Exponential inequalities for the number of triangles in the Erd\"{o}s-R\'{e}nyi random graph

TL;DR

This paper derives exponential tail bounds for the number of triangles in Erdős-Rényi graphs, providing probabilistic inequalities that quantify the likelihood of deviations from the expected number.

Contribution

It introduces new exponential inequalities for triangle counts in Erdős-Rényi graphs, formulated via random fields, advancing probabilistic understanding of subgraph counts.

Findings

Established upper exponential tail bounds for triangle counts.

Quantified the probability of large deviations in triangle numbers.

Formulated results in the framework of random fields.

Abstract

Upper exponential inequalities for the tail probabilities of the centered and normalized number of triangles in the Erd\"{o}s-R\'{e}nyi graph are obtained, where the probability of every edge is fixed. The result is formulated in terms of random fields.