TY - GEN

T1 - On the Number of Graphs with a Given Histogram

AU - Ioushua, Shahar Stein

AU - Shayevitz, Ofer

N1 - Publisher Copyright:
© 2022 IEEE.

PY - 2022

Y1 - 2022

N2 - 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 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.

AB - 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 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.

UR - http://www.scopus.com/inward/record.url?scp=85136305283&partnerID=8YFLogxK

U2 - 10.1109/ISIT50566.2022.9834629

DO - 10.1109/ISIT50566.2022.9834629

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AN - SCOPUS:85136305283

T3 - IEEE International Symposium on Information Theory - Proceedings

SP - 456

EP - 461

BT - 2022 IEEE International Symposium on Information Theory, ISIT 2022

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 2022 IEEE International Symposium on Information Theory, ISIT 2022

Y2 - 26 June 2022 through 1 July 2022

ER -