Full Connectivity: Corners, edges and faces

TL;DR

This paper introduces a cluster expansion method to analyze the probability of full connectivity in high-density random networks within confined spaces, emphasizing the importance of boundary effects and demonstrating universality through analytical formulas and examples.

Contribution

It presents a novel analytical approach that accounts for boundary effects in high-density network connectivity, extending percolation theory concepts.

Findings

Boundary effects are dominant in high-density connectivity.

Universal analytical formulas are derived and confirmed numerically.

The approach is simple and applicable to various models.

Abstract

We develop a cluster expansion for the probability of full connectivity of high density random networks in confined geometries. In contrast to percolation phenomena at lower densities, boundary effects, which have previously been largely neglected, are not only relevant but dominant. We derive general analytical formulas that show a persistence of universality in a different form to percolation theory, and provide numerical confirmation. We also demonstrate the simplicity of our approach in three simple but instructive examples and discuss the practical benefits of its application to different models.