Epidemic Threshold in Structured Scale-Free Networks

We analyze the spreading of viruses in scale-free networks with high clustering and degree correlations, as found in the Internet graph. For the susceptible-infected-susceptible model of epidemics the prevalence undergoes a phase transition at a finite threshold of the transmission probability. Comparing with the absence of a finite threshold in networks with purely random wiring, our result suggests that high clustering (modularity) and degree correlations protect scale-free networks against the spreading of viruses. We introduce and verify a quantitative description of the epidemic threshold based on the connectivity of the neighborhoods of the hubs.

Figure 1

Prevalence $\rho$ (fraction of infected individuals in the stationary state) as a function of the spreading rate $\lambda$ for highly clustered scale-free networks, with $⟨k⟩=4$ (circles), 6 (squares), and 10 (diamonds), and for random scale-free networks with $⟨k⟩=6$ (solid curve). The simulations have been run in networks containing ${10}^5$ nodes and averaging over 100 different realizations. Inset: Survival probability, $P_s$, for a localized infection after $t$ time steps. Parameter values (from bottom to top) $\lambda =0.15$, 0.18, 0.2, 0.22, 0.25; and $⟨k⟩=6$.

Figure 2

Average degree of the neighbors of a node with connectivity $k$ in the structured networks with $⟨k⟩=4$ (circles), 6 (open squares), and 10 (diamonds). The asymptotic values for large $k$ are $3.0\pm 0.1$, $5.1\pm 0.3$, and $9\pm 1$ to be compared with the theoretical prediction $⟨k_h^{nn}⟩=⟨k⟩-1=3$, 5, and 9, respectively [cf. Eq. (3)]. The filled squares are the average degrees of the neighbors in random scale-free networks with $⟨k⟩=6$.

Figure 3

Dependence of the average degree of the neighbors of a node with system size $N$. For the case of highly clustered scale-free networks, the value has been obtained averaging for nodes with $k>1000$. The theoretical predictions ($⟨k⟩-1$) are also plotted (dash-dotted line). For the case of the randomly wired networks, the values are the average over the full range of available connectivities. The theoretical prediction $⟨k⟩/4\ln N$ is also plotted (dashed line).

Figure 4

Prevalence $\rho$ as a function of the spreading rate $\lambda$ for the Internet graph at three different times. The large filled symbols indicate the transmission threshold calculated according to the secondary reproductive number [Eq. (6)]. The value of $⟨k_h^{nn}⟩$ has been obtained as an average over the two largest hubs.