Localization and Spreading of Diseases in Complex Networks

Using the susceptible-infected-susceptible model on unweighted and weighted networks, we consider the disease localization phenomenon. In contrast to the well-recognized point of view that diseases infect a finite fraction of vertices right above the epidemic threshold, we show that diseases can be localized on a finite number of vertices, where hubs and edges with large weights are centers of localization. Our results follow from the analysis of standard models of networks and empirical data for real-world networks.

Figure 1

Prevalence $\rho$ versus the infection rate $\lambda$ in real networks. (a) astro-phys (upper line) and cond-mat-2005 (lower line) weighted networks [from Eq. (2)]. The eigenstate ${\Lambda }_1$ is localized in the cond-mat-2005 network and delocalized in the astro-phys and karate-club networks. (b) Karate-club network. The lower curve accounts for only the eigenstate ${\Lambda }_1$ in Eq. (4). Accounting for eigenstates ${\Lambda }_1$ and ${\Lambda }_2$, we find the higher curve and so on. The most upper curve is the exact $\rho$.

Figure 2

(a) Regular Bethe lattice with degree $k=3$. (b) Bethe lattice with one hub of degree $q>k$. This hub is connected to neighbors by edges having the same weight $w\ge 1$ (red or gray lines). (c) Bethe lattice with two vertices of degrees $q_1$ and $q_2$ connected by an edge with a weight $w\ge 1$ (red or gray line).

Figure 3

(a) Prevalence $\rho$ versus $\lambda$ in a scale-free network of ${10}^5$ vertices generated by the static model with $\gamma =4$, $⟨q⟩=10$. The lowest curve accounts for only the principal eigenstate in Eq. (4), the next one accounts ${\Lambda }_1$ and ${\Lambda }_2$, and so on. (b) Zoom of the prevalence at $\lambda$ near ${\lambda }_c=1/{\Lambda }_1$.