Note on edge expansion and modularity in preferential attachment graphs

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 $h$.