Note on edge expansion and modularity in preferential attachment graphs

TL;DR

This paper investigates edge expansion and modularity in preferential attachment graphs, providing new bounds for these parameters which are crucial for understanding network robustness, community structure, and information flow.

Contribution

It establishes new bounds on edge expansion and modularity in preferential attachment models with multiple edges added per step, advancing theoretical understanding.

Findings

New bounds for edge expansion in preferential attachment graphs.

Upper bounds for modularity in small subsets of these graphs.

Insights into network robustness and community detection.

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 $h$.