Clustering determines the dynamics of complex contagions in multiplex networks

We present the mathematical analysis of generalized complex contagions in a class of clustered multiplex networks. The model is intended to understand spread of influence, or any other spreading process implying a threshold dynamics, in setups of interconnected networks with significant clustering. The contagion is assumed to be general enough to account for a content-dependent linear threshold model, where each link type has a different weight (for spreading influence) that may depend on the content (e.g., product, rumor, political view) that is being spread. Using the generating functions formalism, we determine the conditions, probability, and expected size of the emergent global cascades. This analysis provides a generalization of previous approaches and is especially useful in problems related to spreading and percolation. The results present nontrivial dependencies between the clustering coefficient of the networks and its average degree. In particular, several phase transitions are shown to occur depending on these descriptors. Generally speaking, our findings reveal that increasing clustering decreases the probability of having global cascades and their size, however, this tendency changes with the average degree. There exists a certain average degree from which on clustering favors the probability and size of the contagion. By comparing the dynamics of complex contagions over multiplex networks and their monoplex projections, we demonstrate that ignoring link types and aggregating network layers may lead to inaccurate conclusions about contagion dynamics, particularly when the correlation of degrees between layers is high.

Figure 1

Illustration of multilayer and multiplex network representations of our model. In (a), we see a multilayer network (e.g., a physical communication layer and an online social network layer) with overlapping vertex sets; vertical dashed lines represent nodes corresponding to the same individual. In (b), we see the equivalent representation of this model by a multiplex network, i.e., an edge-colored multigraph. Edges from online social networks are shown in red and edges from the physical network are shown in blue; although not shown in this particular example, it is possible for a pair of nodes to be connected by both blue and red links, rendering the resulting representation a multigraph.

Figure 2

Illustration of three cases that would be counted as $m_{rs}, m_{rt1}$, and $m_{rt2}$, respectively, for the number of active nodes. Nodes shown in filled (green) circles are active while those shown in nonfilled circles are inactive.

Figure 3

Simulations for doubly Poisson degree distributions. In (a), we set the content parameter $c=0.25$, the threshold as $\tau =0.18$, and $\alpha =0.5$, and vary the degree parameters. In (b), we fix $\tau =0.18, {\lambda }_{r,1}={\lambda }_{r,2}={\lambda }_{b,1}={\lambda }_{b,2}=0.3$, and $\alpha =0.5$ while varying content parameter $c$.

Figure 4

Illustration of the effect of clustering coefficient on the expected cascade and probability of global cascades. We fix $\tau =0.18, c=0.25$, and $\alpha =0.5$, then vary the degree parameter $\lambda$ defined in Table 1. We see (a) the probability to trigger a global cascade; (b) the global clustering coefficient described in Sec. 2a; and (c) the expected cascade size.

Figure 5

We show the cascade regions in the degree parameter threshold plane when $\alpha =0.5, \tau =0.18$, and both networks follow doubly Poisson distributions as described in Table 1. Clustering increases as $\eta$ increases.

Figure 6

Comparison between monoplex networks and multiplex networks with limited assortativity. In (a) and (b), we fix the threshold $\tau =0.15$, the content parameter $c=1$, then vary the degree parameters in (13). For the networks obtained by projected theory and the networks in multiplex theory with $\alpha =0.99$, assortativity is negligible. However, when $\alpha =0.1$, the assortativity coefficient of the networks in the multiplex theory become significant, e.g., it can be up to 0.21.

Figure 7

Comparison between monoplex networks and multiplex networks with assortativity. Similar with the observation in Fig. 6, networks in the projected theory and in the multiplex theory with $\alpha =0.99$ have negligible assortativity coefficients. However, for the networks of multiplex theory with $\alpha =0.1$, assortativity coefficient ranges from 0.19 to 0.79. In general, assortativity increases with increasing ${\lambda }_r$ and ${\lambda }_b$ in the multiplex theory.

Figure 8

Demonstration of multiple phase transitions.