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Are Large Random Graphs Always Safe to Hide?

Sourav Chakraborty, Sujata Ghosh, Smiha Samanta

TL;DR

The paper investigates the cops and robber game on large random graphs, leveraging the zero-one law for first-order logic to show that, for FO-expressible winning conditions, one player almost surely dominates as graph size grows. It first reviews foundational zero-one results and extension axioms, then applies them to constant and varying edge-probability random graphs, analyzing several variants (traps, roadblocks, edge-set differences, tandem-cops). Across constant p, the robber almost surely wins for standard and several variants, while tandem-cops restore a winning strategy for the cops in large graphs; under varying p(N), threshold functions and regime-based outcomes are derived, including for complementary-edge scenarios. The work highlights a deep link between logic, probability, and pursuit-evasion games, and outlines directions for future exploration of non-monotone cases and broader random graph models.

Abstract

We discuss winning possibilities of players in various variants of cops and robber game played on large random graphs, a testbed for various kinds of network queries, search problems in particular. We explore the use of logic frameworks to investigate such results; in particular, we show that whenever a winning condition for either player can be expressed as a certain kind of formula in first-order logic, that player almost always wins. In the process, we obtain more insight into the logic-game connection from the zero-one law perspective.

Are Large Random Graphs Always Safe to Hide?

TL;DR

The paper investigates the cops and robber game on large random graphs, leveraging the zero-one law for first-order logic to show that, for FO-expressible winning conditions, one player almost surely dominates as graph size grows. It first reviews foundational zero-one results and extension axioms, then applies them to constant and varying edge-probability random graphs, analyzing several variants (traps, roadblocks, edge-set differences, tandem-cops). Across constant p, the robber almost surely wins for standard and several variants, while tandem-cops restore a winning strategy for the cops in large graphs; under varying p(N), threshold functions and regime-based outcomes are derived, including for complementary-edge scenarios. The work highlights a deep link between logic, probability, and pursuit-evasion games, and outlines directions for future exploration of non-monotone cases and broader random graph models.

Abstract

We discuss winning possibilities of players in various variants of cops and robber game played on large random graphs, a testbed for various kinds of network queries, search problems in particular. We explore the use of logic frameworks to investigate such results; in particular, we show that whenever a winning condition for either player can be expressed as a certain kind of formula in first-order logic, that player almost always wins. In the process, we obtain more insight into the logic-game connection from the zero-one law perspective.

Paper Structure

This paper contains 9 sections, 15 theorems, 11 equations, 4 figures.

Table of Contents

  1. Introduction
  2. On zero-one laws in first-order logic
  3. Hiding on random graphs with constant edge probability
  4. Cops and robber games with traps and roadblocks
  5. Cops and Robbers with different edge set
  6. Cops and robber game with tandem-cops
  7. Hiding on random graphs with varying edge probability
  8. Other variants
  9. Concluding remarks

Key Result

Theorem 1

Any sentence in first-order logic with only a binary relation symbol in the vocabulary is either almost surely true or almost surely false.

Figures (4)

  • Figure 1: $EA_{3,8}$ : for all possible choice of vertices $x_1, x_2, \ldots x_8$ we can always find a vertex $z$ that is adjacent to only 3 of those 8 vertices.
  • Figure 2: The robber at $y$ has a move to a vertex $z$ avoiding the cop at $x.$
  • Figure 3: Petersen Graph
  • Figure 4: The tandem-cops can catch the robber in a single move.

Theorems & Definitions (31)

  • Definition 1
  • Definition 2
  • Theorem 1: Rfagin1976
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 1
  • Theorem 4
  • ...and 21 more