Eigenvector Localization as a Tool to Study Small Communities in Online Social Networks

Abstract

We present and discuss a mathematical procedure for identification of small "communities" or segments within large bipartite networks. The procedure is based on spectral analysis of the matrix encoding network structure. The principal tool here is localization of eigenvectors of the matrix, by means of which the relevant network segments become visible. We exemplified our approach by analyzing the data related to product reviewing on Amazon.com. We found several segments, a kind of hybrid communities of densely interlinked reviewers and products, which we were able to meaningfully interpret in terms of the type and thematic categorization of reviewed items. The method provides a complementary approach to other ways of community detection, typically aiming at identification of large network modules.

Localization in random bipartite graphs: Numerical and empirical study

“…The ratio L/N is, for all three sizes, L/N ≃ 20. As we have already shown in [46], the degree distribution is power-law on both the A and B sides, P > A,B (k) ∼ k −γA,B , but the exponents slightly differ: we found γ A ≃ 1.2 and γ B ≃ 1.35. The density of eigenvalues is shown in Fig.…”

“…As a complement to the study of artificial bipartite graphs, we analyzed also one empirical bipartite graph, namely the network of reviewers and items on the amazon.com server. In our previous study [46] we observed the scale-free nature of this graph and by extracting the most localized eigenvectors we found small communities with sensible semantic information. Here we wanted to check if the model of Goh et al is useful also in describing the spectra of the graphs and properties of its eigenvectors.…”

“…The third reason for studying adjacency matrix lies in our previous empirical study of Amazon network [46], which was done using adjacency matrix. Small communities in the network were successfully found by studying localization.…”

“…We already studied several electronic-commerce networks in the past [46][47][48]. Here we shall reexamine the Amazon network [46]. It is a bipartite graph of reviewers on one side and items offered for sale on the other side.…”

Dynamics of User Networks in on-Line Electronic Auctions

“…Here we concentrate on social networks which emerge without explicit intention of the agents. We recently investigated the properties of a hybrid human-inanimate network composed of books, reviews and reviewers in the Amazon.com site [17]. The users do not interact directly through links as "friends" or "I like", but through the products of common interest.…”