2008/09/30 by Marija Mitrović Dankulov, Marija Mitrović, Bosiljka Tadić · 1 citation
Physics and Astronomy · #Complex Network Analysis Techniques #Opinion Dynamics and Social Influence #Theoretical and Computational Physics #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.80.026123
published as Physical Review E 80, 026123 (2009)
openalex publication_date 2009/08/24 · arxiv created 2009/08/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study structure, eigenvalue spectra, and random-walk dynamics in a wide class of networks with subgraphs (modules) at mesoscopic scale. The networks are grown within the model with three parameters controlling the number of modules, their internal structure as scale-free and correlated subgraphs, and the topology of connecting network. Within the exhaustive spectral analysis for both the adjacency matrix and the normalized Laplacian matrix we identify the spectral properties, which characterize the mesoscopic structure of sparse cyclic graphs and trees. The minimally connected nodes, the clustering, and the average connectivity affect the central part of the spectrum. The number of distinct modules leads to an extra peak at the lower part of the Laplacian spectrum in cyclic graphs. Such a peak does not occur in the case of topologically distinct tree subgraphs connected on a tree whereas the associated eigenvectors remain localized on the subgraphs both in trees and cyclic graphs. We also find a characteristic pattern of periodic localization along the chains on the tree for the eigenvector components associated with the largest eigenvalue lambda(L)=2 of the Laplacian. Further differences between the cyclic modular graphs and trees are found by the statistics of random walks return times and hitting patterns at nodes on these graphs. The distribution of first-return times averaged over all nodes exhibits a stretched exponential tail with the exponent sigma approximately 1/3 for trees and sigma approximately 2/3 for cyclic graphs, which is independent of their mesoscopic and global structure.