Epidemic spreading on activity-driven networks with attractiveness

We study SIS epidemic spreading processes unfolding on a recent generalization of the activity-driven modeling framework. In this model of time-varying networks, each node is described by two variables: activity and attractiveness. The first describes the propensity to form connections, while the second defines the propensity to attract them. We derive analytically the epidemic threshold considering the time scale driving the evolution of contacts and the contagion as comparable. The solutions are general and hold for any joint distribution of activity and attractiveness. The theoretical picture is confirmed via large-scale numerical simulations performed considering heterogeneous distributions and different correlations between the two variables. We find that heterogeneous distributions of attractiveness alter the contagion process. In particular, in the case of uncorrelated and positive correlations between the two variables, heterogeneous attractiveness facilitates the spreading. On the contrary, negative correlations between activity and attractiveness hamper the spreading. The results presented contribute to the understanding of the dynamical properties of time-varying networks and their effects on contagion phenomena unfolding on their fabric.

Figure 1

Degree and edge weight distribution for time-integrated ADA and AD. Both distributions show long tails. We used an activity distribution $F\left(a\right)\propto a^{-2.4}$ for all three networks. For the uncorrelated ADA, the attractiveness distribution is also $G\left(b\right)\propto b^{-2.4}$. For the correlated case, we set $b_i\propto a_i$ for each node $i$. We run our simulations on networks of ${10}^5$ nodes with $m=5$, integrated over ${10}^3$ time steps and averaged over 100 runs.

Figure 2

Value of $1/T_{\text{ADA}}$ for the case of uncorrelated distributions: $H(a,b)=F\left(a\right)G\left(b\right)$, where both variables are distributed according to a power law with values in the range $[{10}^{-3},1]$: $F\left(a\right)\propto a^{-{\gamma }_a}$ and $G\left(b\right)\propto b^{-{\gamma }_b}$. The threshold depends on the exponents ${\gamma }_a$ and ${\gamma }_b$ via the moments of the two distributions. We plot the reciprocal of the epidemic threshold, so that the larger the plotted value is, the easier it is for the disease to spread. $T_{\text{ADA}}$ has the same dependence on the two exponents and it is symmetric around its maximum in ${\gamma }_a={\gamma }_b=2$.

Figure 3

Plot of $T_{\text{AD}}/T_{\text{ADA}}$, the ratio of the epidemic threshold in the original model and the epidemic threshold with attractiveness, for the case of uncorrelated distributions: $H(a,b)\propto a^{-{\gamma }_a}{b}^{-{\gamma }_b}$ on the ADA. The activity on the AD in distributed according to the same law as on the ADA: $F\left(a\right)\propto a^{-{\gamma }_a}$. The ratio is a function of the two exponents ($T_{\text{AD}}$ depends on ${\gamma }_a$ only). When ${\gamma }_b$ diverges or tends to zero, the attractiveness distribution loses heterogeneity and the ADA converges to the AD. The difference is maximal for ${\gamma }_b=2$.

Figure 4

Lifetime of SIS processes on an ADA uncorrelated network with power-law distributed activity (${\gamma }_a=1.8$) and attractiveness (top: ${\gamma }_b=2.1$, bottom: ${\gamma }_b=2.8$), plotted for different values of $\beta /\mu$. We let $\lambda$ vary while we keep $\mu =0.01$ fixed, and $\langle k\rangle$ is determined by the relation $\langle k\rangle =2m\langle a\rangle$. The theoretical epidemic threshold (dotted line) is appreciably lower than the threshold for the AD (dashed line), and it is in accordance with the simulations. We used $m=2,\epsilon ={10}^{-3}$. In the upper plot, we used the following: for $N={10}^4,3000$ simulations (red); for $N={10}^5,1000$ simulations (blue); for $N={10}^6,500$ simulations (green). In the lower plot, we used the following: for $N={10}^4,5000$ simulations (red); for $N={10}^5,500$ simulations (blue); for $N={10}^6,500$ simulations (green). Solid lines with markers and shaded areas represent the mean and $95\%$ confidence interval separately.

Figure 5

For the case of deterministic activity-attractiveness correlation of the form $b\propto a^{{\gamma }_c}$, with activity distribution $F\left(a\right)\propto a^{-{\gamma }_a}$, the plot shows the logarithm of $T_{\text{ADA}}^{-1}$ as a function of the two exponents ${\gamma }_a$ and ${\gamma }_c$. Higher values in the plot correspond to a lower epidemic threshold, thus to an easier spreading. In particular, when ${\gamma }_c=0$, which corresponds to the AD case as the attractiveness is constant, the spreading is maximal for ${\gamma }_a=2$ as expected. When ${\gamma }_c$ increases, the threshold value decreases very rapidly, as the most active nodes also become the most popular ones.

Figure 6

Lifetime of an SIS process on an ADA network with identical activity-attractiveness correlation ($b\propto a$), plotted for different values of $\beta /\mu$ and different network sizes; we used a power-law distributed activity with ${\gamma }_a=2.8$ and let $\lambda$ vary while keeping fixed $\mu =0.01$ and $\langle k\rangle =2m\langle a\rangle$. The theoretical epidemic threshold (solid line) is lower than the threshold for the AD (dashed line). The case of identical but uncorrelated distributions (${\gamma }_a={\gamma }_b=2.8$, dotted line) is also significantly different from the correlated case. We used $m=2,\epsilon ={10}^{-3}$, and run the following: for $N={10}^4,3000$ simulations (red); for $N={10}^5,800$ simulations (blue); for $N={10}^6,500$ simulations (green). Solid lines with markers and shaded areas represent the mean and $95\%$ confidence interval separately.

Figure 7

Lifetime of an SIS process on an ADA network with negative deterministic activity-attractiveness correlation of the type $b\propto a^{-1}$, plotted for different values of $\beta /\mu$ and different network sizes; we used a power-law distributed activity with exponent ${\gamma }_a=2.8$ and let $\lambda$ vary while keeping fixed $\mu =0.01$ and $\langle k\rangle =2m\langle a\rangle$. We set $m=2,\epsilon ={10}^{-3}$, and run the following: for $N={10}^4,3000$ simulations (red); for $N={10}^5,2000$ simulations (blue); for $N={10}^6,2500$ simulations (green). Solid lines with markers and shaded areas represent the mean and $95\%$ confidence interval separately. The results match the theoretical prediction for the epidemic threshold (solid line). The comparison with the case of identical correlation shows how the two scenarios produce contrasting effects, with a negative correlation hindering the spreading phenomenon.