Degree correlations amplify the growth of cascades in networks.

Networks facilitate the spread of cascades, allowing a local perturbation to percolate via interactions between nodes and their neighbors. We investigate how network structure affects the dynamics of a spreading cascade. By accounting for the joint degree distribution of a network within a generating function framework, we can quantify how degree correlations affect both the onset of global cascades and the propensity of nodes of specific degree class to trigger large cascades. However, not all degree correlations are equally important in a spreading process. We introduce a new measure of degree assortativity that accounts for correlations among nodes relevant to a spreading cascade. We show that the critical point defining the onset of global cascades has a monotone relationship to this new assortativity measure. In addition, we show that the choice of nodes to seed the largest cascades is strongly affected by degree correlations. Contrary to traditional wisdom, when degree assortativity is positive, low degree nodes are more likely to generate largest cascades. Our work suggests that it may be possible to tailor spreading processes by manipulating the higher-order structure of networks.

Limitations of discrete-time approaches to continuous-time contagion dynamics.

Continuous-time Markov process models of contagions are widely studied, not least because of their utility in predicting the evolution of real-world contagions and in formulating control measures. It is often the case, however, that discrete-time approaches are employed to analyze such models or to simulate them numerically. In such cases, time is discretized into uniform steps and transition rates between states are replaced by transition probabilities. In this paper, we illustrate potential limitations to this approach. We show how discretizing time leads to a restriction on the values of the model parameters that can accurately be studied. We examine numerical simulation schemes employed in the literature, showing how synchronous-type updating schemes can bias discrete-time formalisms when compared against continuous-time formalisms. Event-based simulations, such as the Gillespie algorithm, are proposed as optimal simulation schemes both in terms of replicating the continuous-time process and computational speed. Finally, we show how discretizing time can affect the value of the epidemic threshold for large values of the infection rate and the recovery rate, even if the ratio between the former and the latter is small.

Analytical approach to the dynamics of facilitated spin models on random networks.

Facilitated spin models were introduced some decades ago to mimic systems characterized by a glass transition. Recent developments have shown that a class of facilitated spin models is also able to reproduce characteristic signatures of the structural relaxation properties of glass-forming liquids. While the equilibrium phase diagram of these models can be calculated analytically, the dynamics are usually investigated numerically. Here we propose a network-based approach, called approximate master equation (AME), to the dynamics of the Fredrickson-Andersen model. The approach correctly predicts the critical temperature at which the glass transition occurs. We also find excellent agreement between the theory and the numerical simulations for the transient regime, except in close proximity of the liquid-glass transition. Finally, we analytically characterize the critical clusters of the model and show that the departures between our AME approach and the Monte Carlo can be related to the large interface between blocked and unblocked spins at temperatures close to the glass transition.