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Large deviations for triangles in scale-free random graphs

Clara Stegehuis, Bert Zwart

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 $-α, α\in (1,2)$. We show that upper tail probabilities for triangles undergo a phase transition. For $α<4/3$, 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 $α>4/3$ 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 $α=4/3$, 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.

Large deviations for triangles in scale-free random graphs

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 . We show that upper tail probabilities for triangles undergo a phase transition. For , the upper tail is caused by many vertices of degree of order , and this probability is semi-exponential. In this regime, additional triangles consist of two hubs. For on the other hand, the upper tail is caused by one hub of a specific degree, and this probability decays polynomially in , leading to additional triangles with one hub. In the intermediate case , 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.
Paper Structure (22 sections, 28 theorems, 209 equations, 3 figures)

This paper contains 22 sections, 28 theorems, 209 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Notation.
  3. Model description
  4. Main results and discussion
  5. Organization of the paper
  6. Nonlinear heavy-tailed large deviations
  7. Concentration of weighted empirical distribution functions
  8. Estimating the number of triangles generated by large hubs
  9. $G_n$ cannot be large without a big hub
  10. Proof of Theorem \ref{['thm-gn']}
  11. The boundary case $\alpha=4/3$
  12. Completing the proofs of Theorem \ref{['thm-triangle-singlebigjump']} and \ref{['thm-triangleboundary']}
  13. Proof of Lemma \ref{['lem:expconditionedtausmallnew']}
  14. Proof of Lemma \ref{['lem-lb-additionaltriangles']}
  15. Many dominating hubs
  16. ...and 7 more sections

Key Result

Lemma 1.1

Let $\alpha \in (1,2)$. $m_n = (1+o(1))H n^3 (\bar{F}(\sqrt{n}))^3$, with In particular, $m_n$ is regularly varying with index $3- \frac{3}{2} \alpha$.

Figures (3)

  • Figure 1: Illustration of the events that cause polynomial and exponential deviations.
  • Figure 2: The exponent of Theorem \ref{['thm:thetasmall']} plotted against $\theta$ for several values of $\alpha$.
  • Figure 3: Illustration of the sets $A$, $B$ and $B_1$, $B_2$, $B_3$

Theorems & Definitions (48)

  • Lemma 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 2.1
  • Proposition 2.2
  • Lemma 2.3
  • ...and 38 more