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Source localisation in simple random walks

Ritesh Goenka, Peter Keevash, Tomasz Przybyłowski

TL;DR

This work analyzes the problem of locating the origin of a simple random walk from its trace on infinite vertex-transitive graphs, establishing a clear dichotomy: localisation is possible with positive probability on strongly transient graphs (notably $\mathbb{Z}^d$ for $d\ge5$) using a simple diametral-path endpoint estimator, while localisation fails on recurrent graphs ($d\le2$). The authors develop a robust framework based on cut-edges, two-sided walk embeddings, and coupling arguments to obtain both lower bounds (via positive liminf probabilities) and sharp upper bounds (via intricate couplings) that match in high dimensions, with a detailed treatment of infinite-trace, range-input, and vertex-trace variants. A suite of variants demonstrates that high-accuracy localisation is achievable with a constant-size candidate set, and that meaningful localisation persists even when only partial trace information is available, with probabilities intricately tied to the dimension through quantities like $c(d)$ and $\tilde c(d)$. The results illuminate the information content of SRW traces for source inference on lattices and vertex-transitive graphs, with implications for diffusion-root detection in networks and related growth models.

Abstract

We consider the problem of locating the source (starting vertex) of a simple random walk, given a snapshot of the set of edges (or vertices) visited in the first $n$ steps. Considering lattices $\mathbb{Z}^d$, in dimensions $d \geq 5$, we show that the source can be identified (a) with probability bounded away from $0$ using one guess, and (b) with probability arbitrarily close to $1$ using a constant number of guesses. On the other hand, for dimensions $d \leq 2$, we show that one cannot locate the source with positive constant probability. Our arguments apply more generally to strongly transient and recurrent simple random walks on vertex-transitive graphs.

Source localisation in simple random walks

TL;DR

This work analyzes the problem of locating the origin of a simple random walk from its trace on infinite vertex-transitive graphs, establishing a clear dichotomy: localisation is possible with positive probability on strongly transient graphs (notably for ) using a simple diametral-path endpoint estimator, while localisation fails on recurrent graphs (). The authors develop a robust framework based on cut-edges, two-sided walk embeddings, and coupling arguments to obtain both lower bounds (via positive liminf probabilities) and sharp upper bounds (via intricate couplings) that match in high dimensions, with a detailed treatment of infinite-trace, range-input, and vertex-trace variants. A suite of variants demonstrates that high-accuracy localisation is achievable with a constant-size candidate set, and that meaningful localisation persists even when only partial trace information is available, with probabilities intricately tied to the dimension through quantities like and . The results illuminate the information content of SRW traces for source inference on lattices and vertex-transitive graphs, with implications for diffusion-root detection in networks and related growth models.

Abstract

We consider the problem of locating the source (starting vertex) of a simple random walk, given a snapshot of the set of edges (or vertices) visited in the first steps. Considering lattices , in dimensions , we show that the source can be identified (a) with probability bounded away from using one guess, and (b) with probability arbitrarily close to using a constant number of guesses. On the other hand, for dimensions , we show that one cannot locate the source with positive constant probability. Our arguments apply more generally to strongly transient and recurrent simple random walks on vertex-transitive graphs.
Paper Structure (18 sections, 33 theorems, 146 equations, 1 figure)

This paper contains 18 sections, 33 theorems, 146 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Localisation for the SRW
  3. Overview of ideas and methods
  4. Further results
  5. Organisation
  6. Proofs of the main results
  7. Notation
  8. Localisation for strongly transient walks
  9. Optimal bounds in large dimensions
  10. No localisation for recurrent walks
  11. Variants of localisation for the SRW
  12. High accuracy source localisation
  13. Infinite trace
  14. Range input
  15. Vertex trace only
  16. ...and 3 more sections

Key Result

Theorem 1.1

For any infinite vertex-transitive strongly transient graph $\mathbb{H}$ of finite degree, In particular, source localisation is possible if $\mathbb{H} = \mathbb{Z}^d$ for $d \geqslant 5$.

Figures (1)

  • Figure 1: Two walks producing the same trace. The left and right walks start at $u$ and $v$, respectively.

Theorems & Definitions (72)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 62 more