Bond Percolation on Multiplex Networks

We present an analytical approach for bond percolation on multiplex networks and use it to determine the expected size of the giant connected component and the value of the critical bond occupation probability in these networks. We advocate the relevance of these tools to the modeling of multilayer robustness and contribute to the debate on whether any benefit is to be yielded from studying a full multiplex structure as opposed to its monoplex projection, especially in the seemingly irrelevant case of a bond occupation probability that does not depend on the layer. Although we find that in many cases the predictions of our theory for multiplex networks coincide with previously derived results for monoplex networks, we also uncover the remarkable result that for a certain class of multiplex networks, well described by our theory, new critical phenomena occur as multiple percolation phase transitions are present. We provide an instance of this phenomenon in a multiplex network constructed from London rail and European air transportation data sets.

Popular Summary

In a multiplex network, nodes are connected by edges and links of different types; if the nodes represent cities, for example, then the edges of different types could be various transportation modes (e.g., air, road, rail) that connect the cities. Bond percolation is the process of randomly removing the edges of the network and asking whether the remaining edges can keep the network as a whole connected. Here, we present an analytical approach for bond percolation on multiplex networks and use it to examine the importance of the multiplex nature of the networks as opposed to the “projected” network where the differences between edge types are ignored.

We show that in many cases the theory for projected networks is sufficient to describe bond percolation on multiplex networks, but we also identify a class of multiplex networks in which multiple phase transitions are present. We analyze a combined multiplex of the European Union air ($nodes=450$) and London rail transportation systems ($nodes=369$), and we connect the layers by considering the ten nodes in common between the air and rail systems. We show that the percolation degradation of the multiplex is induced by the fragility of the rail network, giving a clear illustration of how the interconnectivity of multiplex structures may suffer from fragilities that depend on the most vulnerable edge types. In other words, multiplex networks can be more susceptible to overall fragility than their constituent parts.

We expect that our findings will have implications for the robustness of real-life transportation and communication networks.

Figure 1

Expected size of the GCC as a function of $p$ for (a) a maximally correlated two-layer multiplex network made of two Poisson random networks and (b) an uncorrelated three-layer multiplex network made of one Poisson random network and two scale-free networks. Numerical simulations are averaged over 100 realizations, and $N={10}^5$. Peaks in the expected size of the SLCC indicate percolation transitions.

Figure 2

Expected size of the GCC as a function of $p$ for (a) a two-layer multiplex network made of two Poisson random networks and (b) a three-layer multiplex network made of three Poisson random networks. Both networks are highly anticorrelated. Numerical simulations are averaged over 100 realizations, $N={10}^5$, and the overlap has 100 nodes. Peaks in the expected size of the SLCC indicate percolation transitions.

Figure 3

Expected size of the GCC as a function of $p$ for a three-layer multiplex network made of three Poisson random networks. Each layer contains $N={10}^4$ nodes. Layers 1 and 2 have an overlap of ${10}^3$ nodes, as do layers 2 and 3. Peaks in the expected size of the SLCC indicate percolation transitions.

Figure 4

Expected size of the GCC as a function of $p$ for a two-layer multiplex network constructed from the London rail transportation network and the EU air transportation network. Numerical simulations are averaged over 1000 realizations. Peaks in the expected size of the SLCC indicate percolation transitions.