Temporal Networks: Slowing Down Diffusion by Long Lasting Interactions

Naoki Masuda,Víctor M Eguíluz,Konstantin Klemm

doi:10.1103/physrevlett.111.188701

Abstract

Interactions among units in complex systems occur in a specific sequential order, thus affecting the flow of information, the propagation of diseases, and general dynamical processes. We investigate the Laplacian spectrum of temporal networks and compare it with that of the corresponding aggregate network. First, we show that the spectrum of the ensemble average of a temporal network has identical eigenmodes but smaller eigenvalues than the aggregate networks. In large networks without edge condensation, the expected temporal dynamics is a time-rescaled version of the aggregate dynamics. Even for single sequential realizations, diffusive dynamics is slower in temporal networks. These discrepancies are due to the noncommutability of interactions. We illustrate our analytical findings using a simple temporal motif, larger network models, and real temporal networks.

Highlights

Interactions among units in complex systems occur in a specific sequential order, affecting the flow of information, the propagation of diseases, and general dynamical processes
In the context of numerical simulations of population dynamics, including networked dynamical systems, the comparison of aggregate and temporal dynamics is tantamount to the choice of synchronous or asynchronous numerical schemes for updating states of the agents
A comparison is made with the corresponding aggregate dynamics where all interactions are present permanently

Summary

Introduction

Interactions among units in complex systems occur in a specific sequential order, affecting the flow of information, the propagation of diseases, and general dynamical processes. We show that diffusive dynamics is slower for the temporal network (i.e., asynchronous update) than for the aggregate network (i.e., synchronous update) and find qualitatively different effects, even after averaging over random temporal sequences of purely linear interactions. For the temporal dynamics averaged over random sequences with replacement, Eq (12) results in ^ 1⁄4 0, fðÀ1=2; Þ, and fðÀ3=2; Þ.

Results

Conclusion