Are Large Random Graphs Always Safe to Hide?

TL;DR

This paper investigates the cops and robber game on large random graphs, revealing that logical expressibility of winning conditions correlates with almost certain victory, and provides insights into the logic-game relationship through zero-one laws.

Contribution

It demonstrates that winning conditions expressible in first-order logic almost always determine the outcome in large random graphs, linking logic and game theory.

Findings

Winning conditions in first-order logic lead to almost certain wins.

Provides a new perspective on the logic-game connection via zero-one laws.

Enhances understanding of network query robustness in large graphs.

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.