Spatiotemporal Activity-Driven Networks

TL;DR

A spatial activity-driven model in which short-range contacts are more frequent and this model is analytically tractable and captures the joint effects of space and time, which enables systematic exploration of dynamical processes on spatiotemporal networks.

Abstract

Temporal-network models have provided key insights into how time-varying connectivity shapes dynamical processes such as spreading. Among them, the activity-driven model is a widely used, analytically tractable benchmark. Yet many temporal networks, such as those of physical proximity, are also embedded in space, and spatial constraints are known to affect dynamics unfolding on the networks strongly. Despite this, there is a lack of similar simple and solvable models for spatiotemporal contact structures. Here, we introduce a spatial activity-driven model in which short-range contacts are more frequent. This model is analytically tractable and captures the joint effects of space and time. We show analytically and numerically that the model reproduces several characteristic features of social and contact networks, including strong and weak ties, clustering, and triangles having weights above the median. These traits can be attributed to space acting as a form of memory. Simulations of spreading dynamics on top of the model networks further illustrate the role of space, highlighting how localisation slows down spreading. Furthermore, the framework is well-suited for modelling social distancing in a principled way as an intervention measure aimed at reducing long-range links. We find that, unlike for non-spatial networks, even a small spatially targeted reduction in the total number of contacts can be very effective. More broadly, by offering a tractable framework, the model enables systematic exploration of dynamical processes on spatiotemporal networks.

Figures (11)

• Figure 1: Visualisation of the spatially preferential contact process. $N$ nodes are positioned randomly in a 2D toroidal space, and their activity potential is represented by their colour, such that the lighter a node, the higher its potential. The plot consists of three panels, each showcasing an active node initiating contacts according to the model's mechanism. A red disc of radius $R$ is centred at the selected active node $i$, the $m$ nodes for contacts are sampled from the nodes $j$ within a cut-off distance $R$ with probability $p_{ij}$ inversely proportional to their distance to the active node $i$. Note that if there are fewer than $m$ nodes within the cut-off distance, then only $m_{true}=min\{m, m_{reachable}\}$ contacts occur. For the simulation above, the parameter values were $N=30, R=0.2, m=3$.
• Figure 2: Comparison of theoretical $P_{avg}(w)$ and empirical (simulated) link weight distributions of the aggregated network over $T=2000$ time steps. Further simulation parameters were $N=10^3, R=0.2, m=3$. The analytical result is supported up to the highest observed link weight in the simulation data.
• Figure 3: Analytical approximation $P_{w_{\triangle}}$ of the empirical triangle weight distribution. Simulation parameters were $N=10^3, T=2000, R=0.2, m=3$. The approximation was obtained by a Gaussian kernel density estimate of the expected triangle weights (Equation \ref{['eq: exp_trigweight']}) calculated for each triangle of the aggregated network. The grey dotted and dashed lines indicate the median and mean link weights of the network, respectively. $74\%$ of triangle weights are higher than the median link weight ($54\%$ are higher than the mean link weight, but note the exponential distribution).
• Figure 4: Correlation (linear regression) results for spatiality and activity of triangles with triangle weight. Blue dots denote triangles in the simulated aggregated network, with parameters $N=10^3, T=2000, R=0.2, m=3$. Similar results were obtained for different $R$ and $m$ values. The regression surface shows moderate negative correlation (Pearson, Spearman $< -0.4$) with average triangle distance $\bar{d_{\triangle}}$, and strong positive correlation (Pearson, Spearman $> 0.7$) with average activity of nodes $\bar{a}_{\triangle}$, verifying our hypothesis.
• Figure 5: Evolution of link density for different $R,m$ parameter values on networks of $N=10^3$ nodes. Links are aggregated and density values are evaluated at checkpoints in time from $t_0=100$ up to $T=5000$. $d_{max}(R)$ values are marked with grey dashed lines.