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Sparse random graphs with many triangles

Suman Chakraborty, Remco van der Hofstad, Frank den Hollander

TL;DR

This work analyzes the sparse Erdős-Rényi model with $p_n=\lambda/n$ to quantify how rare it is to have many triangles and how many vertices participate in triangles. Using nonlinear large deviations, variational methods, and a seed/core decomposition, it derives sharp upper and lower bounds for the probability of observing $T\ge a k_n$ triangles and shows conditioning on many triangles concentrates triangles in a near-clique while preserving the local $G_{n,p_n}$ structure. It also establishes a distinct large-deviation rate for the event $V_T(G_{n,p_n})\ge k_n$, namely $\log \mathbb{P}(V_T\ge k_n) \approx -\tfrac{1}{3} k_n \log k_n$, and discusses the implications for real-world sparse networks and exponential random graphs, including a phase transition in partition-function scaling when modeling by $V_T$. The results illuminate how sparse clustering can arise and be modeled, and they provide rigorous tools for analyzing sparse ERGMs and phase transitions in network models. Overall, the paper advances understanding of extreme clustering in sparse graphs and offers a rigorous basis for studying sparse, highly clustered networks via nonlinear large deviations and variational principles.

Abstract

In this paper we consider the Erdős-Rényi random graph in the sparse regime in the limit as the number of vertices $n$ tends to infinity. We are interested in what this graph looks like when it contains many triangles, in two settings. First, we derive asymptotically sharp bounds on the probability that the graph contains a large number of triangles. We show that conditionally on this event, with high probability the graph contains an almost complete subgraph, i.e., the triangles form a near-clique, and has the same local limit as the original Erdős-Rényi random graph. Second, we derive asymptotically sharp bounds on the probability that the graph contains a large number of vertices that are part of a triangle. If order $n$ vertices are in triangles, then the local limit (provided it exists) is different from that of the Erdős-Rényi random graph. Our results shed light on the challenges that arise in the description of real-world networks, which often are sparse, yet highly clustered, and on exponential random graphs, which often are used to model such networks.

Sparse random graphs with many triangles

TL;DR

This work analyzes the sparse Erdős-Rényi model with to quantify how rare it is to have many triangles and how many vertices participate in triangles. Using nonlinear large deviations, variational methods, and a seed/core decomposition, it derives sharp upper and lower bounds for the probability of observing triangles and shows conditioning on many triangles concentrates triangles in a near-clique while preserving the local structure. It also establishes a distinct large-deviation rate for the event , namely , and discusses the implications for real-world sparse networks and exponential random graphs, including a phase transition in partition-function scaling when modeling by . The results illuminate how sparse clustering can arise and be modeled, and they provide rigorous tools for analyzing sparse ERGMs and phase transitions in network models. Overall, the paper advances understanding of extreme clustering in sparse graphs and offers a rigorous basis for studying sparse, highly clustered networks via nonlinear large deviations and variational principles.

Abstract

In this paper we consider the Erdős-Rényi random graph in the sparse regime in the limit as the number of vertices tends to infinity. We are interested in what this graph looks like when it contains many triangles, in two settings. First, we derive asymptotically sharp bounds on the probability that the graph contains a large number of triangles. We show that conditionally on this event, with high probability the graph contains an almost complete subgraph, i.e., the triangles form a near-clique, and has the same local limit as the original Erdős-Rényi random graph. Second, we derive asymptotically sharp bounds on the probability that the graph contains a large number of vertices that are part of a triangle. If order vertices are in triangles, then the local limit (provided it exists) is different from that of the Erdős-Rényi random graph. Our results shed light on the challenges that arise in the description of real-world networks, which often are sparse, yet highly clustered, and on exponential random graphs, which often are used to model such networks.
Paper Structure (35 sections, 29 theorems, 137 equations)

This paper contains 35 sections, 29 theorems, 137 equations.

Table of Contents

  1. Introduction and main results
  2. Background and motivation
  3. Triangles.
  4. Subgraphs.
  5. Back to triangles.
  6. Real-world networks.
  7. Organisation of this section.
  8. Introductory notation
  9. Probability to have many triangles
  10. Graph structure conditionally on having many triangles
  11. Large deviations for the number of vertices in triangles
  12. Discussion
  13. Large deviations for the number of triangles.
  14. Large deviations for the number of vertices in triangles.
  15. Extension: Local limit Erdős-Rényi conditioned on many vertices in triangles.
  16. ...and 20 more sections

Key Result

Theorem 1.2

[Large deviations for the number of triangles] Suppose that $\lim_{n\to\infty} \varepsilon_n = 0$. Fix a sequence of positive reals $(w_n)_{n\in\mathbb{N}}$ such that $\lim_{n\to\infty} w_n = 0$, and such that $w_nk_n \geq {n \choose 3} (p_n)^C$ for some constant $C>0$, $\lim_{n\to\infty} (\log{n})/ for a sequence $\psi_n = o(1)$, where, for $\lim_{n\to\infty}$$w_nk_n^{1/3}=\infty$,

Theorems & Definitions (63)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Corollary 1.10
  • ...and 53 more