On the Number of Graphs with a Given Histogram

Let \(G\) be a large (simple, unlabeled) dense graph on \(n\) vertices. Suppose that we only know, or can estimate, the empirical distribution of the number of subgraphs \(F\) that each vertex in \(G\) participates in, for some fixed small graph \(F\). How many other graphs would look essentially th...

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Bibliographic Details
Published inarXiv.org
Main Authors Shahar Stein Ioushua, Shayevitz, Ofer
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 07.08.2023
Subjects
Apexes
Computer Science - Information Theory
Graph theory
Graphs
Histograms
Lower bounds
Mathematics - Information Theory
Maximum entropy
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Summary:Let \(G\) be a large (simple, unlabeled) dense graph on \(n\) vertices. Suppose that we only know, or can estimate, the empirical distribution of the number of subgraphs \(F\) that each vertex in \(G\) participates in, for some fixed small graph \(F\). How many other graphs would look essentially the same to us, i.e., would have a similar local structure? In this paper, we derive upper and lower bounds on the number of graphs whose empirical distribution lies close (in the Kolmogorov-Smirnov distance) to that of \(G\). Our bounds are given as solutions to a maximum entropy problem on random graphs of a fixed size \(k\) that does not depend on \(n\), under \(d\) global density constraints. The bounds are asymptotically close, with a gap that vanishes with \(d\) at a rate that depends on the concentration function of the center of the Kolmogorov-Smirnov ball.
ISSN:2331-8422
DOI:10.48550/arxiv.2202.01563