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

Silvano Coletti; Francesca Fallucchi

cs.LGMar 30, 2026

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

Authors: Silvano Coletti, Francesca Fallucchi

Organizations: Universit`a degli Studi Guglielmo Marconi, Roma, Italy · Chelonia SA, Lugano, Switzerland

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.

Figures & tables

Expert	Simplicial Architecture	Primary Inputs	Temporal Scale
Physio-Expert	Dynamic Simplicial GCN (DS-GCN)	HR, HRV, SpO 2 , EDA, skin temp	Minutes
Behav-Expert	Simplicial Convolutional RNN (S-CRNN)	GPS, sleep staging, activity	Hours–days
Env-Expert	Temporal Simplicial Attention (T-SAT)	PM 2.5 , temp, noise, UV	Minutes
Ext-Expert	Higher-order / Simplicial Transformer	EHR, PRS, surveillance (opt.)	Hours–static

Table 1: Expert decomposition of

F_{\theta}

aligned with simplicial UAT requirements.

Component	Property	Mechanistic Meaning	Intervention
$\nabla_{\!\mathcal{K}}\varphi$ (Gradient)	$\mathrm{curl}_{\mathcal{K}}(\nabla_{\!\mathcal{K}}\varphi)=\mathbf{0}$	Risk flows directionally along a potential gradient. Identifies sources (emitting nodes) and sinks (accumulating nodes).	Reduce source intensity (e.g., lower environmental exposure, improve physiological recovery).
$\mathrm{curl}_{\mathcal{K}}^{*}\psi$ (Curl)	$\mathrm{div}_{\mathcal{K}}(\mathrm{curl}_{\mathcal{K}}^{*}\psi)=\mathbf{0}$	Risk circulates in self-sustaining closed loops. Signature of cyclic coupling (e.g., sleep-stress-recovery loops).	Break the cycle at its weakest link (sleep hygiene, pharmacological disruption).
$\mathbf{h}$ (Harmonic)	$\Delta_{1}\mathbf{h}=\mathbf{0}$	Risk locked by global network topology. Dimension $=\beta_{1}(\mathcal{K}(t))$ . Not reducible by local interventions.	Structural intervention: modify network topology.

Table 2: HHD components with mechanistic interpretation and intervention strategy.

HHD Component	Modality Class	Structural Role	Why this correspondence
$\nabla_{\!\mathcal{K}}\varphi$ (Gradient / exact)	Clinical & genomic ( $\mathcal{X}^{\mathrm{ext}}$ )	Scalar potential $\varphi$ encodes baseline, constitutive risk: fixed biological or clinical state driving a directional outflow.	Clinical scores and genomic risk factors define a baseline potential landscape that changes slowly and governs the direction of risk flow.
$\mathrm{curl}_{\mathcal{K}}^{*}\psi$ (Curl / co-exact)	Wearable & behavioural ( $\mathcal{X}^{\mathrm{phys}},\,\mathcal{X}^{\mathrm{beh}}$ )	Stream function $\psi$ encodes dynamic, oscillatory coupling: physiological cycles (sleep-wake, HRV rhythm, activity-rest) generate closed-loop risk circulation.	Wearable time series directly observe cyclical physiological dynamics; their high temporal resolution makes them the natural carrier of the curl component.
$\mathbf{h}$ (Harmonic)	Environmental ( $\mathcal{X}^{\mathrm{env}}$ )	Topology-locked risk determined by the global structure of $\mathcal{K}(t)$ , which is built from spatial co-presence, dwell times, and sensor adjacency.	The complex topology itself—driven by environmental sensor placement, spatial proximity, and shared exposure zones—determines the harmonic dimension $\beta_{1}(\mathcal{K}(t))$ and hence the harmonic risk subspace.

Table 3: Natural correspondence between HHD components and modality classes. Each modality class contributes most directly to one geometric mode of risk; the contribution is not exclusive but reflects the structural role of each data type. To the best of our knowledge, no prior framework establishes this correspondence (cf. [ 6 , 7 , 8 ] ).

