Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Note on edge expansion and modularity in preferential attachment graphs

Colin McDiarmid, Katarzyna Rybarczyk, Fiona Skerman, Małgorzata Sulkowska

TL;DR

The paper addresses edge expansion and modularity in two standard preferential attachment graphs $G_n^h$ and $\tilde{G}_n^h$ (with $h\ge2$). It derives a whp, linear-in-$h$ lower bound on edge expansion for both models and uses small-set expansion to obtain whp upper bounds on modularity, notably $q^*(G) \le 0.92383$ for both models across $h\ge2$. A general bound connecting expansion to modularity is established: $q^*(G) \le 1 - \inf_{0<u\le 1/2} (\delta_u/(2+\delta_u) + u/2)$ with $\delta_u = \min\{\alpha_u(G)/h, 1\}$, enabling sharper modularity controls than prior work. Together, these results illuminate how network growth via preferential attachment constrains community structure and robustness, with practical implications for clustering and network design in graphs with high-degree hubs.

Abstract

Edge expansion is a parameter indicating how well-connected a graph is. It is useful for designing robust networks, analysing random walks or information flow through a network and is an important notion in theoretical computer science. Modularity is a measure of how well a graph can be partitioned into communities and is widely used in clustering applications. We study these two parameters in two commonly considered models of random preferential attachment graphs, with $h \geq 2$ edges added per step. We establish new bounds for the likely edge expansion for both random models. Using bounds for edge expansion of small subsets of vertices, we derive new upper bounds also for the modularity values for small $h$.

Note on edge expansion and modularity in preferential attachment graphs

TL;DR

The paper addresses edge expansion and modularity in two standard preferential attachment graphs and (with ). It derives a whp, linear-in- lower bound on edge expansion for both models and uses small-set expansion to obtain whp upper bounds on modularity, notably for both models across . A general bound connecting expansion to modularity is established: with , enabling sharper modularity controls than prior work. Together, these results illuminate how network growth via preferential attachment constrains community structure and robustness, with practical implications for clustering and network design in graphs with high-degree hubs.

Abstract

Edge expansion is a parameter indicating how well-connected a graph is. It is useful for designing robust networks, analysing random walks or information flow through a network and is an important notion in theoretical computer science. Modularity is a measure of how well a graph can be partitioned into communities and is widely used in clustering applications. We study these two parameters in two commonly considered models of random preferential attachment graphs, with edges added per step. We establish new bounds for the likely edge expansion for both random models. Using bounds for edge expansion of small subsets of vertices, we derive new upper bounds also for the modularity values for small .
Paper Structure (13 sections, 15 theorems, 69 equations)

This paper contains 13 sections, 15 theorems, 69 equations.

Table of Contents

  1. Introduction and statement of results
  2. Introduction
  3. Basic definitions
  4. The preferential attachment models
  5. Expansion of the preferential attachment models.
  6. Modularity of the preferential attachment models.
  7. Proofs
  8. Proof of Theorem \ref{['thm:our_u_expansion_fussy']} and Corollary \ref{['cor:exp_linear_in_h']}: edge expansion.
  9. Proofs of Lemma \ref{['lem:us_upper_bound']}, Theorem \ref{['thm:general_upper_bound_mod_2']} and Corollary \ref{['cor.q_upper_bound']}: modularity.
  11. Appendix
  12. Proof of Lemma 2 of mihail2006journal for $\mathbf{G_n^h}$
  13. Proof of Lemma \ref{['lem:dull']}

Key Result

theorem 1.1

Let $h \geq 2$ and $G \sim \tilde{G}_n^h$. Then for any $c$ with $0<c<2(h-1)-1$ if $\,\hat{\alpha} < \min\left\{ \frac{h-1}{2}-\frac{c+1}{4}, \frac{1}{5}, \frac{(h-1)\ln 2 -\frac{2}{5}\ln 5}{2(\ln h +\ln 2 +1)}\right\}$ then

Theorems & Definitions (28)

  • definition 1.1: Expansion and $u$-bounded expansion
  • definition 1.2: Modularity, NeGi04
  • theorem 1.1: Theorem 1 of mihail2006journal
  • theorem 1.2
  • corollary 1.1
  • proposition 1.1: prokhorenkova2017modularity_internet_Math
  • lemma 1.1
  • remark 1.1
  • theorem 1.3
  • corollary 1.2
  • ...and 18 more