Velocity and Hierarchical Spread of Epidemic Outbreaks in Scale-Free Networks

We study the effect of the connectivity pattern of complex networks on the propagation dynamics of epidemics. The growth time scale of outbreaks is inversely proportional to the network degree fluctuations, signaling that epidemics spread almost instantaneously in networks with scale-free degree distributions. This feature is associated with an epidemic propagation that follows a precise hierarchical dynamics. Once the highly connected hubs are reached, the infection pervades the network in a progressive cascade across smaller degree classes. The present results are relevant for the development of adaptive containment strategies.

Figure 1

(a) Average density of infected individuals versus time in a BA network of $N={10}^4$ with $m=2$. The inset shows the exponential fit obtained in the early times (lines) and the numerical curves $i(t)$ for networks with $m=4$, 8, 12, and 20 (from bottom to top). (b) Measured time scale $\tau$ in BA networks as obtained from exponential fitting versus the theoretical prediction for different values of $m$ and $N$ corresponding to different levels of heterogeneity.

Figure 2

(a) Time behavior of the average degree of the newly infected nodes for SI outbreaks in BA networks of size $N={10}^4$. Time is rescaled by $\tau$. Reference lines are drawn at the asymptotic values $⟨k^2⟩/⟨k⟩$ for $t\ll \tau$ and $m$ for $t\gg \tau$. The two curves are for $m=4$ (bottom) and $m=14$ (top). (b) Inverse participation ratio $Y_2$ versus time for BA network of size $N={10}^4$ with minimum degree $m=4$, 6, 8, 10, 12, 14, and 20, from top to bottom. Time is rescaled with $\tau$. The reference line indicates the minimum of $Y_2$ around $t/\tau \simeq 6.5$.