Aging and equilibration in bistable contagion dynamics

We analyze the late-time relaxation dynamics for a general contagion model. In this model, nodes are either active or failed. Active nodes can fail either “spontaneously” at any time or “externally” if their neighborhoods are sufficiently damaged. Failed nodes may always recover spontaneously. At late times, the breaking of time-translation invariance is a necessary condition for physical aging. We observe that time-translational invariance is lost for initial conditions that lie between the basins of attraction of the model's two stable stationary states. Based on corresponding mean-field predictions, we characterize the observed model behavior in terms of a phase diagram spanned by the fractions of spontaneously and externally failed nodes. For the square lattice, the phases in which the dynamics approaches one of the two stable stationary states are not linearly separable due to spatial correlation effects. Our results provide insights into aging and relaxation phenomena that are observable in a model of social contagion processes.

Figure 1

Schematic of model dynamics. Nodes are arranged in a square lattice. Active nodes ($A$) can fail if their neighborhoods are sufficiently damaged (external failure) with rate $r$ or spontaneously (spontaneous failure) with rate $p$. A node fails externally if less than or equal to $m$ of its neighbors are active. The recovery rate of spontaneously failed nodes is $q$ and that of externally failed nodes is $q^{\prime }$. We use $X$ and $Y$ to indicate that nodes failed spontaneously and externally, respectively.

Figure 2

Phase portrait and relaxation dynamics. (a) The numerically determined phase portrait of the fraction of active nodes $n\left(t\right)$ in the general contagion model. Simulations were performed on a square lattice with $N=512\times 512$ sites using ${10}^3$ samples for each data point. (b) The time evolution of $n\left(t\right)$ for different initial conditions on a square lattice with $N=1024\times 1024$ sites. We use $n\left(0\right)\in \{0.1,0.2,...,0.9\}$ as initial densities of active nodes. (c) The relaxation dynamics of the general contagion model on a square lattice with $N=1024\times 1024$ nodes. The chosen initial densities are close to $n^{*}\left(0\right)\approx 0.62$. For initial densities $n\left(0\right)>n^{*}\left(0\right)$, we observe that $n\left(t\right)$ moves towards the upper stationary state, while for $n\left(0\right)<n^{*}\left(0\right)$ the densities move towards the lower stationary state. We use $n\left(0\right)\in \{0.610,0.612,0.614,...,0.630\}$ as initial densities. Since the time axis begins at $t={10}^{-1}$ and not at $t=0$, the densities have already slightly decreased. (d) The time evolution of $n\left(t\right)$ for different initial conditions on a random regular graph with $N={10}^5$ nodes and degree $k=4$ (blue solid lines) and corresponding mean-field results (orange dashed lines). The initial conditions are the same as in (b), but the relaxation is significantly faster than on a square lattice. In all simulations, we set $m=1, r=0.95, p=1.0, q=1.0$, and $q^{\prime }=0.1$. The initial density of spontaneously failed nodes is zero and we use uniformly distributed externally failed nodes such that $v\left(0\right)=1-n\left(0\right)$.

Figure 3

Relaxation for different fractions of externally and internally failed nodes. (a) The relaxation of the model dynamics in the $u-v$ plane for different initial conditions. All trajectories first approach an “equilibration line” (stage I) and then slowly relax towards one of the two stable stationary states that we indicate by black crosses (stage II). Blue and orange trajectories approach the stationary states with densities $n_{\mathrm{st}}^1$ and $n_{\mathrm{st}}^3$, respectively. Simulations were performed on a square lattice with $N=1024\times 1024$ nodes and for parameters $m=1, r=0.95, p=1.0, q=1.0$, and $q^{\prime }=0.1$. (b) Corresponding mean-field trajectories (black solid lines) in the $u-v$ plane for different initial conditions and $m=1, r=5.0, p=0.1, q=3.5$, and $q^{\prime }=1.0$. Within the orange (blue) region, all trajectories approach the stationary state with density $n_{\mathrm{st}}^1$ ($n_{\mathrm{st}}^3$). To obtain the mean-field trajectories, we numerically solve Eqs. (1) and (2). In both panels, grey-shaded regions correspond to densities $n>1$ and are excluded from our analysis. (c) The fractions of CDNs for the two trajectories that cross in (a). For initial conditions $\left(u\right(0)=0,v(0\left)\right)=(0.6,0.2)$ (blue solid line), we observe that the proportions of CDNs are smaller than for $\left(u\right(0.77)=0,v(0.77\left)\right)\approx (0.3,0.3)$. (d) If two mean-field trajectories share the same densities $u$ and $v$, they converge towards the same stationary state.

Figure 4

Correlation and covariance functions for different densities of active nodes. We show the correlation function $\mathrm{\Gamma }(t,s)$ and covariance function $C(t,s)$ [see Eqs. (3) and (4)] as functions of $t-s$. (a–d) We show $\mathrm{\Gamma }(t,s)$ and $C(t,s)$ without an initial relaxation period (i.e., $T=0$). For a comparison with longer initial relaxation times, we set $T=5000$ in (e–h). In (c) and (d) and in (g) and (h), we use an initial condition of $n\left(0\right)=0.8115$ (close to the upper stationary state) and we set $n\left(0\right)=0.62=n^{*}\left(0\right)$ in (a) and (b) and in (e) and (f). In our simulations, we used a square lattice with $N=512\times 512$ sites and generated 1000 samples. As we show in (g), the correlation function $\mathrm{\Gamma }$ converges towards the square of the density $n_{st}^3$ with the value ${(1-n_{st}^3)}^2\approx 0.035$.

Figure 5

Covariance functions at $n^{*}\left(0\right)$ for different relaxation times. We show the covariance function $C(T+t,T+s)$ [see Eq. (4)] as a function of $t-s$. We use an initial density of $n\left(0\right)=0.62=n^{*}\left(0\right)$ and let the system relax for a period $T$ before measuring the correlation functions. The relaxation times were chosen as (a) $T=0$, (b) $T=100$, (c) $T=500$, (d) $T=1000$, (e) $T=2000$, and (f) $T=5000$. In our simulations, we used a square lattice with $N=512\times 512$ sites and generated 1000 samples for each value of $T$.

Figure 6

Correlation and covariance functions at $n^{*}\left(0\right)$. We show the correlation function $\mathrm{\Gamma }(t,s)$ and $C(t,s)$ [see Eqs. (3) and (4)] as functions of $t/s$. The initial density is $n\left(0\right)=0.62=n^{*}\left(0\right)$. All simulations were performed on a square lattice with $N=512\times 512$ sites and without any initial relaxation before measuring the correlation functions (i.e., $T=0$). The number of samples is 1000.