Analysis of complex contagions in random multiplex networks

We study the diffusion of influence in random multiplex networks where links can be of $r$ different types, and, for a given content (e.g., rumor, product, or political view), each link type is associated with a content-dependent parameter $c_i$ in $[0,\infty ]$ that measures the relative bias type $i$ links have in spreading this content. In this setting, we propose a linear threshold model of contagion where nodes switch state if their “perceived” proportion of active neighbors exceeds a threshold $\tau$. Namely a node connected to $m_i$ active neighbors and $k_i-m_i$ inactive neighbors via type $i$ links will turn active if $\sum c_i{m}_i/\sum c_i{k}_i$ exceeds its threshold $\tau$. Under this model, we obtain the condition, probability and expected size of global spreading events. Our results extend the existing work on complex contagions in several directions by (i) providing solutions for coupled random networks whose vertices are neither identical nor disjoint, (ii) highlighting the effect of content on the dynamics of complex contagions, and (iii) showing that content-dependent propagation over a multiplex network leads to a subtle relation between the giant vulnerable component of the graph and the global cascade condition that is not seen in the existing models in the literature.

Figure 1

The fractional size of the giant vulnerable cluster, $S_v$, the final cascade size $S$, and the triggering probability $P_{\mathrm{trig}}$ are plotted for $n=5\times {10}^5$, $\alpha =0.5$, where $\tau$ is fixed at ${\tau }^{☆}=0.18$ and $\mathbb{F}$ and $\mathbb{W}$ are Erdös-Rényi networks with mean degrees $z_1$ and $z_2$, respectively. The content parameter is taken to be $\left(a\right)$ $c=0.25$ and $\left(b\right)$ $c=1$. The lines correspond to the analytical results obtained from Eqs. (8) and (13), whereas symbols represent experimental results averaged over 100 independent realizations for $S_v$ and $S$ and over 5000 realizations for $P_{\mathrm{trig}}$. (Insets) The same plots at a higher resolution on $z_1=z_2$ near the lower phase transition.

Figure 2

We see the variation of cascade probability $P_{\mathrm{trig}}$ and cascade size $S$ with respect to the content parameter $c$, when $\alpha =0.5$, $\tau ={\tau }^{☆}=0.18$, and $\mathbb{F}$ and $\mathbb{W}$ are Erdös-Rényi networks with mean degrees $z_1$ and $z_2$, respectively. We set $\left(a\right)$ $z_1=1.5$, $z_2=5.5$, $\left(b\right)$ $z_1=z_2=0.7$, and $\left(c\right)$ $z_1=6.0$, $z_2=1.5$.

Figure 3

(a) We see the cascade windows for several $c$ values when $\alpha =0.5$, $\tau ={\tau }^{☆}$, and $\mathbb{F}$ and $\mathbb{W}$ are Erdös-Rényi networks with mean degrees $z_1$ and $z_2$, respectively. In other words, the lines enclose the region of the $({\tau }^{☆},z_1=z_2)$ plane for which the global cascade condition $\sigma \left(\mathbf{J}{}_p\right)>1$ is satisfied; outside the boundary, we have $\sigma \left(\mathbf{J}{}_p\right)\le 1$ and global cascades cannot take place. (b) Global cascade size $S$ when $\mathbb{F}$ and $\mathbb{W}$ are random clustered networks with degree distributions given by (14). We take $n={10}^5$, $\alpha =0.5$, $\tau ={\tau }^{☆}=0.18$, and $c=0.5$. The line corresponds to analytical prediction from (13), whereas symbols are obtained from simulations by averaging over 50 independent realizations. (Inset) Average clustering coefficient [45] versus $2z_1=2z_2$.