Method of Graph Mining based on the Topological Anomaly Matrix and its Application for Discovering the Structural Peculiarities of Complex Networks

The article introduces the mathematical concept of the topological anomaly matrix providing the foundation for the qualitative assessment of the topological organization underlying the large-scale complex networks. The basic idea of the proposed concept consists in translating the distributions of the individual vertex-level characteristics (such as the degree, closeness, and betweenness centrality) into the integrative properties of the overall graph. The article analyzes the lower bounds imposed on the items of the topological anomaly matrix and obtains the new fundamental results enriching the graph theory. With a view to improving the interpretability of these results, the article introduces and proves the theorem regarding the smoothness of the closeness centrality distribution over the graph's vertices. By performing the series of experiments, the article illustrates the application of the proposed matrix for evaluating the topology of the real-world power grid network and its post-attack damage.