Percolation Theory on Interdependent Networks Based on Epidemic Spreading

We consider percolation on interdependent locally treelike networks, recently introduced by Buldyrev et al., Nature 464, 1025 (2010), and demonstrate that the problem can be simplified conceptually by deleting all references to cascades of failures. Such cascades do exist, but their explicit treatment just complicates the theory – which is a straightforward extension of the usual epidemic spreading theory on a single network. Our method has the added benefits that it is directly formulated in terms of an order parameter and its modular structure can be easily extended to other problems, e.g. to any number of interdependent networks, or to networks with dependency links.

💡 Research Summary

The paper revisits percolation on interdependent networks—a problem that gained prominence after the seminal work of Buldyrev et al. (Nature 2010)—and shows that the whole phenomenon can be captured without invoking the cumbersome “cascade of failures” formalism. Starting from the well‑known epidemic (SIR‑type) spreading description on a single random network, the authors write the self‑consistency equations for the probability S that a randomly chosen node belongs to the giant component and for the probability S′ that a node reached by following a random edge belongs to the giant component. Using generating functions G₀(x)=∑ₖp(k)xᵏ and G₁(x)=G₀′(x)/z, they recover the classic result S=1−G₀(1−S′) and S′=1−G₁(1−S′). For Erdős‑Rényi graphs this reduces to the simple equation S=1−e^{−zS}, which yields a continuous (second‑order) transition at average degree z=1.

The authors then consider two fully interdependent networks A and B, where each node in A depends on exactly one node in B and vice‑versa. A node belongs to the “AB‑cluster” only if it is simultaneously connected to the giant component through A‑links and through B‑links. This leads to a product form for the order parameter: S=(1−G₀^A(1−S′^A))(1−G₀^B(1−S′^B)). The edge‑based probabilities satisfy S′^A=(1−G₁^A(1−S′^A))(1−G₀^B(1−S′^B)) and the analogous equation for B. For two ER networks with identical mean degree z the equations collapse to S=(1−e^{−zS})². Solving S−(1−e^{−zS})²=0 together with its derivative yields a discontinuous (first‑order) transition at z_c≈2.455 with a jump size S_c≈0.511, exactly matching previous cascade‑based results.

The framework is then generalized to M interdependent networks. The order parameter becomes the product over all layers, S=∏_{m=1}^{M}