Hyperspherical embedding of graphs and networks in communicability spaces

Let G be a simple connected graph with n nodes and let f"@a"""k(A) be a communicability function of the adjacency matrix A, which is expressible by the following Taylor series expansion: @?"k"="0^[email protected]"kA^k. We prove here that if f"@a"""k(A) is positive semidefinite then the function @h"p","q=(f"@a"""k(A)"p"p+f"@a"""k(A)"q"q-2f"@a"""k(A)"p"q)^1^2 is a Euclidean distance between the corresponding nodes of the graph. Then, we prove that if f"@a"""k(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 @h"p","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.

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