Simplicial Complex Emergence on Directed Hypergraphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908680723628032 |
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| author | Kuehn, Christian Murphy, Fergal |
| author_facet | Kuehn, Christian Murphy, Fergal |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_00043 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simplicial Complex Emergence on Directed Hypergraphs Kuehn, Christian Murphy, Fergal Combinatorics Algebraic Topology 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. |
| title | Simplicial Complex Emergence on Directed Hypergraphs |
| topic | Combinatorics Algebraic Topology |
| url | https://arxiv.org/abs/2512.00043 |