Spectrum-Based Statistical Methods for Directed Graphs with Applications in Biological Data

Graphs often model complex phenomena in diverse fields, such as social networks, connectivity among brain regions, or protein-protein interactions. However, standard computational methods are insufficient for empirical network analysis due to randomness. Thus, a natural solution would be the use of statistical approaches. A recent paper by Takahashi et al. suggested that the graph spectrum is a good fingerprint of the graph's structure. 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. In this paper, we extend their results to directed graphs by analyzing the distribution of complex eigenvalues instead. We show the strength of our methods by performing simulations on artificially generated groups of graphs and finally show a proof of concept using concrete biological data obtained by Project Tycho.