Statistical Methods for Directed Graphs Basead on the Graph Spectrum - Villela, Victor Chavauty (2024)
Graphs are often used to model diverse, complex phenomena in many fields, such as social networks, brain region connectivity, or protein-protein interaction. However, due to the complexity of their structure, standard statistical methods are insufficient in searching for a correlation between populations of graphs. In a recent paper by Takahashi et al. (2012), they suggested that the graph spectrum is a good fingerprint of the graphs structure, and they developed several statistical methods based on this feature. These methods, however, rely on the distribution of the eigenvalues of the graph being real-valued, which is false when graphs are directed or weighted. In this thesis, we extend their results to directed graphs by working with the distribution of complex eigenvalues instead. Then, we constructed a parameter estimator, a model selection approach, a statistical test to compare two populations of graphs, a test for association between a graph and variables of interest, and a clustering algorithm.