Percolation in higher order networks via mapping to chygraphs

TL;DR

This paper introduces a method to analyze percolation in higher order networks by mapping them to chygraphs, simplifying calculations and enabling analytical solutions for complex network resilience problems.

Contribution

It presents a novel mapping of higher order networks to chygraphs, facilitating analytical percolation analysis and extending previous results with symbolic calculations.

Findings

Analytical equations for percolation in chygraphs derived

Mapping simplifies complex higher order network analysis

Code available for symbolic calculations

Abstract

Percolation theory investigates systems of interconnected units, their resilience to damage and their propensity to propagation. For random networks we can solve the percolation problems analytically using the generating function formalism. Yet, with the introduction of higher order networks, the generating function calculations are becoming difficult to perform and harder to validate. Here, I illustrate the mapping of percolation in higher order networks to percolation in chygraphs. Chygraphs are defined as a set of complexes where complexes are hypergraphs with vertex sets in the set of complexes. In a previous work I reported the generating function formalism to percolation in chygraphs and obtained an analytical equation for the order parameter. Taking advantage of this result, I recapitulate analytical results for percolation problems in higher order networks and report extensions to more complex scenarios using symbolic calculations. The code for symbolic calculations can be found at https://github.com/av2atgh/chygraph.