Higher-Order Representations

• Higher-order representations are formal abstractions that capture multi-entity interactions and meta-properties, enabling advanced modeling of complex systems.
• They utilize mathematical structures like hypergraphs and simplicial complexes to simulate multiway dependencies, memory effects, and meta-cognitive processes.
• Their application across fields such as machine learning, neuroscience, and network science enhances predictive accuracy and provides deeper insights into system dynamics.

Higher-order representations are formalized abstractions that encode relationships, dependencies, or meta-properties arising from interactions among three or more entities, or that are "about" lower-level (first-order) representations themselves. These representations extend classical pairwise frameworks—such as dyadic graphs, first-order logic, local message passing, or fixed-order dynamical models—by introducing structures and semantics capable of modeling multiway dependencies, memory, compositionality, or meta-cognition. The paper and application of higher-order representations is critical in fields including physics, network science, machine learning, theoretical computer science, topological data analysis, and neuroscience, where complex systems regularly exhibit interactions or representational requirements that defy naive binary encoding.

1. Formal Structures and Theoretical Foundations

Higher-order representations arise in multiple mathematical frameworks, most notably through hypergraphs, simplicial complexes, cell complexes, higher-order network constructions (such as memory-nodes or higher-order transitions), and higher-order functors in coalgebraic settings.

• Hypergraphs generalize graphs by encoding hyperedges that connect arbitrary subsets of nodes, thus capturing non-pairwise, multi-entity interactions. These are suitable for modeling group collaborations, biochemical complexes, or multi-drug interactions (Srinivasan et al., 2021, Pellegrin et al., 13 Feb 2025).
• Simplicial and Cellular Complexes further extend this idea by encoding not just sets of nodes, but structured inclusion relationships (e.g., triangles and tetrahedra in a simplicial complex) (Carrasco et al., 21 May 2025, Tian et al., 29 Feb 2024).
• Higher-Order Behavioural Functors in coalgebraic theory are defined circularly, where the behavioural functor references its own final coalgebra. The prototypical equation $B \cong \mathbb{F}(|\nu B|, |\nu B|)$ captures the essence of semantic higher-order modeling, in contrast to term-based (syntactic) approaches (Peressotti, 2016).
• Meta-Representations in Neuroscience (higher-order representations of uncertainty or signal reliability) formally distinguish neural states that encode properties "about" first-order representations (FORs), such as signal strength, reliability, or expected noise distributions. These are often modeled using generative Bayesian approaches, where posterior-like HORs integrate likelihood-like (momentary) and prior-like (learned) uncertainty (Peters et al., 23 Jun 2025, Asrari et al., 18 Mar 2025).

2. Methodologies for Construction and Computation

The realization of higher-order representations necessitates algorithms and frameworks that go beyond local, pairwise aggregation or traditional message passing.

• Higher-Order Networks (HON) are constructed by augmenting nodes with contextual information—encoding histories of variable length—to capture sequential dependencies in dynamic systems. The methodology involves rule extraction (using criteria such as Kullback–Leibler divergence) to determine whether memory-augmented nodes are necessary, and network wiring to ensure statistical fidelity (Xu et al., 2015, Coquidé et al., 2021).
• Higher-Order Positional and Structural Encoders (HOPSE) circumvent the combinatorial burden of higher-order message passing by precomputing encodings over augmented Hasse graphs derived from combinatorial complexes, combining graph-theoretic positional features (Laplacian eigenvectors, RWSE, etc.) with learnable functions, and aggregating across neighborhoods efficiently (Carrasco et al., 21 May 2025).
• Efficient Higher-Order Belief Propagation utilizes low-rank tensor decompositions (CP model) to model higher-order factors in probabilistic graphical models, embedding these message-passing updates within neural networks and achieving linear scaling with respect to the number of variables per factor (Dupty et al., 2020).
• Spectral Moments for Higher-Order Networks provide compact, interpretable representations by splitting networks into uniform hypergraph layers, extracting spectral information from random walk transition matrices, and computing moments $m_\ell = (1/n)\sum_{i=1}^n \lambda_i^\ell$ to capture global structural properties such as degree and clustering (Tian et al., 29 May 2025).
• Guiding-Center Theory in Plasma Physics employs perturbative Lie-transform methods to construct higher-order corrections to equations of motion, leading to complementary (symplectic vs. Hamiltonian) higher-order representations with proven equivalence (Brizard et al., 2012).

3. Expressivity, Power, and Theoretical Guarantees

Higher-order representations increase the expressive capacity of models and learning algorithms beyond first-order analogs.

