Contact-based social contagion in multiplex networks

We develop a theoretical framework for the study of epidemiclike social contagion in large scale social systems. We consider the most general setting in which different communication platforms or categories form multiplex networks. Specifically, we propose a contact-based information spreading model, and show that the critical point of the multiplex system associated with the active phase is determined by the layer whose contact probability matrix has the largest eigenvalue. The framework is applied to a number of different situations, including a real multiplex system. Finally, we also show that when the system through which information is disseminating is inherently multiplex, working with the graph that results from the aggregation of the different layers is inaccurate.

Figure 1

Schematic of a two-layer multiplex system where the contagion dynamics takes place. There are actors that take part in more than one layer (green nodes connected by the dotted edges), whereas others are present only in one layer (red nodes). ${\beta }_{1,2}$ is the contagion rate within the same layer whereas ${\gamma }_{1,2}$ represents the probability that the contagion occurs between layers. The right panel shows a small network and its associated $C$ and $A={⨁}_{\alpha }{A}_{\alpha }$.

Figure 2

(a) Density of adopters ($\rho$) at the steady state against the rescaled contagion probability $\frac{\beta }{\mu }$ for a multiplex system composed of two layers with $N={10}^4$ nodes each for different values of the ratio $\eta =\frac{\gamma }{\beta }$. The arrows represent the inverse of the largest eigenvalues of the two layers, whereas the inset shows the case in which both layers are completely disconnected. (b) The same quantity of (a), for $\eta =2.0$, is represented but computed at each layer. The inset is a zoom around the critical point. See the text for further details.

Figure 3

Dependence of the largest eigenvalues of the contact probability matrices, $R_{\alpha }$'s, on ${\lambda }_{\alpha }$ for the system in Fig. 2. As it can be seen, there might be a crossover signaling that the dominant layer changes. This crossover occurs only if the activity of the topologically dominant layer is small enough: In the example, it should be smaller than ${\lambda }_1=32$.

Figure 4

Density of adopters ($\rho$) at the steady state as a function of the rescaled contagion probability $\frac{\beta }{\mu }$ for a multiplex system composed of two layers with $N={10}^4$ nodes each (lines with symbols) and the corresponding aggregated graph (dotted lines). Different curves represent different values of the ratio $\eta =\frac{\gamma }{\beta }$ as indicated. The inset is a zoom of the region around the contagion threshold.