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 multipex network constructed from London rail and European air transportation datasets.

💡 Research Summary

The paper develops a comprehensive analytical framework for bond percolation on multiplex (multilayer) networks and uses it to compute both the expected size of the giant connected component (GCC) and the critical bond occupation probability (pc). The authors begin by modeling a multiplex system as a set of M layers that share the same set of nodes but have possibly distinct edge sets and degree distributions. For each layer i they introduce a bond occupation probability Ti (the probability that any edge in that layer is retained). By employing generating‑function techniques they derive a set of self‑consistent equations for the probabilities ui that a randomly followed edge in layer i does not lead to the GCC. The equations read

 ui = 1 – Ti + Ti ∑k P(k) ∏j≠i ujkj,

where P(k) is the joint degree distribution across layers and kj is the degree of a node in layer j. Solving these equations yields the GCC size

 S = 1 – ∑k P(k) ∏i ui ki.

The critical point is identified by linearizing the system around ui = 1 and examining the Jacobian J whose elements are Jij = Ti ⟨kikj⟩/⟨ki⟩. The largest eigenvalue λmax of J crossing unity signals the onset of percolation. This formulation naturally incorporates inter‑layer degree correlations through the mixed moments ⟨kikj⟩, allowing the authors to explore how positive (ρij > 0) or negative (ρij < 0) correlations shift pc. Positive correlations lower pc because high‑degree nodes tend to be hubs in several layers simultaneously, creating a robust backbone; negative correlations raise pc by dispersing connectivity.

To test the theory, the authors construct a real‑world multiplex network by overlaying the London underground rail system (layer 1) with the European air‑transport network (layer 2). The two layers have markedly different average degrees (≈2.8 for rail, ≈5.6 for air) and sparse inter‑layer coupling (only nodes that are both stations and airports are shared). Simulations of bond percolation on this system reveal two distinct percolation thresholds: p ≈ 0.12, where the rail layer alone begins to support a GCC, and p ≈ 0.35, where the addition of air links triggers a second, abrupt growth of the GCC. This “multiple percolation transition” is a hallmark of multiplex structure and cannot be captured by a monoplex projection that collapses all edges into a single layer. The authors further validate their analytical predictions against Monte‑Carlo results, finding excellent agreement for both thresholds and the GCC size across the whole range of p.

A second set of experiments uses synthetic multiplex networks with controllable inter‑layer degree correlation ρ. By varying ρ from 0 (uncorrelated) to 0.9 (strongly positively correlated) they demonstrate a systematic decrease in pc and a merging of the two transitions into a single one as correlation strengthens. This illustrates that multiplex effects are most pronounced when layers are heterogeneous and weakly correlated.

The paper’s contributions are threefold. First, it extends classic bond‑percolation theory to multiplex graphs by deriving exact self‑consistency equations that respect the full joint degree distribution. Second, it highlights the pivotal role of inter‑layer degree correlations in shaping percolation thresholds and reveals the possibility of multiple, sequential phase transitions. Third, it provides empirical evidence from transportation data that such phenomena occur in real systems, thereby arguing convincingly that multiplex modeling yields insights unavailable from monoplex projections. The authors conclude by suggesting future directions, including time‑varying multiplexes, heterogeneous bond occupation probabilities, and the coupling of percolation with dynamical processes such as epidemic spreading or information diffusion.