Nutritional Graph Router (NG-Router)

• NG-Router is an advanced framework for nutritional question answering that integrates multi-agent collaboration with graph neural networks for personalized dietary health analysis.
• It utilizes a heterogeneous knowledge graph combining nutritional entities, queries, and agent nodes to optimize multi-hop relational reasoning across complex nutritional data.
• Empirical validation shows NG-Router improves accuracy by over 50% in sparse settings, offering robust, scalable, and personalized dietary recommendations.

The Nutritional-Graph Router (NG-Router) is an advanced framework for nutritional question answering (QA) built upon heterogeneous knowledge graphs and supervised multi-agent collaboration. By integrating agent nodes and leveraging graph neural networks (GNNs), NG-Router optimizes multi-hop, relational reasoning over complex nutritional data, producing robust and domain-aware answers to questions about personalized dietary health.

1. Multi-Agent Collaboration Supervised by Knowledge Graphs

NG-Router formalizes nutritional QA as a supervised, knowledge-graph-guided multi-agent collaboration problem (Shi et al., 10 Oct 2025). The knowledge graph (KG) in this paradigm is heterogeneous, comprising nutritional entities (foods, nutrients, health conditions), query nodes (semantic embeddings of individual diet-related questions), and agent nodes (representing candidate LLMs—LLMs—each with distinct prompting strategies). Edges within the KG are classified as:

• Query–entity edges: Ground queries to nutritional evidence.
• Agent–entity edges: Encode the informational focus of each agent.
• Adaptive query–agent edges: Carry learned routing signals for agent participation.

Supervision is derived from empirical agent performance, and the router learns to assign soft probability weights to agents. The final answer is produced through weighted aggregation, directly incorporating domain structure into collaborative reasoning and obviating the need for handcrafted coordination rules. The routing score for each query–agent pair is formalized by

$s(q,a) = f_\theta(q, a; \mathcal{G})$

and normalized via softmax over the agent set.

2. Graph Neural Network Mechanisms

A heterogeneous GNN underpins NG-Router, enabling type-sensitive message passing across the extended KG. Nodes (queries, agents, entities) are projected into shared latent spaces through type-specific projections ($\text{Proj}_{\tau(v)}$). For edge $(u \to v)$ with relation type $\psi$, the message passing update is

$$m^{(l,\psi)}_{u \to v} = \text{Proj}(W^{(l)}_{\psi} \cdot h_u^{(l-1)})$$

Messages from neighbors are aggregated per edge type and gated using learnable coefficients ($w^{(l)}_{\psi}$). The node update proceeds as

$$h^{(l)}_{v} = U^{(l)}_{\tau(v)} ( h^{(l-1)}_{v} \parallel \sum_{\psi \in \Psi(v)} w^{(l)}_{\psi} \cdot \tilde{m}^{(l,\psi)}_{v} )$$

where $\parallel$ denotes concatenation. After $L$ layers, MLPs compute routing scores, powering soft agent selection and fusion for answer generation.

3. Gradient-Based Subgraph Retrieval

To address contextual overload from large nutritional graphs, NG-Router employs a gradient-based subgraph retrieval technique. For each entity node $v_i$ with embedding $h_i$, entity importance is quantified by the norm of the gradient of the KL-divergence loss

$$\alpha_i = \| \nabla_{h_i} \mathcal{L}_{KL}(q) \|_2$$

Entities whose $\alpha_i$ exceed a predefined threshold ($\tau$) are retained, yielding the induced subgraph $\mathcal{G}^*$. This procedure selectively prunes noisy nodes and edges, thus enhancing multi-hop and relational reasoning by improving signal-to-noise ratio in retrieved context.

4. Empirical Validation and Performance

Across several nutritional QA benchmarks with varying complexity (Sparse, Standard, Complex) (Shi et al., 10 Oct 2025), NG-Router demonstrates consistent superiority to single-agent and ensemble baselines. The gradient-based subgraph retrieval module yields marked performance gains—over 50% improvement in accuracy in sparse settings versus models lacking context pruning. Experiments conducted with multiple backbone LLMs (e.g., Llama-3.2-3B-Instruct, Qwen2.5-7B-Instruct, Mistral-7B-Instruct, GPT-4o-mini) show that adaptive agent weighting yields increased robustness and transferability. Both binary classification and text generation tasks benefit directly from learned graph supervision and context retrieval mechanisms.

5. Relation to Graph-Based Nutritional Reasoning

NG-Router operationalizes advances in nutritional graph reasoning set forth in the NGQA benchmark (Zhang et al., 2024). NGQA advocates knowledge graphs constructed from curated nutrition datasets (e.g., NHANES, FNDDS), where nodes represent foods, nutrients, user health indicators, and dietary habits. Edges formalize relationships such as "match" or "contradict," supporting multi-label and multi-hop reasoning about dietary suitability per user profile. NG-Router extends these concepts, embedding agent collaboration directly within the graph context and optimizing paths through adaptive weighting. Methods such as subgraph retrieval (akin to KAPING or ToG) ensure evidence selection aligns with personalized health nuances, trade-offs, and explanatory needs.

6. Integration into Personalized Nutrition Systems

NG-Router's architecture provides a blueprint for domain-aware, multi-agent systems in the personalized nutrition space. The framework's modularity—separating health profile mapping, nutritional property analysis, and agent routing—suggests it can be expanded to accommodate:

• Individual medical indicators and dietary behaviors
• Personalized nutrient thresholds and conflicting requirements
• Binary recommendation and textual explanation tasks for interactive applications

The system's aggregation function ($\phi$), applied to weighted agent outputs, flexibly supports voting, reranking, or fusion strategies to produce actionable dietary guidance. By supporting context-sensitive reasoning, NG-Router transcends generic food recommendations, dynamically interpreting nutrition graphs in light of user-specific constraints.

7. Mathematical Foundations and Algorithmic Summary

Key elements of NG-Router's approach include:

Component	LaTeX Formula / Principle	Functionality
Routing score	$s(q,a) = MLP( h^{(L)}_q \parallel h^{(L)}_a )$	Scores agent relevance per query
Probability	$$p_\theta(a \mid q, \mathcal{G}) = \text{softmax}_{a \in \mathcal{A}} ( s(q,a) )$$	Soft agent weighting for answer aggregation
Final output	$$\hat{y}(q) = \phi(\{ y_a(q), w_a(q) : a \in \mathcal{A} \})$$	Weighted answer computation
Gradient salience	$$\alpha_i = \| \nabla_{h_i} \mathcal{L}_{KL}(q) \|_2$$	Subgraph node selection based on routing loss impact

These formalizations underpin NG-Router's principled handling of nutritional QA, supporting scalable, interpretable, and personalized multi-agent reasoning in health-aware dietary systems.