Large deviations for triangles in scale-free random graphs

TL;DR

This paper analyzes the probability of observing many triangles in scale-free random graphs with heavy-tailed degree distributions, revealing phase transitions and different mechanisms depending on the tail index.

Contribution

It provides the first large deviations estimates for the upper tail of triangle counts in scale-free graphs with degrees following a power law, identifying phase transitions and the role of hubs.

Findings

Upper tail probabilities undergo a phase transition at /3.

For /3, tail decay is polynomial with finitely many hubs.

For \u2212, tail decay is semi-exponential with many hubs.

Abstract

We provide large deviations estimates for the upper tail of the number of triangles in scale-free inhomogeneous random graphs where the degrees have power law tails with index $-\alpha, \alpha \in (1,2)$. We show that upper tail probabilities for triangles undergo a phase transition. For $\alpha<4/3$, the upper tail is caused by many vertices of degree of order $n$, and this probability is semi-exponential. In this regime, additional triangles consist of two hubs. For $\alpha>4/3$ on the other hand, the upper tail is caused by one hub of a specific degree, and this probability decays polynomially in $n$, leading to additional triangles with one hub. In the intermediate case $\alpha=4/3$, we show polynomial decay of the tail probability caused by multiple but finitely many hubs. In this case, the additional triangles contain either a single hub or two hubs. Our proofs are partly based on various concentration inequalities. In particular, we tailor concentration bounds for empirical processes to make them well-suited for analyzing heavy-tailed phenomena in nonlinear settings.