Spectral estimation of the percolation transition in clustered networks

Abstract

There have been several spectral bounds for the percolation transition in networks, using spectrum of matrices associated with the network such as the adjacency matrix and the non-backtracking matrix. However they are far from being tight when the network is sparse and displays clustering or transitivity, which is represented by existence of short loops e.g. triangles. In this work, for the bond percolation, we first propose a message passing algorithm for calculating size of percolating clusters considering effects of triangles, then relate the percolation transition to the leading eigenvalue of a matrix that we name the triangle-non-backtracking matrix, by analyzing stability of the message passing equations. We establish that our method gives a tighter lower-bound to the bond percolation transition than previous spectral bounds, and it becomes exact for an infinite network with no loops longer than 3. We evaluate numerically our methods on synthetic and real-world networks, and discuss further generalizations of our approach to include higher-order sub-structures.