Griffiths effects of the susceptible-infected-susceptible epidemic model on random power-law networks

We provide numerical evidence for slow dynamics of the susceptible-infected-susceptible model evolving on finite-size random networks with power-law degree distributions. Extensive simulations were done by averaging the activity density over many realizations of networks. We investigated the effects of outliers in both highly fluctuating (natural cutoff) and nonfluctuating (hard cutoff) most connected vertices. Logarithmic and power-law decays in time were found for natural and hard cutoffs, respectively. This happens in extended regions of the control parameter space ${\lambda }_1<\lambda <{\lambda }_2$, suggesting Griffiths effects, induced by the topological inhomogeneities. Optimal fluctuation theory considering sample-to-sample fluctuations of the pseudothresholds is presented to explain the observed slow dynamics. A quasistationary analysis shows that response functions remain bounded at ${\lambda }_2$. We argue these to be signals of a smeared transition. However, in the thermodynamic limit the Griffiths effects loose their relevancy and have a conventional critical point at ${\lambda }_c=0$. Since many real networks are composed by heterogeneous and weakly connected modules, the slow dynamics found in our analysis of independent and finite networks can play an important role for the deeper understanding of such systems.

Figure 1

Decay of density of infected vertices for networks with free upper cutoff $(k_{\max }=N)$ using (a) $\gamma =3.5$ and (b) $\gamma =2.7$ for networks with sizes $N={10}^7$ and $N={10}^5$, respectively. The number of samples were up to $\mathcal{N}=500$ and 4000 for $\gamma =3.5$ and 2.7, respectively. Lines are predictions of the optimal fluctuation theory in Sec. 4. The values of $\lambda$ are indicated in the legends.

Figure 2

Decay of density of infected vertices for (a) $\gamma =3.5$ and (b) $\gamma =2.7$ for networks of sizes $N={10}^7$ and ${10}^5$, respectively, using hard cutoff $k_c=k_0{N}^{0.9/(\gamma -1)}$. The numbers of independent networks are $\mathcal{N}=500$ and 4000 for $\gamma =3.5$ and $2.7$, respectively. The values of $\lambda$ are indicated in the legends.

Figure 3

Decay of density of infected vertices for (a) $\gamma =2.7$ and (b) $\gamma =2.3$ for networks of size $N={10}^5$ using structural cutoff $k_c=N^{1/2}$. The values of $\lambda$ are indicated in the legends.

Figure 4

Sample-to-sample fluctuation of the evolution of SIS on PL networks: (a) $\gamma =3.5, N={10}^7$, free cutoff, and $\lambda =0.04$; (b) $\gamma =3.5$, hard cutoff $k_c=k_0{N}^{0.9/(\gamma -1)}, N={10}^7$, and $\lambda =0.09$; (c) $\gamma =2.3$, structural cutoff $k_c=N^{1/2}, N={10}^5$, and $\lambda =0.026$. Curves for 50 independent networks are shown. Thick lines represent the average of $\mathcal{N}=500$ and 2000 samples for $\gamma =3.5$ and $2.3$, respectively.

Figure 5

Finite-size analysis of the density decay against time for networks with $\gamma =3.5$ and hard cutoff $k_c=k_0{N}^{0.9/(\gamma -1)}$ for infection rate $\lambda =0.088$. The number of network samples is 100 for the two largest sizes and 500 for the others. The network sizes are indicated in the legends.

Figure 6

Finite-size analysis of the QS state for networks with $\gamma =3.5$. The [(a) and (b)] QS density $\rho$, [(c) and (d)] log-derivative of $\rho$, and [(e) and (f)] dynamical susceptibility are shown in left and right panels for free and hard cutoffs, respectively. The number of samples were at least $\mathcal{N}=50$. The network sizes are indicated in the legend.

Figure 7

Critical quantities against networks size for $\gamma =3.5$: (a) threshold, (b) critical density, and (c) critical susceptibility at $\lambda ={\lambda }_i$. Solid lines are PL regressions while the dashed lines are ${\lambda }_1\sim N^{-0.2}$ and $N^{-0.18}$ predicted by the QMF theory for free and hard cutoffs, respectively.

Figure 8

Finite-size scaling of IPR for $\gamma =2.7, k_0=1,2,3$, and structural and $k_0=2$ with free cutoff.