Scale-dependent measure of network centrality from diffusion dynamics

Abstract

Classic measures of graph centrality capture distinct aspects of node importance, from the local (e.g., degree) to the global (e.g., closeness). Here we exploit the connection between diffusion and geometry to introduce a multiscale centrality measure. A node is defined to be central if it breaks the metricity of the diffusion as a consequence of the effective boundaries and inhomogeneities in the graph. Our measure is naturally multiscale, as it is computed relative to graph neighborhoods within the varying time horizon of the diffusion. We find that the centrality of nodes can differ widely at different scales. In particular, our measure correlates with degree (i.e., hubs) at small scales and with closeness (i.e., bridges) at large scales, and also reveals the existence of multicentric structures in complex networks. By examining centrality across scales, our measure thus provides an evaluation of node importance relative to local and global processes on the network.

Unfolding the multiscale structure of networks with dynamical Ollivier-Ricci curvature

“…In turn, one may infer the network structure by observing properties of their evolution. We focus on Markov diffusion processes 8,[22][23][24] , a class of linear dynamical systems which is rich enough to capture several properties of nonlinear processes on networks 25,26 . On a connected network G weighted by pairwise distances w ij , the continuous time diffusion is constructed by the standard procedure 27 of defining the normalised graph Laplacian matrix L := K −1 (K − A), where K is the diagonal matrix of node degrees with K ii = j A ij and A is the weighted adjacency matrix encoding similarities between nodes.…”

HCGA: Highly comparative graph analysis for network phenotyping

“…Among many others, examples include linking the structural motifs of proteins and their function, 2 , 3 , 4 aiding the diagnosis of diseases using fMRI data, 5 understanding structural properties of organic crystal structures for electron transport, 6 or modeling network flows, e.g., city traffic, 7 information (or misinformation) spread in a social network, 8 , 9 or topic affinity in a citation network. 10 The growing importance of such network data has driven the development of a multitude of methods for investigating and revealing relevant topological, combinatorial, statistical, and spectral properties of graphs, e.g., node centralities, 11 , 12 assortativity, 13 , 14 path-based properties, 15 graph distance measures, 16 , 17 connectivity, 18 or community detection, 19 , 20 to name but a few in the highly interdisciplinary area of network science.…”

hcga: Highly Comparative Graph Analysis for network phenotyping

“…Among many others, examples include linking the structural motifs of proteins and their function [2,3,4], aiding the diagnosis of diseases using fMRI data [5], understanding structural properties of organic crystal structures for electron transport [6], or modelling network flows, e.g., city traffic [7], information (or misinformation) spread in a social network [8,9], or topic affinity in a citation network [10]. The growing importance of such network data has driven the development of a multitude of methods for investigating and revealing relevant topological, combinatorial, statistical and spectral properties of graphs, e.g., node centralities [11,12], assortativity [13,14], path-based properties [15], graph distance measures [16,17], connectivity [18], or community detection [19,20], to name but a few in the highly interdisciplinary area of network science.…”