On the Number of Graphs with a Given Histogram

Shahar Stein Ioushua, Ofer Shayevitz

Research output: Contribution to journalArticlepeer-review


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 distribution at the center of the Kolmogorov-Smirnov ball.

Original languageEnglish
Pages (from-to)6909-6921
Number of pages13
JournalIEEE Transactions on Information Theory
Issue number11
StatePublished - 1 Nov 2023


  • Graph theory
  • anticoncentration
  • maximum entropy
  • method of types
  • regularity lemma

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Library and Information Sciences
  • Computer Science Applications


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