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Graph Vector Field: A Unified Framework for Multimodal Health Risk Assessment from Heterogeneous Wearable and Environmental Data Streams

Silvano Coletti, Francesca Fallucchi

Abstract

Digital health research has advanced dynamic graph-based disease models, topological learning on simplicial complexes, and multimodal mixture-of-experts architectures, but these strands remain largely disconnected. We propose Graph Vector Field (GVF), a framework that models health risk as a vector-valued field on time-varying simplicial complexes, coupling discrete differential-geometric operators with modality-structured mixture-of-experts. Risk is represented as a vector-valued cochain whose evolution is parameterised with Hodge Laplacians and discrete exterior calculus operators, yielding a Helmholtz-Hodge decomposition into potential-driven (exact), circulation-like (coexact), and topologically constrained (harmonic) components linked to interpretable propagation, cyclic, and persistent risk mechanisms. Multimodal inputs from wearable sensors, behavioural/environmental context, and clinical/genomic data are incorporated through a bundle-structured mixture-of-experts in which modality-specific latent spaces are attached as fibres to the base complex. This separates modality-specific from shared contributions and offers a principled route toward modality-level identifiability. GVF integrates geometric dynamical systems, higher-order topology (enforced indirectly via geometric regularisation and Hodge decomposition), and structured multimodal fusion into a single framework for interpretable, modality-resolved risk modelling. This paper develops the mathematical foundations, architectural design, and formal guarantees; empirical validation is the subject of ongoing work.

Graph Vector Field: A Unified Framework for Multimodal Health Risk Assessment from Heterogeneous Wearable and Environmental Data Streams

Abstract

Digital health research has advanced dynamic graph-based disease models, topological learning on simplicial complexes, and multimodal mixture-of-experts architectures, but these strands remain largely disconnected. We propose Graph Vector Field (GVF), a framework that models health risk as a vector-valued field on time-varying simplicial complexes, coupling discrete differential-geometric operators with modality-structured mixture-of-experts. Risk is represented as a vector-valued cochain whose evolution is parameterised with Hodge Laplacians and discrete exterior calculus operators, yielding a Helmholtz-Hodge decomposition into potential-driven (exact), circulation-like (coexact), and topologically constrained (harmonic) components linked to interpretable propagation, cyclic, and persistent risk mechanisms. Multimodal inputs from wearable sensors, behavioural/environmental context, and clinical/genomic data are incorporated through a bundle-structured mixture-of-experts in which modality-specific latent spaces are attached as fibres to the base complex. This separates modality-specific from shared contributions and offers a principled route toward modality-level identifiability. GVF integrates geometric dynamical systems, higher-order topology (enforced indirectly via geometric regularisation and Hodge decomposition), and structured multimodal fusion into a single framework for interpretable, modality-resolved risk modelling. This paper develops the mathematical foundations, architectural design, and formal guarantees; empirical validation is the subject of ongoing work.

Paper Structure

This paper contains 59 sections, 4 theorems, 16 equations, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Evolution of AI for Multimodal Health Risk Modelling
  3. Graph- and sequence-based models for wearable-driven risk
  4. Dynamic graphs for disease progression
  5. Geometric and topological learning in health
  6. Multimodal fusion, foundation models, and mixture-of-experts
  7. Mathematical Substrate
  8. From Pairwise Graph to Simplicial Complex
  9. Discrete Exterior Calculus Operators
  10. Inner product and Hodge star
  11. Constructing Simplicial Complexes from Multimodal Health Data
  12. 0-simplices (nodes)
  13. 1-simplices (edges)
  14. 2-simplices (triangles) and biological justification
  15. Practical construction from raw streams
  16. ...and 44 more sections

Key Result

Theorem 5.4

For any risk flow $\mathbf{F}\in C^1(\mathcal{K}(t);\mathbb{R}^m)$, there exists a unique orthogonal decomposition: where: The three components are mutually $L^2$-orthogonal with respect to the inner product induced by the Hodge star on $\mathcal{K}(t)$. The curl and harmonic components are generically non-zero because $\mathbf{F}$ is not constrained to lie in $\mathrm{im}(\nabla_{\!\mathcal{K}}

Theorems & Definitions (23)

  • Definition 3.1: Multimodal Interaction Simplicial Complex $\mathcal{K}(t)$
  • Remark 3.2
  • Definition 4.1: Risk Vector Bundle $\varepsilon(\mathcal{K})$
  • Definition 4.2: GVF Operator $F_\theta$
  • Remark 4.3: Graceful Degradation under Missing Modalities
  • Remark 5.1: Why HHD requires a separate 1-cochain
  • Definition 5.2: Risk Flow Field $\mathbf{F}$
  • Remark 5.3: Role of 2-simplices in the learning pipeline
  • Theorem 5.4: Discrete HHD of the Risk Flow Field
  • proof
  • ...and 13 more