Message passing and cyclicity transition

Message passing, also known as belief propagation, is a versatile framework for analyzing models defined on graphs. Its most prototypical application is percolation; yet, the interpretation of the message passing formulation of percolation remains elusive. We show that the message passing solutions commonly associated with the probability of belonging to the giant component actually identify reachability from cycles. This interpretation generally applies to bond and site percolation on any directed or undirected networks. Our findings highlight the distinction between transition in cyclicity and the emergence of the giant component.