Feature	GVF (Proposed)	Scalar Dynamic GNN	Vector MoE (no bundle)	Clinical Scores (SOFA, NEWS)	Scalar LSTM/Transf.
Time complexity	$\mathcal{O}(\|V\|{+}\|\mathcal{K}^{1}\|{+}\|\mathcal{K}^{2}\|)$	$\mathcal{O}(\|V\|{+}\|\mathcal{K}^{1}\|)$	$\mathcal{O}(\|V\|{\cdot}m)$	$\mathcal{O}(\|V\|)$	$\mathcal{O}(\|V\|{\cdot}d)$
Space complexity	$\mathcal{O}(\|\mathcal{K}^{2}\|{\cdot}m)$	$\mathcal{O}(\|\mathcal{K}^{1}\|{\cdot}d)$	$\mathcal{O}(\|V\|{\cdot}m)$	$\mathcal{O}(\|V\|)$	$\mathcal{O}(\|V\|{\cdot}d)$
Vector-valued risk output	✓	$\times$	✓	$\times$	$\times$
Gradient component	✓	$\times$	$\circ$	$\times$	$\times$
Curl component (cyclic risk)	✓	$\times$	$\times$	$\times$	$\times$
Harmonic component (topology)	✓	$\times$	$\times$	$\times$	$\times$

Table 4: Extended comparison of GVF against representative baseline classes. ✓ = supported by construction;

\circ

= partial or heuristic;

\times

= not supported. Complexity:

|V|

nodes,

|\mathcal{K}^{1}|

edges,

|\mathcal{K}^{2}|

2-simplices,

m

output dimension,

d

scalar hidden size. All comparisons are architectural, not empirical.

Component	Signal	Interpretation	Intervention
Gradient $\nabla_{\!\mathcal{K}}\varphi$	$\mathrm{DPS}_{i}>0$	Net risk source; directional propagation	Reduce source (exposure, shift pattern)
Curl $\mathrm{curl}_{\mathcal{K}}^{*}\psi$	$\mathrm{CRI}_{i}>0$	Self-sustaining loop; persists if source removed	Break cycle (sleep hygiene, pharmacological)
Harmonic $\mathbf{h}$	$\dim\ker\Delta_{1}>0$	Topology-locked; local actions ineffective	Restructure network (scheduling, zoning)

Table 5: HHD components and their operational intervention targets.

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May 18, 2026J/K move · Enter open · S save

A Unified Framework for Structured Flow Modeling: From Representation to Verification and Model Discovery

Diego Casadei

Fly High Engineering AG, Badenerstrasse 13, 5200 Brugg AG, Switzerland

Dynamical Systems Hodge Decomposition Scientific Discovery Reduced-Order Modeling

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Many dynamical systems can be described in terms of structured flows combining source/sink behavior, cyclic dynamics, and topology-constrained transport. These features arise across a wide range of physical, engineered, and data-driven systems. The objective of this work is to establish a unified perspective on such systems, to identify modeling approaches that balance expressivity, interpretability, computational complexity, and data requirements, and to investigate how highly expressive models can be used to uncover the dominant mechanisms underlying observed dynamics. Starting from the Helmholtz-Hodge decomposition of continuous vector fields, we review the recently proposed Graph Vector Field (GVF) framework and its discrete representation on simplicial complexes. We then introduce a hierarchy of alternative approaches, including parametric conditional models, linear graph dynamical systems, and reduced Hodge representations. Finally, we propose a verification and validation methodology based on benchmark datasets from well-understood physical systems and on systematic model-reduction and ablation studies. The resulting family of structured-flow models within a common framework, ranging from low-dimensional parametric representations to full GVF formulations, supports a diagnostic methodology in which gradient, curl, harmonic, and topological contributions are systematically assessed through ablation studies. This process enables the identification of dominant mechanisms underlying the observed dynamics and guides the construction of simplified models tailored to the available data and operational constraints. By separating structural verification, behavioral verification, and domain-specific validation, the proposed approach provides a foundation for scalable and interpretable analysis of complex dynamical systems across multiple application domains.

Abstract

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