Networks arise frequently in the study of complex systems, since interactions amongst their components are critical. Networks can act as a substrate for dynamical process, such as the diffusion of information or disease throughout populations. Network structure can determine the temporal evolution of a dynamical process, including the characteristics of the steady state.
The simplest representation of a complex system is an undirected, unweighted, single layer graph. In contrast, real systems exhibit heterogeneity of interaction strength and type. Such systems are frequently represented as weighted multiplex networks, and in this work we in- corporate these heterogeneities into a master equation formalism in order to study their effects on spreading processes. We also carry out simulations on synthetic and empirical networks, and show that spread- ing dynamics, in particular the speed at which contagion spreads via threshold mechanisms, depend non-trivially on these heterogeneities. Further, we show that an important family of networks undergo re-entrant phase transitions in the size and frequency of global cascades as a result of these interactions.
A challenging feature of real systems is their tendency to evolve over time, since the changing structure of the underlying network is critical to the behaviour of overlying dynamical processes. We show that one aspect of temporality, the observed “burstiness” in interaction patterns, leads to non-monotonic changes in the spreading time of threshold driven contagion processes.
The above results shed light on the effects of various network heterogeneities, with respect to dynamical processes that evolve on these networks.