• Expressivity Beyond 1-WL: The incorporation of hypergraph-level encodings—such as Hodge Laplacian Positional Encodings (H-LAPE), curvature profiles, or higher-order random walk features—provably increases the distinguishability of message passing GNNs beyond that of the 1-Weisfeiler–Leman (1-WL) test on graphs. For example, certain graph pairs indistinguishable by 1-WL can be separated when hypergraph-level features are introduced (Pellegrin et al., 13 Feb 2025).
• Permutation and Local-Isomorphism Invariance: Modern hypergraph neural network (HGNN) architectures leverage injective set functions in update rules to guarantee invariance to permutations and to preserve local-isomorphism, ensuring robustness and faithful representation of higher-order structures (Srinivasan et al., 2021).
• Comparative Evaluation: Systematic comparisons show graph-level models equipped with hypergraph-level encodings often outperform native hypergraph message passing architectures, even on data naturally parameterized as hypergraphs, underscoring the practical potency of embedding higher-order information within familiar graph neural network pipelines (Pellegrin et al., 13 Feb 2025, Srinivasan et al., 2021).

4. Applications Across Scientific Domains

Higher-order representations have demonstrated significant practical impact across diverse domains.

• Complex Network Analysis: HONs enable faithful modeling of real-world sequential processes (shipping routes, web clickstreams, user mobility), improving tasks such as random walk simulation, prediction accuracy, clustering resolution, and ranking with direct compatibility to classical algorithms (e.g., PageRank) (Xu et al., 2015, Coquidé et al., 2021).
• Relational Learning and Molecule Modeling: In molecular sciences, higher-order factors (e.g., functional groups, multi-bond constraints) are essential for accurately capturing chemical properties. Neural architectures that explicitly encode such higher-order motifs achieve superior predictive accuracy and compact representations (Dupty et al., 2020, Srinivasan et al., 2021).
• Quantum Optimization: Direct higher-order (HUBO) representations in the Quantum Approximate Optimization Algorithm (QAOA) substantially reduce qubit requirements and improve solution quality compared to quadratic (QUBO) encodings, despite deeper circuits. Factoring methods can further mitigate gate depth and improve performance on hardware (Bell et al., 24 Sep 2025).
• Neuroscience and Uncertainty Representation: Generative models (such as the NERD RL-diffusion model) reveal that brains construct higher-order representations of their own uncertainty—integrating online estimates with prior expectations—in a manner consistent with Bayesian inference, with direct implications for metacognition and learning (Asrari et al., 18 Mar 2025, Peters et al., 23 Jun 2025).
• Explanations in Graph Learning: Lifting graphs to cell complexes for explanation purposes (e.g., HOGE/FORGE) enhances interpretability by aligning GNN explanations with meaningful higher-order structures (cycles, rings), yielding improved agreement with ground truth and explanatory fidelity (Sinha et al., 5 Jun 2024).

5. Computational and Scalability Considerations

The combinatorial complexity inherent in higher-order structures presents significant computational challenges.

• Avoidance of Message Passing Explosion: The message passing–free design of HOPSE overcomes the factorial growth of message routes in higher-order domains (combinatorial complex of rank $R$ yields a minimum of $(R+1)!$ routes) by encoding relevant structure in preprocessing (Carrasco et al., 21 May 2025).
• Spectral Methods: Computing only a handful of spectral moments per uniform layer provides linear scalability, robust global features, and avoids the parameter explosion of motif- and message-passing-based approaches (Tian et al., 29 May 2025).
• Integration with Existing Pipelines: Numerous frameworks (HONE, HON, HOGE, GCN-TULHOR) are explicitly constructed to be compatible with existing network analysis, GNN training, or explanatory pipelines, enabling practitioners to leverage higher-order representations without algorithmic redesign (Rossi et al., 2018, Xu et al., 2015, Sinha et al., 5 Jun 2024, Tran et al., 14 Sep 2025).

6. Future Directions and Open Challenges

The maturation of higher-order representations opens avenues for both theoretical and applied advances.

• Benchmark and Dataset Limitations: Widespread adoption is hindered by the scarcity of truly higher-order-structured datasets; many available benchmarks are permutations or expansions of dyadic graphs (Pellegrin et al., 13 Feb 2025).
• Expressivity Beyond Current Frameworks: Extending hypergraph-level encoding techniques to richer topological domains (e.g., polyhedral complexes, CW complexes, persistent homology) and systematically quantifying their expressivity beyond standard tests remain ongoing challenges (Tian et al., 29 Feb 2024, Carrasco et al., 21 May 2025).
• Interpretability and Human-Like Uncertainty Processing: Combining analytical approaches (probabilistic coding, RL-diffusion, generative modeling) with neuroscientific data aims to clarify how higher-order uncertainty is encoded, updated, and exploited, with implications for both cognitive science and machine learning (Peters et al., 23 Jun 2025, Asrari et al., 18 Mar 2025).
• Optimization in Quantum and Combinatorial Domains: Continued development of factoring and encoding methods is critical to balance qubit efficiency and circuit depth, especially for NISQ-era quantum devices (Bell et al., 24 Sep 2025).

In summary, higher-order representations unify a set of mathematical, computational, and empirical techniques that enable the modeling, analysis, and interpretation of complex systems whose fundamental structure, dependency, or meta-cognitive properties transcend first-order pairwise frameworks. Their development is integral to advancing data-driven discovery in science and engineering, offering both formal guarantees and practical utility.