Spectral estimation of the percolation transition in clustered networks
This work addresses the challenge of accurately predicting percolation transitions in sparse, clustered networks, which is important for network science and applications like epidemiology, but it is incremental as it builds on prior spectral methods.
The authors tackled the problem of estimating the percolation transition in clustered networks, where existing spectral bounds are not tight, by proposing a message passing algorithm and relating the transition to the eigenvalue of a new triangle-non-backtracking matrix, resulting in a tighter lower-bound that becomes exact for networks with no loops longer than 3.
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.