Technology Category
Reasoning under Uncertainty: Graphical ModelsData Mining & Knowledge Management: Graph Mining, Social Network Analysis & CommunityMachine Learning: Graph-based Machine Learning
Functionality of Random Graphs
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
Random Graphs
Vertex Labeling
Graph Properties
Innovation
Methods, ideas, or system contributions that make the work stand out.
Functional Concepts in Graphs
Random Graphs G(n,p)
Vertex Labeling Solutions
🔎 Similar Papers
No similar papers found.
