Spectral and dynamical properties in classes of sparse networks with mesoscopic inhomogeneities

Abstract

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, 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 λ 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 σ ≈ 1/3 for trees and σ ≈ 2/3 for cyclic graphs, which is independent on their mesoscopic and global structure.

Co-Evolutionary Mechanisms of Emotional Bursts in Online Social Dynamics and Networks

“…The resulting (monopartite) network is symmetrical, weighted and strongly clustered; a number of such networks obtained from Blogs data and projected onto either user or post partition, have been studied in [10,27,28]. The community structure of the user-projected networks can be readily detected by the eigenvalue spectral analysis [57]; the occurrence of communities is strongly related with the weights of the links. By identifying all users within a given community, and then filtering out their actions and texts of their comments from the original data, one can find two prevailing patterns of activity [10,12,27].…”

“…The eigenvector localization is visualized as a characteristic branched structure of the scatter-plot in the space of these eigenvectors. This property of the eigenvectors is then utilized to identify the nodes of the network that belong to each community [10,12,26,57]). The scatter-plot of three eigenvectors belonging to the three lowest eigenvalues of the Laplacian is shown in Figure 8a.…”

“…In order to determine the community structure, we perform the spectral analysis of the normalized Laplacian operator [57,61] which is related to the weighted agent-projection network. Its matrix elements C W ij represent the common number of posts per pair of agents, including the multiplicity of user-post connections, which is indicated by the superscript.…”

How the online social networks are used: dialogues-based structure ofMySpace

“…Specifically, we determine several topology measures at global network level (distributions of degree, strength, weight, betweenness) and at local node-neighbourhood level (assortativity measures and tests of social weak-tie hypothesis), as well as the mesoscopic level (community structure). For the community structure analysis in compact and relatively small networks, we apply the eigenvalue spectral analysis of the Laplacian operator related to the weighted symmetrical adjacency matrix [32]. In the case of large graphs, the communities are detected by Gephi software, which uses a maximum modularity approach [33].…”

Randomized Algorithms for Matrices and Data

“…See, e.g., [189,190,191,192] for detailed discussions and empirical evaluations. Depending on whether one is considering the adjacency matrix or the Laplacian matrix, localized eigenvectors can correspond to structural inhomogeneities such as very high degree nodes or very small cluster-like sets of nodes.…”