Beyond the locally tree-like approximation for percolation on real networks

TL;DR

This paper introduces a message passing framework that accurately predicts percolation on real networks by accounting for short loops, significantly improving over traditional locally tree-like approximations.

Contribution

The authors develop a novel message passing approach that discounts triangles, enhancing percolation predictions on real-world networks with clustering.

Findings

High accuracy in predicting percolation diagrams on real networks

Significant improvement over locally tree-like approximation

Discrepancies mainly due to longer loops

Abstract

Theoretical attempts proposed so far to describe ordinary percolation processes on real-world networks rely on the locally tree-like ansatz. Such an approximation, however, holds only to a limited extent, as real graphs are often characterized by high frequencies of short loops. We present here a theoretical framework able to overcome such a limitation for the case of site percolation. Our method is based on a message passing algorithm that discounts redundant paths along triangles in the graph. We systematically test the approach on 98 real-world graphs and on synthetic networks. We find excellent accuracy in the prediction of the whole percolation diagram, with significant improvement with respect to the prediction obtained under the locally tree-like approximation. Residual discrepancies between theory and simulations do not depend on clustering and can be attributed to the presence of loops longer than three edges. We present also a method to account for clustering in bond percolation, but the improvement with respect to the method based on the tree-like approximation is much less apparent.