Are Large Random Graphs Always Safe to Hide?

2025-11-27

AI Summary

This study employs the Cops-and-Robbers game as a structured benchmark for network query and search problems, focusing on characterizing winning probabilities for each player on large random graphs. Method: Leveraging first-order logic (FO) to formalize winning conditions, we establish—via the zero-one law—a rigorous correspondence between logical expressibility and asymptotic winning probability: if a player’s winning condition is definable in FO, then that player wins with probability tending to 1 on almost all Erdős–Rényi random graphs (G(n,p)). Our approach integrates tools from random graph theory, probabilistic analysis, combinatorial game theory, and logical semantics. Contribution/Results: This work provides the first formal link between dynamic pursuit-evasion games and first-order definability, yielding a precise logical criterion for network stealth and searchability. It advances the theoretical interface of logic and graph algorithms, offering foundational insights into the expressive limits of logical formalisms in modeling distributed search and adversarial network behavior.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Adversarial Search
• Game Theory and Economic Paradigms: Cooperative Game Theory

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Social mining and social search on the Web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Analyzes cops and robber game on large random graphs
• Uses logic frameworks to study winning conditions
• Explores zero-one law for logic-game connections

Innovation

• Using logic frameworks for game analysis
• First-order logic formulas determine winning conditions
• Zero-one law perspective connects logic and games