Diffusion capacity analysis of complex network based on the cluster distribution

Understanding the influence of structural features on diffusive processes is a significant challenge in network science. Real-world systems often exhibit multiscale structures, in which individuals can be partitioned into densely connected clusters with sparser coupling among neighboring clusters. Such structures hinder the diffusion process across the clusters, leading to the diffusion being trapped within clusters. At the macro scale, diffusion is intimately linked to the size distribution patterns of clusters. This article presents the generalized Herfindahl-Hirschman Index for cluster distribution (GHI-CD), a concept that measures a network's capacity for diffusion. We provide an analysis of its mathematical properties and give the upper bound of the number of clusters and diffusion sources. Theoretical and simulation results demonstrate that a homogeneous distribution of clusters leads to a lower diffusion capacity. Furthermore, based on community detection, a close correlation between the GHI and the magnitude of diffusion is observed on synthetic and real networks, both of which are considerably impacted by cluster-heterogeneity. This GHI-based method can also track the evolution of diffusion capacity during community evolution. Our results demonstrate how cluster distribution offers a new and complementary perspective for analyzing diffusive dynamics on networks.