Hyperspherical embedding of graphs and networks in communicability spaces

Let GG be a simple connected graph with nn nodes and let fαk(A)fαk(A) be a communicability function of the adjacency matrix AA, which is expressible by the following Taylor series expansion: ∑k=0∞αkAk. We prove here that if fαk(A)fαk(A) is positive semidefinite then the function ηp,q=(fαk(A)pp+fαk(A)qq−2fαk(A)pq)12 is a Euclidean distance between the corresponding nodes of the graph. Then, we prove that if fαk(A)fαk(A) is positive definite, the communicability distance induces an embedding of the graph into a hyperdimensional sphere (hypersphere) such as the distances between the nodes are given by ηp,qηp,q. In addition we give analytic results for the communicability distances for the nodes in paths, cycles, stars and complete graphs, and we find functions of the adjacency matrix for which the main results obtained here are applicable. Finally, we study the ratio of the surface area to volume of the hyperspheres in which a few real-world networks are embedded. We give clear indications about the usefulness of this embedding in analyzing the efficacy of geometrical embeddings of real-world networks like brain networks, airport transportation networks and the Internet.