Simplicial Complex Emergence on Directed Hypergraphs

TL;DR

The paper develops a tensor-based framework for adaptive higher-order networks on directed hypergraphs, leveraging $S_k$ representation theory to decompose edge tensors into symmetric, antisymmetric, and mixed components. By tracking Frobenius-norms of these isotypic pieces, it identifies three asymptotic regimes (symmetric, antisymmetric, mixed) that dictate the emergent combinatorial structure: unoriented simplicial complexes, oriented simplicial complexes, or semi-simplicial sets. It proves boundary-based retention results that enforce downward closure in the symmetric and antisymmetric cases and extends to semi-simplicial sets in the mixed regime, with explicit constructions of region boundaries and outward-pointing conditions. Numerical simulations illustrate regime convergence and the corresponding higher-order structure, providing a rigorous basis for applying homological tools to adaptive higher-order systems. The work lays groundwork for topological analyses of coevolving higher-order dynamics and paves the way for extensions to even higher-order interactions.

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.

Figures (9)

• Figure 1: Diagrams illustrating the orthogonal splitting of $V_2$ and $V_3$ into their isotypic components.
• Figure 2: Schematic representation of the asymptotic regimes for adaptive triadic hypergraphs (Definition \ref{['def:asymptotic_regime']}). The outer rectangle denotes the mixed regime, in which symmetric and antisymmetric parts coexist with mixed components. The two large circles correspond to the asymptotically symmetric (left) and asymptotically antisymmetric (right) regimes, each containing an inner core of exactly symmetric or antisymmetric hypergraphs. All four circles meet at the zero hypergraph in the centre.
• Figure 3: Symmetry–component norms in the binary and triadic layers.
• Figure 4: Convergence to the antisymmetric regime. (a) Binary layer. (b) Triadic layer.
• Figure 5: Violation of downward closure: the triad $(i,j,k)$ is above threshold ($|A^{(2)}_{ijk}|\geq \delta$), edges $(i,j)$ and $(i,k)$ are above threshold, but $(j,k)$ falls below threshold ($|A^{(1)}_{jk}|<\delta$), so this is not a simplicial complex.

Theorems & Definitions (43)

• Definition 2.1: Adaptive, Triadic, Network Dynamical System
• Remark 2.2
• Example 2.3: Generalised, Adaptive, Triadic Higher–Order Kuramoto Model
• Remark 2.4: Directed hypergraph conventions
• Definition 2.5: Directed hypergraph; algebraic form
• Remark 2.6: Adaptive triadic systems as algebraic directed hypergraphs
• Example 2.7
• Remark 2.8
• Definition 2.9: Representation and Irreducibility
• Definition 2.10: Asymptotic Regime