Note on edge expansion and modularity in preferential attachment graphs
🤖 AI Summary
This study investigates the theoretical bounds on edge expansion and modularity in preferential attachment random graphs where each new node introduces $h \geq 2$ edges, aiming to elucidate their connectivity and community structure properties. Employing probabilistic graph theory and combinatorial analysis, the work establishes novel probabilistic bounds on the edge expansion of small vertex subsets for small values of $h$, and leverages these results to derive a tighter global upper bound on modularity. These findings advance the theoretical understanding of structural evolution in dynamic networks and provide new analytical foundations for assessing network robustness and community detection.
Application Category
📝 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 $h$.
Problem
Research questions and friction points this paper is trying to address.
edge expansion
modularity
preferential attachment graphs
random graphs
graph connectivity
Innovation
Methods, ideas, or system contributions that make the work stand out.
edge expansion
modularity
preferential attachment graphs
random graph models
community detection
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