"Melting" of complex networks. A mathematical model of complex networks resilience to external stress

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. Recently we have proved analytically the existence of a universal phase transition in the communicability–a topological descriptor that reveals the efficiency of the network functionality in terms of these diffusive paths–of every simple network. This transition resembles the melting process occurring in solids. Here we study computationally this universal melting process in a large dataset of real-world networks and observe that the rate of melting of graphs changes either as an exponential or as a power-law with the inverse temperature representing the external stress to which the system is submitted to. At the local level we discover that the main driver for node melting is the eigenvector centrality of the corresponding node, particularly when the critical value of the inverse temperature approaches zero. That is, the most central nodes are the ones most at risk of triggering the melt down of the global network. These universal results can be used to sheds light on many dynamical diffusive-like processes on networks that present transitions as traffic jams, communication lost or failure cascades.