Communication Melting in Graphs and Complex Networks

Abstract

Complex networks are the representative graphs of interactions in many complex systems. Usually, these interactions are abstractions of the communication/diffusion channels between the units of the system. Real complex networks, e.g. traffic networks, reveal different operation phases governed by the dynamical stress of the system. In the case of traffic networks the archetypical transition is from free flow to congestion. A revolutionary approach to ascertain how these transitions emerge is that of using physical models that could account for diffusion process under stress. Here we show how, communicability, a topological descriptor that reveals the efficiency of the network functionality in terms of these diffusive paths, could be used to reveal the transitions mentioned. By considering a vibrational model of nodes and edges in a graph/network at a given temperature (stress), we show that the communicability function plays the role of the thermal Green's function of a network of harmonic oscillators. After, we prove analytically the existence of a universal phase transition in the communicability structure of every simple graph. This transition resembles the melting process occurring in solids. For instance, regular-like graphs resembling crystals, melts at lower temperatures and display a sharper transition between connected to disconnected structures than the random spatial graphs, which resemble amorphous solids. Finally, we study computationally this graph melting process