Probability theory and mathematical statistics

Note on normal approximation for number of triangles in heterogeneous Erdős-Rényi graph

A. V. Logachov^ab, A. A. Mogulskii^a, A. A. Yambartsev<sup>c

a</sup> Lab. of Probability Theory and Math. Statistics, Sobolev Institute of Mathematics, pr. Koptyuga, 4, 630090, Novosibirsk, Russia
^b Dep. of Computer Science in Economics, Novosibirsk State Technical University pr. K. Marksa, 20, 630073, Novosibirsk, Russia
^c Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão, 1010, CEP o5508-090, São Paulo, SP, Brazil

Abstract: We obtain a bound for the convergence rate in the central limit theorem for the number of triangles in a heterogeneous Erdős-Rényi graphs. Our approach is reminiscent of Hoeffding decomposition (a common technique in the theory of U-statistics). We show that the centered and normalized number of triangles asymptotically behaves as the normalized sum of centered independent random variables when the number of vertices increases. The proposed method is simple and intuitive.

Keywords: Erdős-Rényi random graphs, central limit theorem, large deviations principle.

UDC: 519.21

MSC: 05C80, 60F05, 60F10

Received March 7, 2024, published November 1, 2024

Language: English

DOI: 10.33048/semi.2024.21.060