Percolation Transitions Are Not Always Sharpened by Making Networks Interdependent

We study a model for coupled networks introduced recently by Buldyrev et al., [Nature (London) 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 dimensions, 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.