On the Number of Graphs with a Given Histogram

TL;DR

This paper investigates the number of large dense graphs sharing similar local subgraph participation distributions, providing bounds based on a maximum entropy approach under global density constraints.

Contribution

It introduces a novel maximum entropy framework to estimate the count of graphs with similar local subgraph distributions, extending understanding of graph local structure equivalence.

Findings

Derived asymptotic bounds on graph counts with similar local distributions

Connected bounds to a maximum entropy problem with density constraints

Quantified how bounds tighten as the Kolmogorov-Smirnov distance decreases

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.