Percolation in Random Graphs: A Finite Approach

TL;DR

This paper introduces an exact finite-size method for determining the percolation threshold in Erdős-Rényi digraphs, connecting finite and infinite graph percolation phenomena through algebraic adjacency matrix analysis.

Contribution

It presents a novel, exact finite-size analytical approach to calculate the percolation threshold using minimal Hamiltonian cycles, applicable to various graph models.

Findings

Exact percolation threshold for finite Erdős-Rényi digraphs derived

Method scales with graph size, matching infinite graph results

Applicable to all models with algebraic adjacency matrices

Abstract

We propose an approach to calculate the critical percolation threshold for finite-sized Erdos-Renyi digraphs using minimal Hamiltonian cycles. We obtain an analytically exact result, valid non-asymptotically for all graph sizes, which scales in accordance with results obtained for infinite random graphs using the emergence of a giant connected component as marking the percolation transition. Our approach is general and can be applied to all graph models for which an algebraic formulation of the adjacency matrix is available.