Exact solution of bond percolation on small arbitrary graphs

We introduce a set of iterative equations that exactly solves the size distribution of components on small arbitrary graphs after the random removal of edges. We also demonstrate how these equations can be used to predict the distribution of the node partitions (i.e., the constrained distribution of the size of each component) in undirected graphs. Besides opening the way to the theoretical prediction of percolation on arbitrary graphs of large but finite size, we show how our results find application in graph theory, epidemiology, percolation and fragmentation theory.

💡 Research Summary

The paper presents a complete analytical framework for bond percolation on small arbitrary graphs, delivering exact results for the distribution of component sizes after random edge removal. The authors begin by highlighting the limitations of classical percolation theory, which largely addresses infinite or regular lattices and provides only macroscopic quantities such as the percolation threshold or average cluster size. Real‑world networks—social contact graphs, infrastructure systems, biological interaction maps—are finite, heterogeneous, and often lack any regular structure, making exact predictions essential for applications ranging from epidemic control to network resilience.

To overcome these challenges, the authors formulate the problem on a graph (G=(V,E)) where each edge is independently retained with probability (p) (or removed with probability (1-p)). For any subset of vertices (S\subseteq V), the event that (S) forms a single connected component is precisely the conjunction of two independent conditions: (i) every edge internal to (S) must survive, and (ii) every edge crossing the boundary of (S) must be removed. This yields the exact probability

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