Analysis of a threshold model of social contagion on degree-correlated networks

We analytically determine when a range of abstract social contagion models permit global spreading from a single seed on degree-correlated undirected random networks. We deduce the expected size of the largest vulnerable component, a network’s tinderboxlike critical mass, as well as the probability that infecting a randomly chosen individual seed will trigger global spreading. In the appropriate limits, our results naturally reduce to standard ones for models of disease spreading and to the condition for the existence of a giant component. Recent advances in the distributed infinite seed case allow us to further determine the final size of global spreading events when they occur. To provide support for our results, we derive exact expressions for key spreading quantities for a simple yet rich family of random networks with bimodal degree distributions.

Figure 1

(a) Three key spreading measures as a function of assortativity $r$ for networks with $P_k=\frac{4}{7}{\delta }_{k3}+\frac{3}{7}{\delta }_{k4}$. Solid lines indicate theoretical curves and symbols represent measurements from simulations. The three curves are (1) the fractional size of the largest vulnerable component $S_{\text{vuln}}$ [Eq. (33), circles], (2) the fractional size of the largest triggering component $S_{\text{trig}}$ [Eq. (35), squares], and (3) the fractional size of global spreading events $S$ [Eq. (22), diamonds]. The upper discontinuous phase transition in $S$ occurs at $r=r_c^{\text{upper}}=0.77485\pm 0.00001$. The data shown were obtained from networks with $N=0.98\times {10}^5$ nodes (correlations were generated using a shuffling algorithm described in [27]) with initial seeds placed at each node of ten sample networks for a total of $9.8\times {10}^5$ samples per each value of $r$. (b) Solutions for ${\theta }_{3,\infty }$ and ${\theta }_{4,\infty }$, the probability that edges leading to degree 3 and 4 nodes will be from nodes in state ${\sigma }_1$ in the long time limit [see Eqs. (36, 37)]. Solid and dashed lines indicate stable and unstable ones, respectively. When $r$ reaches $r_c^{\text{upper}}$, a repeated solution appears for $\left({\theta }_{3,\infty },{\theta }_{4,\infty }\right)$, distinct from (1,1), giving rise to the discontinuous upper phase transition in $S$.