Network Statistics and the Graphs Spectra (André Fujita -- University of São Paulo)

Graphs/networks have become a powerful means of modeling data. If we have a set of items/components related to each other, we can represent them using a graph. Thus, we can find networks everywhere, from brains to social systems. These empirical networks present intrinsic randomness in opposition to deterministic graphs studied in graph theory. For example, functional brain networks are different even among individuals of the same category (e.g., healthy subjects). Besides, their structures change over time. In this case, comparing the presence/absence of edges or centrality measures will falsely distinguish networks belonging to the same group. Thus, the analyses of empirical networks using methods grounded on conventional graph theory seem inappropriate. When data present randomness, a natural solution would be to use statistical methodologies. Statistical approaches for networks are new, with few reports in the literature. One reason is that networks are hardly manipulable from a statistical viewpoint: they are not numbers. Thus, I propose a statistical framework based on the network spectrum. The spectrum is interesting because it codifies the information about the network structure. Based on the graph spectrum, I will show some extensions of well-established statistical tools for graphs: maximum likelihood estimator, model selection, t-test, analysis of variance, correlation, Granger causality, and a supervised classifier.