Congestion Induced by the Structure of Multiplex Networks

Multiplex networks are representations of multilayer interconnected complex networks where the nodes are the same at every layer. They turn out to be good abstractions of the intricate connectivity of multimodal transportation networks, among other types of complex systems. One of the most important critical phenomena arising in such networks is the emergence of congestion in transportation flows. Here, we prove analytically that the structure of multiplex networks can induce congestion for flows that otherwise would be decongested if the individual layers were not interconnected. We provide explicit equations for the onset of congestion and approximations that allow us to compute this onset from individual descriptors of the individual layers. The observed cooperative phenomenon is reminiscent of Braess’ paradox in which adding extra capacity to a network when the moving entities selfishly choose their route can in some cases reduce overall performance. Similarly, in the multiplex structure, the efficiency in transportation can unbalance the transportation loads resulting in unexpected congestion.

Figure 1

Contributions to the size of the queue of each node at each layer of a multiplex network in our standardized transportation model. Arrows mark the flow direction of elements in and out of the node.

Figure 2

Accuracy of the analytical value of ${\rho }_c$ given by Eq. (4) predicting the actual onset of congestion in experimental simulations on 500 random multiplex networks formed by two Erdős-Rényi networks (of 500 nodes) as layers. The inset shows the correlation between the experimentally obtained critical injection rate and the analytical approximation in Eq. (6) where $\lambda$ is approximated by 1. $R^2$ is the coefficient of determination for the linear fits.

Figure 3

Probability of obtaining a multiplex configuration that induces congestion when (a) the multiplex is composed of two Erdős-Rényi layers and (b) the topology is a random geometric multiplex. In both network topologies each layer has 500 nodes. The number of simulations points is ${50}^2$, and for each point we generate ${10}^2$ configurations fixing $⟨k_1⟩$ and $⟨k_2⟩$. The colors indicate the probability of observing that the onset of congestion of the multiplex satisfies ${\rho }_c<\min ({\rho }_c^{(1)},{\rho }_c^{(2)})$. The lines show the accuracy of Eq. (7) in detecting the region where the multiplex structure induces congestion. The solid lines represent the expression when the real value of $\lambda$ is used and the dashed lines when we approximate $\lambda$ by 1.

Figure 4

Sketch of the generation process of a random geometric multiplex. We first choose uniformly at random a set of $N$ points in a bidimensional space, $(x,y)\in [0,1{]}^2$; these are our node locations. We then generate the first layer by adding edges between all locations $i$ and $j$ separated by a distance $d_{ij}$ smaller than a certain radius $R_{L1}\in [0,0.4]$. The second layer is generated by adding edges between all node pairs with distance $R_{L2}^{\mathrm{Min}}<d_{ij}<R_{L2}^{\mathrm{Max}}$. The values of $R_{L2}^{\mathrm{Min}}\in [0,R_{L1}]$ force minimum overlapping between both layers. The value of $R_{L2}^{\mathrm{Max}}=R_{L2}^{\mathrm{Min}}+\mathrm{\Delta }R$ with $\mathrm{\Delta }R\in [0,R_{L1}]$ ensures the range $[R_{L2}^{\mathrm{Min}},R_{L2}^{\mathrm{Max}}]$ does not exceed the radius of the first layer.