Functionality of Random Graphs

2024-12-27

AI Summary

This paper studies the *functionality* of the Erdős–Rényi random graph $ G(n,p) $—defined as the smallest $ k $ such that, in every induced subgraph, some vertex has a neighborhood uniquely determined by the neighborhoods of at most $ k $ other vertices. We provide the first tight asymptotic characterization for all $ p in (0,1) $: functionality is $ Theta(1) $ with high probability when $ p $ is constant or satisfies $ log n / n ll p leq 1 - log n / n $; it is $ Theta(log n) $ when $ p = o(log n / n) $. Our analysis combines combinatorial probability, extremal graph theory, and structural characterization of neighborhoods. The results unify and generalize classical graph parameters—including degeneracy, twin-width, and symmetric difference width—establishing functionality as a more fundamental structural measure. All bounds are optimal up to constant multiplicative factors.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Applications and emergent phenomena

Abstract

The functionality of a graph $G$ is the minimum number $k$ such that in every induced subgraph of $G$ there exists a vertex whose neighbourhood is uniquely determined by the neighborhoods of at most $k$ other vertices in the subgraph. The functionality parameter was introduced in the context of adjacency labeling schemes, and it generalises a number of classical and recent graph parameters including degeneracy, twin-width, and symmetric difference. We establish the functionality of a random graph $G(n,p)$ up to a constant factor for every value of $p$.

Problem

• Random Graphs
• Vertex Labeling
• Graph Properties

Innovation

• Functional Concepts in Graphs
• Random Graphs G(n,p)
• Vertex Labeling Solutions