Phase transition of the susceptible-infected-susceptible dynamics on time-varying configuration model networks

We present a degree-based theoretical framework to study the susceptible-infected-susceptible (SIS) dynamics on time-varying (rewired) configuration model networks. Using this framework on a given degree distribution, we provide a detailed analysis of the stationary state using the rewiring rate to explore the whole range of the time variation of the structure relative to that of the SIS process. This analysis is suitable for the characterization of the phase transition and leads to three main contributions: (1) We obtain a self-consistent expression for the absorbing-state threshold, able to capture both collective and hub activation. (2) We recover the predictions of a number of existing approaches as limiting cases of our analysis, providing thereby a unifying point of view for the SIS dynamics on random networks. (3) We obtain bounds for the critical exponents of a number of quantities in the stationary state. This allows us to reinterpret the concept of hub-dominated phase transition. Within our framework, it appears as a heterogeneous critical phenomenon: observables for different degree classes have a different scaling with the infection rate. This phenomenon is followed by the successive activation of the degree classes beyond the epidemic threshold.

Figure 1

One move of an edge switch to eventually sample the whole of the configuration model space.

Figure 2

Degree-dependent observables for the SIS model in the quasistatic limit ($\omega \to 0$), near the absorbing phase. Solid lines are the predictions of Eqs. (6) and (2); markers are the results of Monte Carlo simulations. To prevent the system from reaching the absorbing state, we have sampled the quasistationary distribution of the system [39, 40, 41]. Averages are made on ${10}^2$ realizations of the configuration model with the same degree sequence to simulate the quasistatic limit. The degree sequences of $N={10}^6$ nodes were drawn from a power-law degree distribution $P\left(k\right)\sim k^{-\gamma }$ with cutoff $k_{\max }<N^{1/2}$. (a) Scaled probability that a degree $k$ node is infected ${\rho }_k^{*}/{\rho }_{k_{\min }}^{*}$. (b) Scaled probability of reaching an infected node from a degree $k$ susceptible node ${\theta }_k^{*}/{\theta }_{k_{\min }}^{*}$.

Figure 3

Ratio ${\tilde{\kappa }}_0/k_{\max }$ against the power-law degree distribution exponent $\gamma$ in the quasistatic limit ($\omega \to 0$), for different values of maximum degree $k_{\max }$. The minimum degree is $k_{\min }=3$. Vertical dashed line corresponds to $\gamma =5/2$. Horizontal dashed line corresponds to ${\tilde{\kappa }}_0/k_{\max }=2k_{\min }$, identified using Eq. (23).

Figure 4

Threshold evaluation for power-law random networks of degree distribution $P\left(k\right)\sim k^{-\gamma }$, minimum degree $k_{\min }=3$ and maximum degree $k_{\max }\le N^{1/2}$. (a, b) Susceptibility against the infection rate for a single network realization. (c, d) Threshold against the number of nodes (averaged over 10 network realizations) estimated by the position of the susceptibility peak ${\lambda }_p\left(N\right)$, our threshold estimate ${\lambda }_c^{\mathrm{RNA}}$ of Eq. (22) for $\omega \to 0$ and the PHMF threshold ${\lambda }_c^{\mathrm{PHMF}}$.

Figure 5

Threshold against the rewiring rate for a regular random network with degree $k_0=6$ and network size $N={10}^5$. The solid line represents the threshold estimated by Eq. (25), and the markers represent the positions of the susceptibility peaks ${\lambda }_p\left(N\right)$. The disparity with the simulations is attributed to a combination of finite size effects and approximations leading to Eq. (21).

Figure 6

Threshold against the rewiring rate for power-law degree distribution $P\left(k\right)\sim k^{-\gamma }$ with exponent $\gamma =2.75$ and maximal degree $k_{\max }=50$. The threshold is evaluated with Eq. (21). The dashed line marks the optimal value ${\omega }_{\mathrm{opt}}$ at which ${\lambda }_c$ is maximized.

Figure 7

Critical exponent $\delta$ associated to the mean infected density ${\rho }^{*}$, for a power-law degree distribution $P\left(k\right)\sim k^{-\gamma }$ in the thermodynamic limit. The bounded (shaded) region and the solid line correspond to the exponent predicted by our approach [Eqs. (37) and (39)] and the dashed line to the HMF exponent. The dashed-dotted line indicates the transition from the collective to the hub-dominated region ($\gamma =5/2$).

Figure 8

Critical exponents ${\eta }_k$ associated to ${\theta }_k^{*}$, for power-law degree distribution $P\left(k\right)\sim k^{-\gamma }$ in the thermodynamic limit. The bounded (shaded) region and the solid lines correspond to the exponents predicted by our approach [Eqs. (48) and (49)] and the dashed line to the HMF exponent. The dashed-dotted line indicates the transition from the collective to the hub-dominated region ($\gamma =5/2$).

Figure 9

Critical behavior for ${\theta }_{k_{\min }}^{*}$ and ${\theta }_{k_{\max }}^{*}$. All curves have been normalized to the value of ${\theta }_k^{*}$ at the origin of the abscissae. Dashed lines and shaded regions correspond to the theoretical scaling predicted in the limit $k_{\max }\to \infty$. Solid lines represent the numerical evaluation of Eq. (6) for bounded degree distribution with $k_{\min }=3$ and $k_{\max }=5\times {10}^5$. (a) Power-law degree distribution with exponent $\gamma =2.5$. (b) Power-law degree distribution with exponent $\gamma =2.75$.

Figure 10

${\theta }_k^{*}/{\theta }_{k_{\min }}^{*}$ near the absorbing phase for power-law degree distribution with exponent $\gamma =3.1$ and maximal degree $k_{\max }=1000$, for different values of rewiring rates $\omega$. The minimal degree is $k_{\min }=3$.

Figure 11

Successive activation of the degree classes for a power-law degree distribution with exponent $\gamma =3.1$, maximal degree $k_{\max }={10}^4$, and rewiring rate $\omega =0$. The minimal degree is $k_{\min }=3$. The vertical dashed lines corresponds to $\tilde{\kappa }=k$ for each degree $k$ involved. (a) Infected density per degree class ${\rho }_k^{*}$ as a function of the infection rate $\lambda$. (b) Derivative of ${\rho }_k^{*}$ with respect to $\lambda , {\zeta }_k^{*}$, as a function of the infection rate $\lambda$.