Models of continuous-time networks with tie decay, diffusion, and convection

The study of temporal networks in discrete time has yielded numerous insights into time-dependent networked systems in a wide variety of applications. However, for many complex systems, it is useful to develop continuous-time models of networks and to compare them to associated discrete models. In this paper, we study several continuous-time network models and examine discrete approximations of them both numerically and analytically. To consider continuous-time networks, we associate each edge in a graph with a time-dependent tie strength that can take continuous non-negative values and decays in time after the most recent interaction. We investigate how the moments of the tie strength evolve with time in several models, and we explore—both numerically and analytically—criteria for the emergence of a giant connected component in some of these models. We also briefly examine the effects of the interaction patterns of continuous-time networks on the contagion dynamics of a susceptible–infected–recovered model of an infectious disease.

Figure 1

An illustration of dynamics in the tie-decay model of Ahmad et al. [6]. The tie strength between a pair of nodes increases by 1 when there is an interaction during a time step, and it decays exponentially when there is no interaction. In the depicted simulation, there are $n=1000$ nodes, a decay rate of $\alpha =0.01$, an interaction probability of $p=0.003$, and $T=1000$ time steps. The vertical axis shows the tie strength of one edge. Six interactions occur between the two nodes that are incident to this edge.

Figure 2

Size of the largest connected component in a tie-decay network that we construct using the model of Ahmad et al. [6] versus the threshold $g$ for different values of the decay rate $\alpha$. In our simulations, there are $n=2000$ nodes; decay parameters of (a) $\alpha =0.001$, (b) $\alpha =0.01$, and (c) $\alpha =0.1$; a total simulation time of $T=1000$; and an interaction probability of $p={10}^{-5}$ (so $\lambda =Tp=0.01$). Each plot is a mean over 200 instantiations. Using Eq. (13), we calculate the critical thresholds $g_{\text{crit}}$ to be (a) 0.9592, (b) 0.6345, and (c) 0.0106.

Figure 3

An illustration of tie-decay dynamics in a simplified version of the back-to-unity model of Jin et al. [12]. The tie strength between two nodes resets to 1 if they interact during a time step. In the depicted simulation, there are $n=1000$ nodes, a decay rate of $\alpha =0.01$, an interaction probability of $p=0.003$, and $T=1000$ time steps. The vertical axis shows the tie strength of one edge. Four interactions occur between the two nodes that are incident to this edge.

Figure 4

Presence versus absence of a GCC in the simplified back-to-unity model. In each panel, we show all components of a network from a single simulation. In both simulations, there are $n=1000$ nodes, an interaction probability of $p=\frac{1}{1.1n}$, a decay parameter of $\alpha =0.01$, and $T=3000$ time steps. (a) We set the threshold to be $g=0.95$, which yields $\mathbf{P}(s\ge g)\approx 0.0054>1/n=0.001$. Therefore, there is a GCC with high probability. (b) We set the threshold to be $g=0.995$, which yields $\mathbf{P}(s\ge g)\approx 9.09\times {10}^{-4}<1/n=0.001$. Therefore, with high probability, there is no GCC.

Figure 5

Size of the largest connected component in networks that we construct using the simplified back-to-unity model. In our simulations, there are $n=1000$ nodes; a decay parameter of (a) $\alpha =0.01$, (b) $\alpha =0.1$, and (c) $\alpha =1$; and a threshold of $g=0.9$. For each value of the interaction probability $p$, we take a mean of our results over 250 realizations. Each realization has a run time of $T=500$. Using Eq. (18), we calculate the critical probabilities $p_{\text{crit}}$ to be (a) $9\times {10}^{-5}$, (b) $0.5\times {10}^{-3}$, and (c) $1\times {10}^{-3}$.

Figure 6

An illustration of tie-strength dynamics for (a) our diffusion model and (b) our bounded convection–diffusion model. In both panels, we use a spatial step of $\delta x=5\times {10}^{-3}$, a time step of $\delta t={10}^{-5}$, and a simulation time of $T=0.03$. For panel (b), the convection parameter is $\beta =5$ and the upper bound of the tie strength is $w=0.8$. The vertical axis of each panel gives the natural logarithm of the tie strength of a single edge. In contrast to our simulations of the Ahmad et al. tie-decay model and the simplified back-to-unity model, the time step $\delta t\ne 1$. The diffusion model has a first-order error in time (as well as in space), so we need the time step to be small.

Figure 7

Comparison of our numerical scheme (red curve) from Eqs. (28) and (32) to a mean of Monte Carlo simulations over 100 realizations of our bounded convection–diffusion network model (gray blocks). This figure gives the probability distribution of the tie strength of an edge in a network. Because we assume that each edge is independent of all other edges, this distribution applies to the tie strength of each edge in the network.

Figure 8

Comparison of (a) the discrete SIR model (54) and (b)–(d) SIR disease spread on simplified back-to-unity tie-decay networks with different values of $\lambda =N_p\mathbf{P}$, where $N_p$ is the population and $\mathbf{P}$ is the probability that a tie strength is larger than or equal to the threshold. We show the fractions of the population in each state (i.e., compartment) as a function of time for (b) $\lambda =1$, (c) $\lambda =3$, and (d) $\lambda =0.3$. In each panel, the horizontal axis is time, the dot-dashed blue curve indicates susceptible individuals, the dashed red curve indicates infected individuals, and the solid yellow curve indicates recovered individuals. The infection rate is $\overline{\beta }=0.6$, the recovery rate is $\overline{\gamma }=0.1$, and the population is $N_p=5000$. Our initial conditions are $S\left(0\right)/N_p=0.998$, $I\left(0\right)/N_p=0.002$, and $R\left(0\right)=0$. Observe the similarity between the plots in panels (a) and (b).