Higher-Order Brain Connectivity

Updated 17 November 2025

Higher-order brain connectivity is a framework that analyzes multivariate, non-reducible interactions among three or more brain regions to capture synergy and redundancy.
It employs advanced methods, including multivariate information metrics, hypergraph and simplicial complex constructions, and topological data analysis to quantify complex neural patterns.
Empirical studies show that these connectivity measures differentiate healthy and clinical states by uncovering dynamic reconfiguration and distributed integrative motifs in the brain.

Higher-order brain connectivity refers to statistical dependencies and organizational motifs among three or more brain regions that cannot be reduced to collections of pairwise (bivariate) interactions. While traditional functional connectivity (FC) analyses focus on the edges of graphs—correlations or mutual information between pairs of regions—higher-order approaches aim to uncover the multivariate, potentially synergistic, and often non-reducible structures underpinning neurocognitive integration. Such structures may manifest as information synergies, redundancy-dominated collective states, cliques and simplices in topological constructs, or explicit hyperedges in hypergraphs. Understanding and quantitatively dissecting higher-order interactions is essential for elucidating the mechanisms underlying distributed computation, emergent cognition, and the pathophysiology of complex brain disorders.

1. Formal Foundations of Higher-Order Dependencies

A higher-order dependency occurs when the joint probability distribution over three or more regional time series cannot be factorized as independent pairwise links. Formally, for random variables $X_1, \ldots, X_N$ representing BOLD activity in $N$ brain regions, a dependency is higher-order if there exists information accessible in $P(X_1,\ldots,X_N)$ that is absent from the collection $\{P(X_i, X_j)\}_{i,j}$ . In information-theoretic terms, this manifests as joint uncertainty being resolved only by considering the multivariate pattern, i.e., some uncertainty about $X_1$ is only resolved by observing both $X_2$ and $X_3$ simultaneously—neither $X_2$ nor $X_3$ alone suffices (Varley et al., 2022).

Standard FC metrics, such as Pearson correlation $r_{ij}$ or mutual information $I(X_i; X_j)$ , are blind to these emergent patterns. The limitations of pairwise FC have driven the development of explicit and implicit formal frameworks for representing and quantifying higher-order connectomics.

2. Mathematical Quantification: Multivariate Information and Complexity Measures

Several multivariate information-theoretic quantities capture different facets of higher-order statistical structure:

Total Correlation (TC):

$TC(X) = \sum_{i=1}^N H(X_i) - H(X_1,\ldots,X_N)$

which measures the redundancy-dominated integration among all variables (Varley et al., 2022, Li et al., 2023).

Dual Total Correlation (DTC)/Binding Information:

$DTC(X) = H(X_1,\ldots,X_N) - \sum_{i=1}^N H(X_i|X_{-i})$

emphasizing information jointly shared but not present in any strict subset.

O-information ( $\Omega$ ) [Rosas et al., 2019]:

$\Omega(X) = TC(X) - DTC(X)$

Interpreted such that $\Omega > 0$ indicates a redundancy-dominated system, while $\Omega < 0$ signals synergy dominance—i.e., information produced only by coalitions (Varley et al., 2022, Li et al., 2023).

Partial Entropy Decomposition (PED): PED decomposes the joint entropy into non-negative “atoms” attributable to redundancy, unique information, or pure synergy, using the Möbius inversion of the partial order of all antichains among sources (Varley et al., 2023). This formalism clarifies how, for $N \geq 3$ , there exist distinct, quantifiable informational "modes" invisible to any bivariate analysis.
Matrix-based Rényi’s Entropy: Employs kernel embeddings to estimate multivariate entropies and concentrations, allowing estimation of higher-order (synergistic) information without explicit density estimation (Li et al., 2023).
Hodge Laplacians and Topological Decomposition: For explicit combinatorial structures such as simplicial complexes, the Hodge Laplacian $L_k$ on $k$ -simplices enables the separation of signals into irrotational (gradient), solenoidal (curl), and harmonic (topological cavity) components and supports spectral analyses of functional integration and segregation (Bispo et al., 10 Apr 2025).

