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Abstract: We study structure, eigenvalue spectra and diffusion dynamics in a wide classof networks with subgraphs modules at mesoscopic scale. The networks aregrown within the model with three parameters controlling the number of modules,their internal structure as scale-free and correlated subgraphs, and thetopology of connecting network. Within the exhaustive spectral analysis forboth the adjacency matrix and the normalized Laplacian matrix we identify thespectral properties which characterize the mesoscopic structure of sparsecyclic graphs and trees. The minimally connected nodes, clustering, and theaverage connectivity affect the central part of the spectrum. The number ofdistinct modules leads to an extra peak at the lower part of the Laplacianspectrum in cyclic graphs. Such a peak does not occur in the case oftopologically distinct tree-subgraphs connected on a tree. Whereas theassociated eigenvectors remain localized on the subgraphs both in trees andcyclic graphs. We also find a characteristic pattern of periodic localizationalong the chains on the tree for the eigenvector components associated with thelargest eigenvalue equal 2 of the Laplacian. We corroborate the results withsimulations of the random walk on several types of networks. Our results forthe distribution of return-time of the walk to the origin autocorrelatoragree well with recent analytical solution for trees, and it appear to beindependent on their mesoscopic and global structure. For the cyclic graphs wefind new results with twice larger stretching exponent of the tail of thedistribution, which is virtually independent on the size of cycles. Themodularity and clustering contribute to a power-law decay at short returntimes.



Author: Marija Mitrović, Bosiljka Tadić

Source: https://arxiv.org/



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