Simplicial Complex Emergence on Directed Hypergraphs

TL;DR

This paper investigates the emergence of simplicial complexes in directed hypergraphs with adaptive couplings, using representation theory to identify conditions for simplicial structure formation and stability.

Contribution

It introduces a novel theoretical framework employing symmetric group representation to analyze when higher-order adaptive networks form simplicial complexes.

Findings

Simplicial complexes emerge under symmetric and antisymmetric limits.

Stability of simplicial structures is certified via boundary tests and drift conditions.

Simulations validate the theoretical conditions for higher-order structure emergence.

Abstract

We study when co-evolving (or adaptive) higher-order networks defined on directed hypergraphs admit a simplicial description. Binary and triadic couplings are modelled by time-dependent weight tensors. Using representation theory of the symmetric group $S_k$, we decompose these tensors into fully symmetric, fully antisymmetric, and mixed isotypic components, and track their Frobenius norms to define three asymptotic regimes and a quantitative notion of convergence. In the symmetric (resp. antisymmetric) limit, we certify emergence and stability of simplicial complexes via a local boundary test and interior drift conditions that enforce downward-closure; in the mixed limit, we show that the minimal faithful object is a semi-simplicial set. We illustrate the theory with simulations that track the isotypic Frobenius norms and the higher-order structure. Practically, our work provides rigorous conditions under which homological tools are justified for adaptive higher-order systems.