2024/03/21 by Ethan Young, Mason A. Porter, Young, Ethan +1
Physics and Astronomy · #Adaptation and Self-Organizing Systems (nlin.AO) #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Opinion Dynamics and Social Influence #Physics and Society (physics.soc-ph) #Social and Information Networks (cs.SI)
paper · pdf · doi:10.48550/arxiv.2403.14584
openalex publication_date 2024/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The leading eigenvalue λ of the adjacency matrix of a graph exerts much influence on the behavior of dynamical processes on that graph. It is thus relevant to relate notions of the importance (specifically, centrality measures) of network structures to λ and its associated eigenvector. We study a previously derived measure of edge importance known as ``dynamical importance'', which estimates how much λ changes when one removes an edge from a graph or adds an edge to it. We examine the accuracy of this estimate for different network structures and compare it to the true change in λ after an edge removal or edge addition. We then derive a first-order approximation of the change in the leading eigenvector. We also consider the effects of edge additions on Kuramoto dynamics on networks, and we express the Kuramoto order parameter in terms of dynamical importance. Through our analysis and computational experiments, we find that studying dynamical importance can improve understanding of the relationship between network perturbations and dynamical processes on networks.