Gaussianization of the spectra of graphs and networks. Theory and applications

Abstract

Matrix functions of the adjacency matrix are very useful for understanding important structural properties of graphs and networks, such as communicability, node centrality, bipartivity, and many more. They are also intimately related to the solution of differential equations describing dynamical processes on graphs and networks. Here, we propose a new matrix function based on the Gaussianization of the adjacency matrix of a graph. This function gives more weight to a selected reference eigenvalue λ ref , which may be located in any region of the graph spectra. We show here that this matrix function can be derived from physical models that consider the interactions between nearest and next-nearest neighbors in the graph. We first obtain a few mathematical results for the trace of this matrix function when λ ref = − 1 ( H − 1 ) for simple graphs as well as for random graphs. We also provide a combinatorial interpretation of this index in terms of subgraphs in the graph, and in terms of the competition pressure among agents in a complex system. Finally, we apply this index to the study of magnetic properties of molecules emerging due to spin interactions as well as to studying the temporal evolution of the international trade network in the period 1992–2002. In both cases we give a clear phenomenological interpretation of the processes described.