## 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 distribution at the center of the Kolmogorov-Smirnov ball.

Original language | English |
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Pages (from-to) | 6909-6921 |

Number of pages | 13 |

Journal | IEEE Transactions on Information Theory |

Volume | 69 |

Issue number | 11 |

DOIs | |

State | Published - 1 Nov 2023 |

## Keywords

- 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