On the accuracy of message-passing approaches to percolation in complex networks

The Message-Passing Approach (MPA) is the state-of-the-art technique to obtain quasi-analytical predictions for percolation on real complex networks. Besides being intuitive and straightforward, it has the advantage of being mathematically principled: it is exact on trees, while yielding generally good predictions on networks containing cycles as do most real complex networks. Here we show that the MPA does not perform its calculations on some ill-defined tree-like approximation of the network, as its formulation leads to believe, but rather considers a random network ensemble in which the original network is cloned and shuffled an infinite number of times. We conclude that the fact that the MPA is exact on trees does not imply that it is nearly exact on tree-like networks. In fact we find that the closer a non-tree network is to a tree, the worse the MPA accuracy becomes.