3. Structural Representations: Hypergraphs, Simplicial Complexes, and Topological Invariants

Hypergraphs generalize pairwise graphs by allowing hyperedges covering any number $k>2$ of nodes, naturally capturing multi-region dependencies. Brain hypergraphs have been constructed using data-driven selection of node ensembles (e.g., by sparse regression (Dolci et al., 2 Aug 2025), MIMR information bottleneck (Qiu et al., 2023), or heuristic methods), and hyperedge weights can be assigned using measures such as algebraic connectivity (the second-smallest eigenvalue of the hypergraph Laplacian) to index the collective cohesion of the hyperedge's constituent regions (Dolci et al., 2 Aug 2025).

Simplicial Complexes arise in algebraic topology and encode collections of $k$ -simplices (e.g., nodes, edges, triangles, tetrahedra) closed under the face operation—every face of a simplex is itself a simplex. The clique complex of a (thresholded) FC graph captures all fully connected subgraphs as simplices. The topology of such complexes is succinctly described by Betti numbers $\beta_k$ (counting $k$ -dimensional cavities or voids), persistent homology diagrams (birth/death of features across thresholds), and topological entropy metrics (quantifying distribution heterogeneity across nodes and simplex orders) (Sizemore et al., 2016, Chung et al., 18 Mar 2025, Andjelkovic et al., 2020).

Co-fluctuation and Signed Complexes: Recent frameworks introduce signed simplicial complexes, encoding both positive and negative higher-order synergies via co-fluctuation measures (e.g., multiplication of temporal derivatives) and leveraging persistent homology to extract disease-relevant structural motifs (Zhao et al., 27 Jul 2025).

4. Computational and Statistical Methodologies

Panels of computational pipelines have been developed to infer, quantify, and model higher-order brain connectivity:

Simulated annealing and combinatorial search to find maximally synergistic ensembles by optimizing O-information over the exponentially many possible region subsets; typical solutions peak at group sizes $k \approx 10$ (Varley et al., 2022).
PED lattice enumeration and Möbius inversion enable direct, atomic decomposition of small region sets (typically $N \leq 4$ ) (Varley et al., 2023).
Random-walk node embeddings (e.g., node2vec): These generate higher-order feature vectors indicative of intrinsic brain characteristics (such as homotopic FC) but are sensitive to the FC construction methodology—partial correlation outperforming bivariate correlation for isolating direct dependencies (Khodabandehloo et al., 9 Jun 2024).
Graph neural networks (GNNs), attention mechanisms, and hypergraph neural networks (HGNNs): State-of-the-art models such as HOGANN, HOI-Brain, and HYBRID incorporate higher-order message passing, explicit multiway neighborhood aggregation, and information bottleneck-based hyperedge discovery. They yield interpretable resultants—subnetwork or circuit motifs whose statistical significance aligns with known neurocognitive circuits or disease markers (Ding et al., 29 Feb 2024, Qiu et al., 2023, Zhao et al., 27 Jul 2025).
Persistent homology on simplicial complexes derived from FC or co-fluctuation networks is used to track topological invariants (cavities, holes) that characterize multiway brain organization (Sizemore et al., 2016, Andjelkovic et al., 2020, Li et al., 1 Nov 2024, Chung et al., 18 Mar 2025).

These approaches face formidable combinatorial and statistical challenges, including the exponential growth of candidate $k$ -tuples, the need for robust multiple testing correction, sample size demands for stabilizing multivariate entropy, and variable interpretability and computational tractability across methods (Salehi et al., 10 Nov 2025).

5. Empirical Findings and Biological Significance

Convergent results across multiple analytical modalities indicate several robust properties of higher-order brain connectivity:

