Alternative expression of message passing on networks

Message passing techniques on networks encompasses a family of related methods that can be employed to ascertain many important properties of a network. It is widely considered to be the state of the art formulation for networked systems and advances in this method have a wide impact across multiple literatures. One property that message passing can yield is the size of the largest connected component in the network following bond percolation. In this paper, we introduce an alternative method of finding this value that differs from the standard approach. Like the canonical approach, our method is exact on trees and an approximation on arbitrary graphs. We show that our method lends itself to the description of a variety of generalisations of bond percolation such as sequential percolation and non-binary percolation and can yield information about the local environment of a node in percolation equilibrium that the traditional approach cannot.