Spectral properties and the accuracy of mean-field approaches for epidemics on correlated power-law networks

We present a comparison between stochastic simulations and mean-field theories for the epidemic threshold of the susceptible-infected-susceptible model on correlated networks (both assortative and disassortative) with a power-law degree distribution $P\left(k\right)\sim k^{-\gamma }$. We confirm the vanishing of the threshold regardless of the correlation pattern and the degree exponent $\gamma$. Thresholds determined numerically are compared with quenched mean-field (QMF) and pair quenched mean-field (PQMF) theories. Correlations do not change the overall picture: The QMF and PQMF theories provide estimates that are asymptotically correct for large sizes for $\gamma <5/2$, while they only capture the vanishing of the threshold for $\gamma >5/2$, failing to reproduce quantitatively how this occurs. For a given size, PQMF theory is more accurate. We relate the variations in the accuracy of QMF and PQMF predictions with changes in the spectral properties (spectral gap and localization) of standard and modified adjacency matrices, which rule the epidemic prevalence near the transition point, depending on the theoretical framework. We also show that, for $\gamma <5/2$, while QMF theory provides an estimate of the epidemic threshold that is asymptotically exact, it fails to reproduce the singularity of the prevalence around the transition.

Figure 1

Average degree of the nearest neighbors as a function of the degree for networks built with the WPCM algorithm [35] for power-law degree distributions with $\gamma =2.3$ (top curves) and $\gamma =3.5$ (bottom curves). The network size is $N={10}^6$ and the lower cutoff is $k_{\text{min}}=3$. The upper cutoff is given by $k_{\text{max}}=2\sqrt{N}$ for $\gamma =2.3$ and $NP\left(k_{\text{max}}\right)=1$ for $\gamma =3.5$.

Figure 2

Threshold as a function of the network size for (a) $\gamma =2.3$ and (b) $\gamma =3.5$ and different values of $\alpha$. The lower cutoff is $k_{\text{min}}=3$ for all curves, while the upper cutoff is $k_{\text{max}}=2\sqrt{N}$ for $\gamma <3$ and $k_{\text{max}}\sim N^{1/\gamma }$ for $\gamma >3$. Curves are averages over ten networks; error bars are smaller than symbols.

Figure 3

Ratio between thresholds of HMF theories (${\lambda }_c^{\text{MF}}$) and simulations (${\lambda }_{\text{c}}$) as a function of the network size for different values of $\gamma$ and $\alpha$. The main panel and the right and left insets correspond to $\gamma =2.3$, 2.8, and 3.5 respectively. An upper cutoff $k_{\text{max}}=2\sqrt{N}$ is considered for $\gamma <3$, while for $\gamma =3.5, k_{\text{max}}\sim N^{1/\gamma }$. Averages correspond to ten network realizations and error bars are smaller than symbols.

Figure 4

(a) Comparison of the QMF and PQMF mean-field theories, (b) IPR, and (c) spectral gap of $A_{ij}$ and $B_{ij}$ against size for $\gamma =2.3$ and different values of $\alpha$. Averages correspond to ten network realizations. In (b), solid lines are power-law decays $Y_4\sim N^{-\nu }$, with $\nu =(3-\gamma )/2$ and $Y_4\sim N^{-1}$ corresponding to localization in the maximum $K$-core and finite set of vertices, respectively. Solid lines and open symbols correspond to the QMF theory and $A_{ij}$ analysis, while dashed lines and closed symbols correspond to the PQMF theory and critical $B_{ij}$.

Figure 5

(a) Comparison of the QMF and PQMF theories, (b) IPR, and (c) spectral gap of $A_{ij}$ and $B_{ij}$ against size for $\gamma =3.5$ and different values of $\alpha$, using an upper cutoff $k_{\text{max}}\sim N^{1/\gamma }$. Averages correspond to ten network realizations. Solid lines and open symbols correspond to the QMF theory and $A_{ij}$ analysis, while dashed lines and closed symbols correspond to the PQMF theory and critical $B_{ij}$.

Figure 6

(a) Comparison of the QMF and PQMF theories, (b) IPR, and (c) spectral gap of $A_{ij}$ and $B_{ij}$ against size for $\gamma =2.8$ and different values of $\alpha$, using an upper cutoff $k_{\text{max}}=2\sqrt{N}$. Averages correspond to ten network realizations. Solid lines and open symbols correspond to the QMF theory and $A_{ij}$ analysis, while dashed lines and closed symbols correspond to the PQMF theory and critical $B_{ij}$.

Figure 7

Scatter plots for a set of 99 real networks (see Appendix). Each point corresponds to a single network. (a) Spectral gap and (b) IPR of the matrix $B_{ij}$ plotted versus the corresponding values for the matrix $A_{ij}$. (c) Relative errors of QMF and PQMF theoretical predictions with respect to the simulation, defined by Eq. (21). Dashed red lines denote the diagonal. Relative errors of the QMF theory are plotted vs (d) the spectral gap, (e) the IPR, and (f) the Pearson coefficient.

Figure 8

Rescaled average density as a function of the distance from the epidemic threshold. Quasistationary simulations for different sizes are indicated in the legend. The solid line is a numerical integration of the QMF theory [Eq. (9)] for $N={10}^7$, while the dashed line is a power law with exponent predicted analytically in Ref. [51]. We used uncorrelated networks ($\alpha =0$) with degree exponent $\gamma =2.3$ and $k_{\text{max}}=2\sqrt{N}$.