Are Percolation Transitions always Sharpened by Making Networks Interdependent?

We study a model for coupled networks introduced recently by Buldyrev et al., Nature 464, 1025 (2010), where each node has to be connected to others via two types of links to be viable. Removing a critical fraction of nodes leads to a percolation transition that has been claimed to be more abrupt than that for uncoupled networks. Indeed, it was found to be discontinuous in all cases studied. Using an efficient new algorithm we verify that the transition is discontinuous for coupled Erdos-Renyi networks, but find it to be continuous for fully interdependent diluted lattices. In 2 and 3 dimension, the order parameter exponent $\beta$ is larger than in ordinary percolation, showing that the transition is less sharp, i.e. further from discontinuity, than for isolated networks. Possible consequences for spatially embedded networks are discussed.

💡 Research Summary

The paper revisits the claim that coupling two interdependent networks always sharpens the percolation transition, making it discontinuous. Using a newly developed algorithm that adds nodes one by one while maintaining separate Union‑Find structures for the two layers, the authors are able to simulate very large systems with high statistical accuracy. For Erdős‑Rényi (ER) graphs, the analytical mean‑field equation S = (1 − e^{−zS})² correctly predicts a critical average degree z_c ≈ 2.4554, and simulations confirm a first‑order transition: the order parameter jumps from zero to S_c ≈ 0.512 at the threshold. In contrast, for fully interdependent diluted lattices in two and three dimensions, finite‑size scaling analyses reveal continuous transitions. In 2‑D, the critical occupation probability is p_c ≈ 0.96025 with exponents β ≈ 0.172, ν ≈ 1.19, and fractal dimension D_f ≈ 1.85; in 3‑D, p_c ≈ 0.871 with β ≈ 0.51, ν ≈ 0.86, D_f ≈ 2.40. These β values are larger than those of ordinary percolation (β = 5/36 ≈ 0.139 in 2‑D, β ≈ 0.417 in 3‑D), indicating that the transition is less abrupt, not more, when the networks are spatially embedded. The study thus demonstrates that the sharpening effect of interdependence is not universal: it holds for mean‑field‑like random graphs but fails for lattices where locality dominates. The authors suggest that real infrastructure networks, which are often embedded in physical space, may behave more like the lattice case, and they outline future directions such as coupling heterogeneous topologies, exploring more than two layers, and investigating semi‑local embeddings.