Spectral Analysis Methods of Social Networks

Abstract

Online social networks (such as Facebook, Twitter, VKontakte, etc.) being an important channel for disseminating information are often used to arrange an impact on the social consciousness for various purposes - from advertising products or services to the full-scale information war thereby making them to be a very relevant object of research. The paper reviewed the analysis methods of social networks (primarily, online), based on the spectral theory of graphs. Such methods use the spectrum of the social graph, i.e. a set of eigenvalues of its adjacency matrix, and also the eigenvectors of the adjacency matrix. Described measures of centrality (in particular, centrality based on the eigenvector and PageRank), which reflect a degree of impact one or another user of the social network has. A very popular PageRank measure uses, as a measure of centrality, the graph vertices, the final probabilities of the Markov chain, whose matrix of transition probabilities is calculated on the basis of the adjacency matrix of the social graph. The vector of final probabilities is an eigenvector of the matrix of transition probabilities. Presented a method of dividing the graph vertices into two groups. It is based on maximizing the network modularity by computing the eigenvector of the modularity matrix. Considered a method for detecting bots based on the non-randomness measure of a graph to be computed using the spectral coordinates of vertices - sets of eigenvector components of the adjacency matrix of a social graph. In general, there are a number of algorithms to analyse social networks based on the spectral theory of graphs. These algorithms show very good results, but their disadvantage is the relatively high (albeit polynomial) computational complexity for large graphs. At the same time it is obvious that the practical application capacity of the spectral graph theory methods is still underestimated, and it may be used as a basis to develop new methods. The work was carried out with the support from the RFBR grant No. 16-29-09517.