Communicability in time-varying networks with memory

Abstract

We develop a first-principles approach to define the communicability between two nodes in a time-varying network with memory. The formulation is based on the time-fractional Schrödinger equation, where the fractional (of Caputo type) derivative accounts for the memory of the system. Using a time-varying Hamiltonian in the tight-binding formalism we propose the temporal communicability as the product of Mittag–Leffer functions of the adjacency matrices of the temporal snapshots. We then show that the resolvent- and exponential-communicabilities of a network are special cases of the proposed temporal communicability when perfect (resolvent) or imperfect (exponential) memory are considered for the system. By using theoretical and empirical evidence we show that real-world systems work out of perfect memory, and with an interrelation between memory-dependent temporal communication and imperfect memory spatial transmission of information. We illustrate our results with the study of trophallaxis interactions in two ant colonies.