Transport on Coupled Spatial Networks

Transport processes on spatial networks are representative of a broad class of real world systems which, rather than being independent, are typically interdependent. We propose a measure of utility to capture key features that arise when such systems are coupled together. The coupling is defined in a way that is not solely topological, relying on both the distribution of sources and sinks, and the method of route assignment. Using a toy model, we explore relevant cases by simulation. For certain parameter values, a picture emerges of two regimes. The first occurs when the flows go from many sources to a small number of sinks. In this case, network utility is largest when the coupling is at its maximum and the average shortest path is minimized. The second regime arises when many sources correspond to many sinks. Here, the optimal coupling no longer corresponds to the minimum average shortest path, as the congestion of traffic must also be taken into account. More generally, results indicate that coupled spatial systems can give rise to behavior that relies subtly on the interplay between the coupling and randomness in the source-sink distribution.

Figure 1

A system made of two coupled networks where the nodes of network 2 form a subset of the nodes of network 1. Edges of network 1 are shown in black, edges of network 2 are shown in red (gray offline), and nodes in common to both networks are considered to be coupled (shown by dashed lines). Highlighted in green (gray offline), we represent a path between two nodes, the “source” $i$ and the “sink” $j$.

Figure 2

Each instance of the system is generated according to the following process: (a) First, $N_1$ nodes (here $N_1=30$) are positioned at random within the unit circle and the Delaunay triangulation is produced; (b) the second network is then generated by drawing $N_2$ (here $N_2=10$) nodes uniformly from the existing ones ($N_2\le N_1$) and, once again, computing the Delaunay triangulation; (c) the combined system is no longer planar and can be represented as the top-down view of Fig. 1 (where zero weights are assigned to the dotted interconnecting lines).

Figure 3

Representations of OD matrices where each arrow corresponds to an entry in $T_{ij}$ and which relates to the set of points of Fig. 2. (a) A monocentric OD matrix. (b) A monocentric OD matrix randomly rewired with probability $p=0.5$.

Figure 4

Simulation results for the average shortest path and the Gini coefficient [$N_1=100$, $N_2=20$, and $p$ values: 0 (purple dots), 0.2 (blue squares), 0.4 (green diamonds), 0.6 (orange triangles), and 0.8 (red inverted triangles)]. When the coupling increases, the average shortest path decreases and the Gini coefficient can increase for large enough disorder.

Figure 5

Colormaps showing normalized edge flows—plotted at the midpoint of each edge—over many instances of the system. Colors are assigned from lightest to darkest, starting with white (for zero flow) and moving through yellow, orange and red for higher values of flow, until reaching black (maximum flow). Each subfigure corresponds to the following parameter values: (a) $p=0.2$, $\alpha =0.9$; (b) $p=0.2$, $\alpha =0.1$; (c) $p=0.8$, $\alpha =0.9$; (d) $p=0.8$, $\alpha =0.1$.

Figure 6

Existence of an optimal coupling: (2) Simulation results for $\mu =10$, $N_1=100$, $N_2=20$, and $p$ values: 0 (purple dots), and 0.8 (red inverted triangles) (only two values of $p$ are shown to ensure the lines of the best fit can be seen clearly). (b) Minima of quadratic best-fit curves for different values of $p$. We obtain $p^{*}\simeq 0.32$ using a straight-line approximation between the two closest points, above and below, ${\lambda }^{*}=1$. (The error bars shown are those of the closest data point to the minimum of the best-fit curve.)