Nested-Utility Graph Neural Network

• NestGNN is a framework that generalizes the nested logit model by integrating deep learning with explicit graph-based modeling of inter-alternative correlations.
• It employs message-passing and flexible aggregation functions to capture both within-nest proportionality and cross-nest non-proportional substitution patterns.
• Empirical results on travel mode choice data show a 9.2% improvement in log-likelihood and higher accuracy, validating its enhanced predictive performance.

Nested-Utility Graph Neural Network (NestGNN) is a graph neural network framework designed to generalize the classical Nested Logit (NL) model for discrete choice analysis. This methodology explicitly captures inter-alternative correlations within the context of travel mode choice and related decision modeling by constructing an "alternative graph" that encodes the relationships among alternatives, thus synthesizing the substitution and nesting structure of NL models with the representational flexibility of deep neural networks (Zhou et al., 8 Sep 2025).

1. Foundations and Motivation

NestGNN originates from the need to overcome the limited expressiveness and manual utility specification of the NL model, which, despite its widespread use in discrete choice analysis (such as travel mode, automobile ownership, and location specification), can only capture correlations through predefined nests and an additive log-sum-exponential (LSE) aggregation. Conventional deep neural network (DNN) models introduced for discrete choice prediction are able to approximate more complex relationships but lack mechanisms for explicit inter-alternative correlation modeling. NestGNN introduces the concept of the "alternative graph," where each node represents a choice alternative and edges encode relationships (e.g., shared unobserved utility components or behavioral similarities), thereby providing a flexible and principled substrate for generalizing nested substitution patterns.

2. Mathematical Structure and Theoretical Guarantees

NestGNN generalizes the NL model by embedding both the random utility maximization principle and the key two-layer substitution structure within a GNN framework. In this formalism:

• Nodes represent alternatives, each initialized by attributes such as travel time, cost, and individual characteristics.
• Edges encode relationships among alternatives (e.g., joint belonging to a nest).

The theoretical contribution is twofold:

1. Model Specification: NestGNN is characterized by a hyperparameter set $H = \{L, M, A, U, R\}$:
   • $L$: number of (message-passing) layers,
   • $M$: message function producing edge information from node attributes and edge types,
   • $A$: aggregation function (including options such as mean, max, or LSE),
   • $U$: node update function,
   • $R$: readout function for utility prediction.
2. Substitution Patterns:
   • Within-nest substitution is proportional: for alternatives $i$ and $j$ in the same nest, $$\frac{P_{ni}}{P_{nj}} = \frac{\exp(\phi(x_{ni}))}{\exp(\phi(x_{nj}))}$$, preserving classical NL behavior.
   • Cross-nest substitution is non-proportional, emerging via flexible aggregation $A$ over neighbor nodes and corresponding edge messages: $P_{ni} \propto \exp(\phi(x_{ni}) + A(\{m_{i,j}\}))$.

Through these formulations, NestGNN formally subsumes NL, Multinomial Logit (MNL), and alternative-specific DNNs (ASU-DNN) as special cases, thereby inheriting interpretability while performing high-capacity modeling.

3. Alternative Graph Construction and Message Passing

The core innovation is the alternative graph, where the following aspects are emphasized:

• For classical NL equivalence, all alternatives within a nest form a complete subgraph, each sharing incoming messages representative of nest-level shared utility.
• For generalizations, alternatives may exhibit richer connectivity, supporting arbitrary behavioral correlations (for instance, grouping automobile and transit together while treating walking and bike as either separate nodes or a composite nest).

The message passing algorithm is restricted to graph-defined neighborhoods, transmitting information only among related alternatives—which operationalizes substitution and correlation inherent in the NL model, but can be extended to arbitrary graph structures for more nuanced behavior modeling.

Example model configuration:

• One message-passing layer ($L=1$),
• Linear message function $M$,
• LSE aggregation function $A$,
• Element-wise addition as $U$,
• MLP readout $R$.

Such a specification precisely recovers the mathematical structure of NL, while multi-layer, non-linear, or more expressive choices yield models beyond the expressivity of classical discrete choice theory.

4. Empirical Evaluation and Substitution Patterns

NestGNN has been empirically validated on the London Passenger Dataset, using a grid search across model architectures and alternative graph structures. Key findings include:

• Predictive Performance: The best NestGNN outperformed the classical NL model by 9.2% in log-likelihood and achieved 5.5–6.3% higher accuracy. This confirms the value added by explicit inter-alternative correlation modeling in the graph domain.
• Elasticity and Substitution Visualization: Elasticity tables demonstrate that, unlike MNL and ASU-DNN which enforce uniform substitution elasticities (reflecting the independence of irrelevant alternatives, IIA), NestGNN reproduces NL-like two-layer substitution patterns—higher elasticity within nests and reduced, variable elasticity across nests.
• Model Specification Impact: The empirical paper reveals that the structure of the alternative graph is critical: grouping automobile and transit results in improved fit, mirroring real-world behavioral similarity. The framework supports extensive flexibility in graph specification, layering, and message aggregation.

5. Interpretability and Generalization

NestGNN strikes a balance between interpretability and predictive power. By parameterizing the aggregation function $A$ as LSE, one can retain economic interpretability regarding nest-specific substitution patterns (mirroring NL) while benefiting from the expanded modeling capacity through deep learning techniques. This flexibility is particularly advantageous for policy analysis and behavioral interpretation because it enables specific structural hypotheses about similarity and substitutability to be encoded directly into the graph, and subsequently tested using observed data.

Moreover, NestGNN can be extended to other domains where alternative correlation and complex choice structures are central—including but not limited to automobile ownership, residential location decisions, and other discrete choice contexts where behavioral similarity, inferential flexibility, and predictive accuracy are simultaneously required.

6. Applications and Broader Implications

The explicit representation and flexible modeling of inter-alternative correlations in NestGNN are particularly salient for travel mode choice analysis, where alternatives exhibit latent similarities and high-dimensional interactions. Compared to rigid nest configurations in classical NL, NestGNN supports richer model specification, individualization, and potential for learning dynamic or personalized alternative graphs.

This generalization from nested logit to nested-utility graph neural networks enables nuanced substitution modeling and interpretability, representing an integration of theoretical economics with high-capacity function approximation. While the primary application showcased is travel mode choice modeling, the framework suggests wider applicability wherever discrete choices are influenced by structured relationships between alternatives.

A plausible implication is the expansion of discrete choice modeling to settings previously inaccessible due to the limitations of classical modeling frameworks. The ability to specify, learn, and interpret alternative graphs opens new directions for both theory and practice in econometrics, machine learning, and behavioral modeling.

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Summary Table: NestGNN Characteristics

Component	Classical NL Model Structure	NestGNN Generalization
Alternative Representation	Flat or nest-based list	Graph-based with explicit connectivity
Substitution Pattern	Two-layer: within-nest proportional	Two-layer, extendable, learned via GNN
Aggregation Function	LSE (log-sum-exp)	LSE, mean, max, or any learnable function
Expressiveness	Limited by nest and utility spec	Supports arbitrary utility and correlation
Interpretability	Strong	Tunable, recoverable, and extendable

NestGNN generalizes discrete choice analysis by embedding nested substitution structure within a graph neural network architecture, supporting both the interpretability of classical economic models and the predictive power of modern deep learning (Zhou et al., 8 Sep 2025).