Communicability in time-varying networks with memory

Ernesto Estrada
New Journal of Physics 24, 063017 (1-16) (2022)

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 Schr¨odinger
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.

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