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 new one when perfect (resolvent) or imperfect (exponential) memory are considered for the system. By using theoretical and empirical evidence we show that real-world systems works out of perfect memory, and with a trade off of 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.

Detecting the driver nodes of temporal networks

“…It is increasingly recognized that real-world systems display intricate dynamical patterns which can not be captured by static networks. Indeed, some important results have been reported regarding temporal networks [8][9][10][11] (also known as time-varying networks, evolving networks, or dynamic networks), where the structure, as well as the attributes associated with nodes and edges, change over time.…”