Fundamentals of spreading processes in single and multilayer complex networks

Abstract

Spreading processes have been largely studied in the literature, both analytically and by means of large-scale numerical simulations. These processes mainly include the propagation of diseases, rumors and information on top of a given population. In the last two decades, with the advent of modern network science, we have witnessed significant advances in this field of research. Here we review the main theoretical and numerical methods developed for the study of spreading processes on complex networked systems. Specifically, we formally define epidemic processes on single and multilayer networks and discuss in detail the main methods used to perform numerical simulations. Throughout the review, we classify spreading processes (disease and rumor models) into two classes according to the nature of time: (i) continuous-time and (ii) cellular automata approach, where the second one can be further divided into synchronous and asynchronous updating schemes. Our revision includes the heterogeneous mean-field, the quenched-mean field, and the pair quenched mean field approaches, as well as their respective simulation techniques, emphasizing similarities and differences among the different techniques. The content presented here offers a whole suite of methods to study epidemiclike processes in complex networks, both for researchers without previous experience in the subject and for experts. 6

Diffusive behavior of multiplex networks

“…These systems are well described by multiplex networks, where links of different type are embedded into separate layers of interactions [12][13][14]. Diffusion processes were among the first dynamics introduced in the context of multiplex networks, where diffusion can occur both within and across layers [15][16][17][18]. In this scenario, flows not only take place through nodes connected at a given layer, but also across two replicas of the same node belonging to two different layers.…”

Heterogeneous behavioral adoption in multiplex networks

“…For evolutionary games in multiplex network, interactions between layers influence evolution of cooperation [20][21][22]. Researchers found spreading dynamics on multiplex network also behaves distinctively from the single-layer network [23][24][25][26][27][28][29][30][31][32][33][34][35]. A few interlayer links can induce the global outbreak of disease, although the epidemic cannot outbreak on a single-layer network [36].…”

Focus on multilayer networks

“…Therein, the traditionally used adjacency matrices, which are useful to describe traditional single-layer networks, were replaced by a tensorial framework that allows the generalization of several important quantities and dynamical processes on multilayer networks, including degree centrality, clustering coefficients, eigenvector centrality, modularity, Von Neumann entropy, and diffusion. As previously for single-layer and isolated networks, popular and likewise highly cited reviews dedicated to temporal, multiplex, and multilayer networks attest to the vibrancy of this field of research [12][13][14], which has since found applications for better understanding epidemic spreading [15,16], vaccination [17], evolution of cooperation [18], information cascades [19], and biological organization at different scales [20].…”