Graph Neural Network (GNN)

The core computational mechanism where nodes iteratively aggregate feature information from their neighbors to update their own representations. Each layer performs: 1) Message computation — a learnable function transforms neighbor features; 2) Aggregation — a permutation-invariant operation (sum, mean, or max) combines incoming messages; 3) Update — the node's state is revised using the aggregated message and its previous state. In constellation graphs, this allows a signal point to incorporate context from adjacent symbols, capturing local geometric structure that distinguishes QPSK clusters from 16-QAM grids.

The adjacency matrix defines the graph's connectivity, encoding which signal points are considered neighbors. Construction strategies include: k-Nearest Neighbors (k-NN) — connecting each node to its k closest points in the complex plane; Radius-based — linking all nodes within a distance threshold ε; Delaunay triangulation — forming edges that maximize the minimum angle, producing a planar graph that respects spatial proximity. The choice of adjacency function directly impacts the receptive field of each node and the classifier's sensitivity to local versus global constellation geometry.

GNNs are inherently invariant to node ordering — the output for a graph remains identical regardless of how nodes are indexed. This property is critical for modulation classification because: constellation points have no natural ordering; the classifier must recognize the modulation scheme independent of symbol sequence; and aggregation functions like sum or mean are symmetric operations. This contrasts with CNNs, which assume a fixed spatial grid structure. Permutation invariance ensures the model learns topological patterns rather than positional artifacts.

After message passing, a readout function aggregates all node embeddings into a single graph-level representation for classification. Common approaches: Global mean pooling — averages all node features, simple but loses distributional information; Global max pooling — captures the most prominent features; Attention-based pooling — learns to weight nodes by importance; Set2Set — uses an LSTM to iteratively aggregate a fixed-size representation. The readout feeds into a fully connected classifier head that outputs modulation class probabilities.

Beyond node features (IQ coordinates), GNNs can incorporate edge attributes that encode relational information between signal points. In constellation graphs, edge features may include: Euclidean distance between points; phase difference; relative amplitude ratio; or learned similarity scores. These edge features are processed alongside node features during message passing, enabling the model to distinguish modulation schemes with identical point clouds but different transition probabilities between states, such as differentiating π/4-QPSK from standard QPSK.

GNNs implement convolution in two paradigms: Spectral methods (e.g., ChebNet, GCN) define convolution via the graph Laplacian's eigendecomposition, operating in the Fourier domain — mathematically principled but computationally expensive and dependent on fixed graph structure. Spatial methods (e.g., GraphSAGE, GAT) define convolution directly in the node domain by aggregating neighbor features — more scalable, supports inductive learning on unseen graphs, and is the dominant approach for modulation classification where constellation graphs vary per signal sample.

The foundational concept of applying GNNs to modulation recognition. A constellation diagram is reinterpreted as a graph where each received symbol is a node, and edges are defined by spatial proximity in the complex plane. This transforms a Euclidean scatter plot into a non-Euclidean structure that captures local density, cluster shapes, and geometric relationships. GNNs then learn to pass messages between neighboring symbols, aggregating information to classify the global modulation scheme. This approach is inherently robust to phase rotations and small frequency offsets that distort absolute positions but preserve relative geometry.