffice for fast triangle discovery even without unique IDs. Notably, the tightness and optimality of this result are established by lower bounds and impossibility theorems discussed below.

Collapse of the Weak Model Hierarchy

Perhaps the most significant structural finding is that in the $\mathcal{M}$9 model, for $1-o(1)$0 above the connectivity threshold, the entire hierarchy of weak anonymous broadcast models collapses to the power of the broadcast model with IDs—apart from a vanishing subset of adversarial graphs. Concretely, any problem solvable by a deterministic uniform algorithm in the ID-based model can also be solved (up to a constant round slack) by an anonymous algorithm in the weak model, effectively nullifying the standard separation results that hold for worst-case inputs.

Further, time hierarchy collapse results are demonstrated: for any $1-o(1)$1, every isomorphism-invariant, component-wise distributed problem can be solved in $1-o(1)$2 rounds a.a.s., essentially matching the diameter of $1-o(1)$3 (often $1-o(1)$4 or $1-o(1)$5).

Impossibility and Limitations: Soundness, Uniformity, and Knowledge

While a.a.s.-correct algorithms ("Monte Carlo" style) enjoy the power described above, soundness (Las Vegas-like guarantee—all failures are explicit and global) provably restricts computability in anonymous models. The paper establishes, via covering space arguments, that neither triangle detection nor Hamiltonicity are possible for sound, uniform, anonymous algorithms—even on almost all random graphs—showing a strict separation between mere high-probability correctness and robust knowledge of success in the model.

When non-uniformity (knowledge of $1-o(1)$6) is allowed, this barrier disappears: every isomorphism-invariant, component-wise problem can be solved by a sound, non-uniform (nodes know $1-o(1)$7), anonymous algorithm in four rounds on $1-o(1)$8 a.a.s. This hierarchy collapse is shown to be optimal, as certain problems cannot be solved by sound, non-uniform, anonymous algorithms in $1-o(1)$9 rounds (even for large $n\to\infty$0).

Implications and Theoretical Significance

Average-Case vs. Worst-Case in Distributed Symmetry and Computability

The results sharply contrast average-case and worst-case behavior for the computability of distributed tasks in anonymous settings. While worst-case input graphs stymie deterministic, anonymous protocols, random instances are typically sufficiently asymmetric such that a minimal, local symmetry-breaking protocol suffices to bootstrap the network into effectively the full power of identifier-based algorithms. This provides the first definitive answer to an open question posed by Gaudio et al. [GaudioRS25]: in random graphs, explicit node IDs are not required for the successful operation of symmetry-breaking-based distributed algorithms.

Robustness and Limitations of Weak Model Simulations

By extending the short-ID canonical labeling protocol to the broadest class of weak models, the work demonstrates robustness of distributed graph algorithm paradigms in random instances, and indicates that the inherent intractability of many classic distributed tasks (leader election, triangle detection, Hamiltonicity) in anonymous settings is, for nearly all graphs, an artifact of pathological symmetries rather than typical instances. However, the sharp impossibility results for sound deterministic algorithms delineate the boundaries of this robustness and underscore the persistent relevance of symmetry even in random settings under strong adversarial guarantees.

Future Directions

Several research avenues are opened by this work:

• Generalization to Local Graph Properties: Which other polynomial-time hard symmetric detection problems (clique-finding, long cycle detection, induced subgraph detection) can be efficiently solved in weak, anonymous models, or do they exhibit average-case hardness even for random graphs?
• Constant-Round Symmetry Breaking near Thresholds: For $n\to\infty$1 near the connectivity threshold, is it possible to generate unique identifiers (or sufficient asymmetry) in $n\to\infty$2 rounds in extremely weak anonymous models?
• Anonymous Models with Additional Resource Constraints: What is the power of sound, non-uniform anonymous algorithms in $n\to\infty$3 or $n\to\infty$4 with stricter (e.g., constant) message size bounds?
• Node-Average Complexity vs. Random Graphs: How do results for node-average complexity (average over nodes in the worst-case input) compare to the average-case analyses provided here for random graphs?

Conclusion

This paper conclusively establishes, through technical results and impossibility theorems, that symmetry is overwhelmingly rare in large random networks, and deterministic, anonymous distributed algorithms are a.a.s. as powerful as in the standard, identifier-equipped setting. The work effectively closes the gap between the worst-case and average-case landscapes for weak distributed computing models, and precisely characterizes the boundary where robust computability is restored or provably precluded in the absence of explicit symmetry-breaking mechanisms.

References:

"What can be computed in average anonymous networks?" (2604.02192)