  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. 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 in some 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. 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. These universal results sheds light on many
dynamical diffusive-like processes on networks that present transitions as
traffic jams, communication lost or failure cascades.
