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arXiv · 2202.01563

On the Number of Graphs with a Given Histogram

Shahar Stein Ioushua·Ofer Shayevitz
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Abstract

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.

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Shahar Stein Ioushua, Ofer Shayevitz. 2022-02-03. On the Number of Graphs with a Given Histogram. https://doi.org/10.1109/tit.2023.3296827

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