Synergy-dominated Resting State: Both O-information, PED, and matrix-based Rényi metrics demonstrate that normal resting-state human cortex is synergy-dominated for typical interaction orders $k=3,4$ —i.e., joint patterns encode information inaccessible to any sub-pair or sub-triad (Varley et al., 2022, Li et al., 2023, Li et al., 2023).
“Shadow Structure” and Integration: Highly synergistic ensembles form a distributed, sparse, and combinatorially rich "shadow structure," especially prominent at the intersection of canonical systems (frontoparietal, DMN, limbic, occipital). These motifs are invisible from standard pairwise FC but likely underpin cross-network integration (Varley et al., 2022, Varley et al., 2023).
Dynamic Reconfiguration: Frame-by-frame analyses of PED reveal that redundancy and synergy dominance alternate dynamically over time, correlating with spontaneous co-fluctuation events and implying a repertoire of transient “information states” that may encode functional flexibility (Varley et al., 2023).
Disease and Individual Differences: Higher-order connectivity reliably distinguishes healthy states from clinical syndromes (AD, schizophrenia, ASD), with disease marked by aberrant synergy or redundancy signatures in domain-specific circuits (e.g., cortico-striatal loops in AD, temporal/higher-cognitive synergies in schizophrenia) (Li et al., 2023, Zhao et al., 27 Jul 2025, Dolci et al., 2 Aug 2025).
Topological Features: Structural connectomes exhibit robust higher-order features such as large cliques and persistent topological cavities (loops, voids) that bridge subcortical “hub” regions to late-evolving cortex; these may encode fundamental constraints on information routing and integration (Sizemore et al., 2016, Andjelkovic et al., 2020).
Modality and Method Sensitivity: The successful detection and interpretability of higher-order patterns strongly depend on the preprocessing, FC metric, parcellation, and inferential pipeline, with partial correlation preferred when isolating direct dependencies (Khodabandehloo et al., 9 Jun 2024).

6. Comparative Methodological Landscape: Implicit and Explicit Approaches

A recent critical review (Salehi et al., 10 Nov 2025) provides a conceptual map, distinguishing:

Implicit Paradigms: Quantify multiway dependence using scalar information-theoretic (e.g., O-information, total correlation) or correlation-of-correlation metrics. These are relatively easy to deploy and yield interpretable summary statistics, but lack explicit combinatorial structure and may conflate true synergy with compositional pairwise effects.
Explicit Paradigms: Construct hypergraphs, simplicial complexes, and apply topological data analysis (persistent homology, Betti numbers, Hodge decompositions). These frameworks provide explicit structural representations but entail severe combinatorial scaling and depend critically on selection, weighting, and thresholding strategies for group interactions. The algebraic and spectral machinery (e.g., hypergraph Laplacian eigenvalues, Hodge Laplacians) supplies principled metrics for hyperedge or cavity “tightness” and functional interpretation (Dolci et al., 2 Aug 2025, Bispo et al., 10 Apr 2025).
Hybrid Approaches: Integrate implicit and explicit perspectives, e.g., by using multivariate information-theoretic scores to select candidate hyperedges or simplices prior to structural analysis, or by projecting explicit motif counts back onto pairwise features for predictive modeling.

Major statistical and computational bottlenecks arise, including multiple comparisons across high-order interaction candidates, requirement for robust null models, and the limited scalability of persistent homology to large $N$ and high $k$ (Salehi et al., 10 Nov 2025, Chung et al., 18 Mar 2025).

7. Outlook: Open Directions and Theoretical Synthesis

Current research trajectories emphasize the integration of information-theoretic, topological, and machine learning–oriented methods:

Principled construction of structure: Synergy metrics can guide explicit structure-building (hyperedges/simplices) to identify meaningful higher-order motifs.
Joint learning architectures: Deep networks that embed explicit higher-order objects (learnable hypergraphs, transformer architectures with motif channels) show promise for end-to-end biomarker discovery (Qiu et al., 2023, Ding et al., 29 Feb 2024, Zhao et al., 27 Jul 2025, Li et al., 27 Aug 2025).
Dynamic analyses and generative models: Development of frameworks for temporal tracking of higher-order connectivity, and causal/biophysical generative models that predict observed patterns from microcircuit interactions or developmental constraints, remain crucial.
Multi-modal and scale-bridging analysis: Connecting higher-order functional connectomics with structural, metabolic, or cellular networks, and integrating multimodal imaging modalities, are emerging as major priorities.

A plausible implication is that higher-order interactions form the mechanistic substrate of integrative cognitive function, adaptive flexibility, and pathological dysconnection in neuropsychiatric disease. Methodologically, the convergence of implicit and explicit approaches—guided by rigorous statistical control and mechanistic interpretability—offers a robust roadmap for future research in brain connectomics.