Graph Neural Network (GNN)

The fundamental computational mechanism where nodes iteratively exchange and aggregate information with their neighbors. Each layer updates a node's hidden state by combining its previous representation with transformed messages from adjacent nodes.

• Aggregation function: Collects neighbor states (mean, sum, max-pooling, or attention-weighted)
• Update function: Combines aggregated neighbor information with the node's own state via a neural network
• Readout function: Produces a graph-level representation by pooling all node embeddings after the final layer

In legal citation networks, message passing allows a case node to incorporate the precedential signals from all cases that cite or are cited by it, propagating authority through the graph structure.

GNNs produce dense, low-dimensional vector representations that encode both a node's intrinsic features and its structural position within the graph. These embeddings serve as the foundation for downstream tasks.

• Feature encoding: Captures the semantic content of the node (e.g., the text of a legal opinion)
• Structural encoding: Encodes the node's role and connectivity patterns in the citation topology
• Positional encoding: Optional signals that capture a node's absolute or relative position within the graph

For legal AI, a case embedding generated by a GNN simultaneously represents the case's doctrinal content and its precedential neighborhood, enabling similarity search that respects both textual and structural relevance.

A specific instantiation of message passing that generalizes the convolution operation from regular grids to irregular graph topologies. Graph convolutions apply a shared transformation to each node's local neighborhood.

• Spectral methods: Operate in the Fourier domain of the graph Laplacian, applying learned filters to the graph's spectral decomposition
• Spatial methods: Define convolution directly on the graph structure by aggregating features from a node's spatial neighbors
• Chebyshev polynomials: Approximate spectral filters efficiently without requiring full eigendecomposition of the Laplacian

In citation networks, graph convolutional layers enable the model to learn that cases cited by highly authoritative precedents should themselves receive elevated influence scores, regardless of their own textual features.

Graph Attention Networks assign learnable importance weights to each neighbor during aggregation, allowing the model to selectively focus on the most relevant connections rather than treating all neighbors equally.

• Attention coefficients: Computed dynamically for each edge based on the features of the connected node pair
• Multi-head attention: Runs multiple independent attention mechanisms in parallel and concatenates or averages their outputs for stability
• Edge feature integration: Incorporates edge attributes (e.g., treatment type, citation sentiment) directly into the attention computation

For legal graphs, attention mechanisms naturally model the reality that not all citations carry equal weight—a case that overrules a precedent should receive different attention than one that merely distinguishes it.

Legal knowledge graphs are inherently heterogeneous, containing multiple node types and edge types. Heterogeneous GNNs extend the message-passing framework to handle this complexity by learning type-specific transformations.

• Type-specific projections: Each node type and edge type has its own learnable weight matrix, projecting different entity types into a shared embedding space
• Relation-aware aggregation: Messages are aggregated separately per relation type before being combined, preserving the semantics of different citation relationships
• Metapath-based sampling: Defines composite relations (e.g., Case → cited_by → Case → decided_by → Court) to capture higher-order structural patterns

This architecture enables a single model to simultaneously reason about cases, statutes, courts, judges, and their distinct interrelationships without collapsing the rich semantics of the legal domain.

Unlike transductive methods that require the entire graph at training time, inductive GNNs learn a function that can generate embeddings for previously unseen nodes and entirely new graphs without retraining.

• Parameter sharing: The same learned aggregation and update functions apply to any node regardless of its position in the graph
• Neighborhood sampling: Enables training on large graphs by sampling subgraphs around target nodes rather than operating on the full adjacency matrix
• Generalization to new cases: A GNN trained on historical citation data can immediately produce embeddings for newly published decisions

This property is critical for production legal AI systems, where new cases are continuously published and must be integrated into the authority graph without costly model retraining cycles.

The fundamental mechanism by which GNNs operate. In each layer, a node aggregates feature vectors from its neighbors and updates its own embedding. In a legal context, a case node receives information from the cases it cites and the cases that cite it, iteratively refining its representation to capture its precedential neighborhood. Common aggregation functions include mean, sum, and max pooling, while attention-based aggregators learn to weight the importance of different citing cases.

The low-dimensional, dense vector representation learned for each node in the graph. A GNN transforms a case's initial features—such as its textual embedding or court metadata—into a final embedding that encodes both its intrinsic properties and its topological context within the citation network. These embeddings serve as the input for downstream tasks like case outcome prediction or citation recommendation, where cases with similar legal function occupy proximate positions in the vector space.

A graph structure containing multiple node and edge types, essential for legal networks. A citation graph is not monolithic: it must simultaneously model cases, statutes, courts, and judges, with distinct relationship types such as 'cites', 'overrules', and 'authored by'. Heterogeneous GNNs, like RGCNs or HANs, use type-specific transformation matrices to learn meaningful representations across these diverse entities and their interactions.

A GNN variant that employs self-attention to assign different weights to a node's neighbors during aggregation. In a citation network, a GAT can learn that a citation from a Supreme Court decision carries more precedential weight than one from a lower court, or that a citation with negative treatment should be processed differently than a supportive one. This dynamic weighting allows the model to focus on the most legally significant connections.

The capability of a GNN to generate embeddings for previously unseen nodes without retraining. Unlike transductive methods that require the entire graph at training time, inductive GNNs like GraphSAGE learn a function that maps a node's local neighborhood to its embedding. This is critical for legal AI, where new cases are continuously decided and must be integrated into the existing authority graph without rebuilding the entire model.

A foundational GNN architecture that performs spectral graph convolution, effectively smoothing a node's features with those of its neighbors. In a citation network, a multi-layer GCN captures increasingly broad context: the first layer aggregates direct citations, while the second layer captures two-hop neighbors—the cases that cite the cases that cite the target. This hierarchical aggregation reveals the broader doctrinal cluster surrounding a decision.