A Study on Relationship between Modularity and Diffusion Dynamics in Networks from Spectral Analysis Perspective

Modular structure is a typical structure that is observed in most of real networks. Diffusion dynamics in network is getting much attention because of dramatic increasing of the data flows via the www. The diffusion dynamics in network have been well analysed as probabilistic process, but the proposed frameworks still shows the difference from the real observations. In this paper, we analysed spectral properties of the networks and diffusion dynamics. Especially, we focus on studying the relationship between modularity and diffusion dynamics. Our analysis as well as simulation results show that the relative influences from the non-largest eigenvalues and the corresponding eigenvectors increase when modularity of network increases. These results have the implication that, although network dynamics have been often analysed with the approximation manner utilizing only the largest eigenvalue, the consideration of the other eigenvalues is necessary for the analysis of the network dynamics on real networks. We also investigated Node-level Eigenvalue Influence Index (NEII) which can quantify the relative influence from each eigenvalues on each node. This investigation indicates that the influence from each eigenvalue is confined within the modular structures in the network. These findings should be made consideration by researchers interested in diffusion dynamics analysis on real networks for deeper analysis. Keywords—complex network; modularity; diffusion; SIS model; graph spectra; eigenvalue and eigenvector