Traffic-driven epidemic spreading in correlated networks

In spite of the extensive previous efforts on traffic dynamics and epidemic spreading in complex networks, the problem of traffic-driven epidemic spreading on correlated networks has not been addressed. Interestingly, we find that the epidemic threshold, a fundamental quantity underlying the spreading dynamics, exhibits a nonmonotonic behavior in that it can be minimized for some critical value of the assortativity coefficient, a parameter characterizing the network correlation. To understand this phenomenon, we use the degree-based mean-field theory to calculate the traffic-driven epidemic threshold for correlated networks. The theory predicts that the threshold is inversely proportional to the packet-generation rate and the largest eigenvalue of the betweenness matrix. We obtain consistency between theory and numerics. Our results may provide insights into the important problem of controlling and/or harnessing real-world epidemic spreading dynamics driven by traffic flows.

Figure 1

For scale-free networks of degree exponent $\gamma =3$ with different values of $c$, the average degree of the nearest neighbors, $\langle k_{nn}\left(k\right)\rangle$, as a function of the degree $k$. Each data point is the result of averaging over 100 independent network realizations. For assortative networks ($c>0$), $\langle k_{nn}\left(k\right)\rangle$ is an increasing function of $k$ (for small $k$ values). For disassortative networks ($c<0$), $\langle k_{nn}\left(k\right)\rangle$ is a decreasing function of $k$. For uncorrelated networks ($c=0$), $\langle k_{nn}\left(k\right)\rangle$ is independent of $k$.

Figure 2

For scale-free networks of different values of degree exponent $\gamma$, epidemic threshold ${\beta }_c$ as a function of the assortativity coefficient $c$. The solid, dashed, and dotted curves are theoretical predictions for the cases $\gamma =2.5$, 3, and 3.5, respectively. The inset shows the critical value $c_{\mathrm{wor}}$ as a function of $\gamma$. The packet-generation rate is $R=1500$. Each data point is result of averaging over 100 random network realizations.

Figure 3

For scale-free networks of degree exponent $\gamma =3$ and different values of the assortativity coefficient $c$, (a) algorithmic betweenness $b_k$ and (b) density of infected nodes ${\rho }_k$ as a function of the degree $k$. For all cases, the packet-generation rate is $R=1500$ and the density of the infected nodes in the whole population is $\rho \approx 0.04$. Each data point results from averaging over 100 random network realizations.

Figure 4

For scale-free networks of degree exponent $\gamma =3$, dependence of the infection probability $p_i$ on the algorithmic betweenness $b_i$ for different nodes. The values of the assortativity coefficients $c$ are $-0.2$, 0, and 0.6 for the left, middle, and right panels, respectively. Other parameters are packet-generation rate $R=1500$ and density of infected nodes $\rho \approx 0.04$.

Figure 5

For scale-free networks of different values of the degree exponent $\gamma$, the maximum nodal algorithmic betweenness $b_{\max }$ versus the assortativity coefficient $c$. Each data point is result of averaging over 100 random network realizations.

Figure 6

For scale-free networks of degree exponent $\gamma =3$ with different values of the assortativity coefficient $c$, epidemic threshold ${\beta }_c$ as a function of the packet-generation rate $R$. The slope of the fitted line is about $-1$. Each data point results from an average over 100 network realizations.

Figure 7

In the case of classical epidemic spreading, the epidemic threshold ${\beta }_c$ as a function of the assortativity coefficient $c$ for different values of degree exponent $\gamma$. Each data point results from an average over 100 network realizations.