Beyond Pairwise Connections: Extracting High-Order Functional Brain Network Structures under Global Constraints

TL;DR

This work tackles the limitation of pairwise functional brain network (FBN) modeling by introducing GCM, a framework that learns high-order FBN structures under four global constraints across four semantic modeling resolutions. GCM uses a prototype graph with Gumbel-Sigmoid relaxation, a Batch Binarization Algorithm to enforce sparsity, a GNN backbone for representation learning, and a multi-objective loss integrating task labels, subject identity, and sparsity priors. The authors provide theoretical arguments showing pairwise models cannot recover high-order interactions, and prove that GCM can approximate arbitrary higher-order rules in principle. Empirically, GCM achieves up to 30.6% relative accuracy improvements and a 96.3% reduction in computation time across five datasets and two task settings, while delivering interpretable, resolution-specific FBNs with strong cross-dataset generalization. This framework offers a foundational approach for end-to-end high-order brain network learning with broad implications for cognitive neuroscience and related interdisciplinary applications.

Abstract

Functional brain network (FBN) modeling often relies on local pairwise interactions, whose limitation in capturing high-order dependencies is theoretically analyzed in this paper. Meanwhile, the computational burden and heuristic nature of current hypergraph modeling approaches hinder end-to-end learning of FBN structures directly from data distributions. To address this, we propose to extract high-order FBN structures under global constraints, and implement this as a Global Constraints oriented Multi-resolution (GCM) FBN structure learning framework. It incorporates 4 types of global constraint (signal synchronization, subject identity, expected edge numbers, and data labels) to enable learning FBN structures for 4 distinct levels (sample/subject/group/project) of modeling resolution. Experimental results demonstrate that GCM achieves up to a 30.6% improvement in relative accuracy and a 96.3% reduction in computational time across 5 datasets and 2 task settings, compared to 9 baselines and 10 state-of-the-art methods. Extensive experiments validate the contributions of individual components and highlight the interpretability of GCM. This work offers a novel perspective on FBN structure learning and provides a foundation for interdisciplinary applications in cognitive neuroscience. Code is publicly available on https://github.com/lzhan94swu/GCM.

Figures (10)

• Figure 1: An illustration of the two paradigms for FBN modeling. Top: Conceptual diagrams illustrating the core theoretical differences. The top-left provides a concrete example of a high-order (XOR) interaction to demonstrate how pairwise models fail to capture patterns that high-order models can identify. The top-right contrasts the scope of Local Constraints (acting on single nodes or edges) versus Global Constraints (acting on the entire network). Bottom: Comparison between the previous pairwise paradigm with our proposed GCM framework, which treats the adjacency matrix as a single, learnable object holistically shaped by global objectives.
• Figure 2: Illustration of the four modeling resolutions defined in this work. Each resolution corresponds to a distinct level of structural abstraction and data aggregation.
• Figure 3: Time consumption of each method.
• Figure 4: Accuracy of GCM using different training strategies.
• Figure 5: FBN of DynHCP$_\text{Gender}$ combining four modeling resolutions, with brain regions as chunks and edges as chords.

Theorems & Definitions (18)

• Lemma 1: Second‑order sufficiency
• Theorem 1: The limitation of Pairwise Network Model
• Corollary 1
• Theorem 2: Existence of a Higher‑Order‑Exact GCM
• Lemma 2: Non‑degenerate variance implies distributional gap
• proof
• Theorem 3: Mutual non‑equivalence
• proof
• proof
• Lemma 3: Finite‑$K$ Concrete Approximation