Vertex-wise graph's spectral density decomposition and its applications

Abstract

The spectral density of a graph is a key concept for quantitatively characterizing empirical networks. It has many applications, including community detection, graph signal processing, spectral embedding, network evolution, brain network analysis, and random graph modeling. The graph's spectral density is also crucial in developing statistical methods for graphs, such as model selection and comparative testing. Despite its broad applicability, a complete understanding of the relationship between a graph's spectral density and structure remains elusive. To advance our understanding of the relationship between graph spectra and their structure, we introduce a vertex-wise decomposition of the graph's spectral density, allowing us to determine each vertex's contribution to specific eigenvalues. We show that the decomposition of distinct isospectral graphs (graphs with identical spectra) can be distinguished by the vertex-wise graph spectra, showing that the proposed new quantities are finer invariants between isomorphic graphs. Finally, we apply these insights to analyze chemical molecules and identify genes associated with normal versus tumoral breast gene interaction networks.