Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Large deviations for subgraphs in inhomogeneous random graphs

Riccardo Michielan, Clara Stegehuis, Bert Zwart

TL;DR

This paper investigates the large deviations of subgraph counts in inhomogeneous random graphs by defining a specific optimization problem that captures the most likely way to generate numerous additional subgraphs.

Abstract

Inhomogeneous random graphs are fundamental models for real-world networks, where prescribed degrees are imposed as soft constraints. A common assumption in such models is that the degree distribution follows a power-law, capturing the heavy-tailed nature observed in many contexts. While various graph functionals have been studied in this setting, inhomogeneity makes their analysis significantly more challenging. The goal of this paper is to investigate the large deviations of subgraph counts in inhomogeneous random graphs. Rare events concerning these functionals translate into quantifying the probability that extremely large hubs appear in the graph. This can be achieved by defining a specific optimization problem that captures the most likely way to generate numerous additional subgraphs. When the expected number of subgraphs is sublinear in the graph size, polynomially large deviations are possible, and in this case, we can derive sharp results on clique counts.

Large deviations for subgraphs in inhomogeneous random graphs

TL;DR

This paper investigates the large deviations of subgraph counts in inhomogeneous random graphs by defining a specific optimization problem that captures the most likely way to generate numerous additional subgraphs.

Abstract

Inhomogeneous random graphs are fundamental models for real-world networks, where prescribed degrees are imposed as soft constraints. A common assumption in such models is that the degree distribution follows a power-law, capturing the heavy-tailed nature observed in many contexts. While various graph functionals have been studied in this setting, inhomogeneity makes their analysis significantly more challenging. The goal of this paper is to investigate the large deviations of subgraph counts in inhomogeneous random graphs. Rare events concerning these functionals translate into quantifying the probability that extremely large hubs appear in the graph. This can be achieved by defining a specific optimization problem that captures the most likely way to generate numerous additional subgraphs. When the expected number of subgraphs is sublinear in the graph size, polynomially large deviations are possible, and in this case, we can derive sharp results on clique counts.
Paper Structure (18 sections, 12 theorems, 115 equations, 1 figure)

This paper contains 18 sections, 12 theorems, 115 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation.
  3. Organization of the paper.
  4. Main results
  5. Model
  6. Large Deviations for Clique Counts
  7. Deviations for subgraph counts
  8. Discussion
  9. Unconditional subgraph deviations.
  10. Other subgraphs.
  11. General $U$-statistics.
  12. Exponential deviations.
  13. LD for conditional number of cliques
  14. Contribution from big hubs
  15. LD for conditional expected cliques
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let $\alpha \in (1,2)$. Then for some constant $H > 0$. In particular, $m_n^{(k)}$ is regularly varying with index $k(2-\alpha)/2$.

Figures (1)

  • Figure 1: Deviations for all subgraphs of size 6 for $\alpha=1.5$, $\gamma = 2$. The colors indicate the optimal $\beta$ values of \ref{['eq:opt_poly_constr']}, and the subgraph captions demonstrate $R(H)$.

Theorems & Definitions (22)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Proposition 4.1
  • Lemma 4.2: Less than $k-2$ hubs
  • proof
  • Lemma 4.3: $k-1$ hubs
  • proof
  • Lemma 4.4: $k-2$ hubs
  • ...and 12 more