Title: 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.