Critical avalanches of susceptible-infected-susceptible dynamics in finite networks

We investigate the avalanche temporal statistics of the susceptible-infected-susceptible (SIS) model when the dynamics is critical and takes place on finite random networks. By considering numerical simulations on annealed topologies we show that the survival probability always exhibits three distinct dynamical regimes. Size-dependent crossover timescales separating them scale differently for homogeneous and for heterogeneous networks. The phenomenology can be qualitatively understood based on known features of the SIS dynamics on networks. A fully quantitative approach based on Langevin theory is shown to perfectly reproduce the results for homogeneous networks, while failing in the heterogeneous case. The analysis is extended to quenched random networks, which behave in agreement with the annealed case for strongly homogeneous and strongly heterogeneous networks.

Figure 1

Survival probability for critical SIS avalanches in homogeneous annealed networks. We set $\gamma =5.3$, $\omega =4.3$, and $k_0=\lfloor k_{\min }(\gamma -1)/(\gamma -2)\rfloor =\lfloor \langle k\rangle \rfloor$ and we consider ${10}^7$ realizations of the process for each value of $N$. Different curves show the survival probability $S\left(t\right)$ as $N$ is varied. The black dashed line scales as $t^{-1}$. In the inset, the same data as in the main panel are rescaled to make the various curves collapse one on the top of the other. The abscissa values are rescaled as $t/t_{\times }\left(N\right)$ with $t_{\times }\left(N\right)=N^{1/2}$, and the ordinate values are rescaled as $tS\left(t\right)$.

Figure 2

Survival probability for critical SIS avalanches in heterogeneous annealed networks. We set $\omega =2$ and $k_0=\lfloor k_{\min }(\gamma -1)/(\gamma -2)\rfloor =\lfloor \langle k\rangle \rfloor$ and we consider ${10}^9$ realizations of the SIS process for each value of the system size $N$. (a) We display the survival probability for $\gamma =2.1$. The inset displays the same data as in the main panel with the abscissa rescaled as $t/t_{\times }\left(N\right)$ and with the ordinate rescaled as $tS\left(t\right)k_{max}$. (b) Same as in panel (a) but for $\gamma =2.5$. (c) Same as in panel (a) but for $\gamma =2.9$.

Figure 3

Survival probability for critical SIS avalanches in heterogeneous annealed networks. We set $\omega =2$, $\gamma =2.1$ and we consider ${10}^9$ realizations of the process for each curve. The process is started from $i_0$ seeds; the sum of the degrees of these seeds is $k_0$. (a) The process is started from $i_0=1$. Different curves correspond to different values of $k_0$. Here the network size is $N={10}^8$. The inset displays the same data as in the panel with the ordinate rescaled as $tS\left(t\right)/k_0$. (b) We start the process from $i_0=1$ seed. We consider different $k_0$ values but also different network sizes $N$. The values of these two parameters are chosen such that the ratio $k_{max}/k_0=100$. The inset displays the same data as in the main panel with the abscissa rescaled as $t/t_{\times }\left(N\right)$, with $t_{\times }\left(N\right)$ given in Eq. (16). We rescale also the ordinate as $tS\left(t\right)k_{max}/k_0$. (c) We start the process from variable values of $i_0$ all with the same degree, but we keep $k_0=60$. The inset displays the same data as in the main panel with the ordinate rescaled as $S\left(t\right)/i_0$. The solid black line is the exponential decay with unitary rate $e^{-t}$.

Figure 4

Survival probability for critical SIS avalanches in quenched networks. We set $k_0=\lfloor k_{\min }(\gamma -1)/(\gamma -2)\rfloor =\lfloor \langle k\rangle \rfloor$. (a) Simulations are performed on a random regular graph with degree $k=10$. We consider ${10}^8$ realizations of the spreading process for each value of $N$. The inset displays the same data as the main panel with the abscissa rescaled as $t/\overline{t}\left(N\right)$ and with the ordinate rescaled as $tS\left(t\right)$. (b) Same as in panel (a) but on configuration model networks with $\gamma =2.1$ and $\omega =2$. We consider ${10}^9$ realizations of the SIS process for each value of $N$. The inset displays the same data as in the main panel with the abscissa rescaled as $t/\overline{t}\left(N_h\right)$ and with the ordinate rescaled as $tS\left(t\right)k_{max}$. (c) Same as in (b) but for $\gamma =2.7$ and $\omega =2$.

Figure 5

Survival probability of the avalanche size for critical SIS avalanches in annealed networks. (a) We set $\gamma =5.3$, $\omega =4.3$, and $k_0=\lfloor k_{\min }(\gamma -1)/(\gamma -2)\rfloor =\lfloor \langle k\rangle \rfloor$ and we consider ${10}^7$ realizations of the SIS process for each value of the system size $N$. The inset displays the same data as in the main panel with the abscissa rescaled as $z/N$ and with the ordinate rescaled as $z^{3/2}S\left(z\right)$. (b) We set $\gamma =2.1$, $\omega =2$, and $k_0=\lfloor k_{\min }(\gamma -1)/(\gamma -2)\rfloor =\lfloor \langle k\rangle \rfloor$ and we consider ${10}^9$ realizations of the SIS process for each value of the system size $N$. The inset displays the same data as in the main panel with the abscissa rescaled as $z/N_h$ and with the ordinate rescaled as $z^{3/2}P\left(z\right)k_c$.

Figure 6

Survival probability for critical SIS avalanches in annealed networks. (a) We set $\gamma =5.3$, $N={10}^6$, and $k_0=\lfloor k_{\min }(\gamma -1)/(\gamma -2)\rfloor =\lfloor \langle k\rangle \rfloor$ and we consider ${10}^7$ realizations of the SIS process for each value of $\omega$. The inset displays the same data as in the main panel with the abscissa rescaled as $t/t_c\left(N\right)$ and with the ordinate rescaled as $tS\left(t\right)$. (b) We set $\gamma =2.1$, $N={10}^6$, and $k_0=\lfloor k_{\min }(\gamma -1)/(\gamma -2)\rfloor =\lfloor \langle k\rangle \rfloor$ and we consider ${10}^7$ realizations of the SIS process for each value of $\omega$. The inset displays the same data as in the main panel with the abscissa rescaled as $t/t_c\left(N_h\right)$ and with the ordinate rescaled as $tS\left(t\right)k_{\max }$.

Figure 7

Trajectories of the order parameter $\mathrm{\Theta }$. We use here $N={10}^8$ and $\omega =2$. Each realization is initialized with $k_0=\lfloor k_{\min }(\gamma -1)/(\gamma -2)\rfloor =\lfloor \langle k\rangle \rfloor$. Solid lines correspond to a single trajectory of the order parameter for different values of $\gamma$. Dashed lines correspond to the bound $1/{\lambda }_c{k}_{\max }$.

Figure 8

Critical point determination and its importance. (a) The susceptibility for the random regular graph with $N={10}^6$ and $k=10$ is computed for different values of $\lambda$, with a resolution $2.5\times {10}^{-5}$. (b) Rescaled survival probability for the random regular graph with $k=10$. The abscissa is rescaled as $t/t_c\left(N\right)$ and the ordinate is rescaled as $tS\left(t\right)$. The inset shows the same data but the axes are not rescaled. The two black curves represent $N={10}^4$ and $N={10}^5$. The red, blue, and green lines represent data for $N={10}^6$ obtained using the three values of the $\lambda$ marked by the dashed lines of the same color in panel (a).