P-DivGNN: Diversified Graph Neural Networks

• P-DivGNN is a diversified graph neural network architecture that integrates low-pass aggregation with high-pass diversification to mitigate over-smoothing and retain node uniqueness.
• It employs a two-channel filtering mechanism where outputs from LP and HP branches are combined via learned coefficients, enhancing discrimination in structurally diverse and heterophilic graphs.
• Extensions using layer-diverse negative sampling and physics-informed divergence constraints demonstrate improved robustness, expressivity, and computational efficiency in both classification and mechanics applications.

P-DivGNN refers to a family of Diversification-augmented Graph Neural Network architectures wherein standard aggregation operations are augmented with a “diversification” or “high-pass” channel to counteract over-smoothing, preserve node identity, and, in some instantiations, incorporate negative sampling or physical constraints. Multiple methodological lines have employed this concept: the two-channel filterbank GNNs (“Complete the Missing Half,” also called FB-GNN), layer-diverse negative sampling schemes, and physics-informed GNNs with divergence constraints in computational mechanics. Each line applies the underlying principle of balancing local feature aggregation with mechanisms that explicitly differentiate or diversify node representations.

1. Principle of Diversification in Graph Neural Networks

Traditional GNN layers perform local neighbor aggregation, which mathematically implements a low-pass (LP) graph filter. This enforces smoothness in node embeddings, aligning representations for homophilic graphs but resulting in oversmoothing and information loss for graphs where sharp distinctions are required. The core innovation of P-DivGNN is supplementing LP aggregation with its dual: a high-pass (HP) filter that amplifies node-wise (high-frequency) differences, thus preserving per-node identity and improving discriminative power on heterophilic or structurally diverse graphs (Luan et al., 2022, Luan et al., 2020).

In the spectral formalism, the LP filter is typically a normalized adjacency or random-walk matrix (with self-loops), while the HP/“diversification” filter is the complementary operator (e.g., $H = I - \bar{A}$). These satisfy a perfect-reconstruction property: the original signal can be reconstructed losslessly by combining both filters, supporting richer representations unattainable by LP alone.

2. Two-Channel Filtering Architectures

P-DivGNN architectures instantiate a parallel two-channel filterbank at each layer. Specifically, input features are first transformed in parallel (linear map plus nonlinearity) to two branches: one subjected to LP aggregation, the other to HP diversification. The outputs are recombined via learned convex coefficients $\alpha_L, \alpha_H \in [0,1]$, followed by a nonlinearity. This framework is applicable in both spectral (matrix-multiply) and spatial (message-passing) forms:

• Spectral variant:
  • $U_L = \text{ReLU}(H^{\ell-1} W_L^{\ell-1})$, $U_H = \text{ReLU}(H^{\ell-1} W_H^{\ell-1})$
  • $H_L^\ell = \bar{A} \cdot U_L$, $H_H^\ell = (I - \bar{A}) \cdot U_H$
  • $$H^\ell = \text{ReLU}(\alpha_L^\ell H_L^\ell + \alpha_H^\ell H_H^\ell)$$
• Spatial variant:
  • At node $i$,

  $$u_{i,L}^\ell = \text{ReLU}(W_L^{\ell-1} h_i^{\ell-1})$$, $$u_{i,H}^\ell = \text{ReLU}(W_H^{\ell-1} h_i^{\ell-1})$$ - Aggregation: sum of $\alpha_L, \alpha_H \in [0,1]$0 or $\alpha_L, \alpha_H \in [0,1]$1 over neighbors (with weights) - Output: $\alpha_L, \alpha_H \in [0,1]$2

The two-channel mechanism is compatible with standard GNN architectures and training procedures; only the forward pass is structurally modified (Luan et al., 2022, Luan et al., 2020).

3. Theoretical Analysis and Implications

The LP and HP filters can be interpreted in the spectral domain as extracting, respectively, the low- and high-frequency components of the input signal. The LP filter attenuates the Dirichlet energy $\alpha_L, \alpha_H \in [0,1]$3; the HP amplifies it. The proportion of non-smooth (high-frequency) content is quantified by $\alpha_L, \alpha_H \in [0,1]$4.

The perfect-reconstruction property ($\alpha_L, \alpha_H \in [0,1]$5) ensures that the two-channel P-DivGNN is, in principle, lossless regarding input features. The HP branch preserves node uniqueness and is particularly beneficial on graphs or tasks where node labels or target outputs are not smooth over the graph.

Empirical ablations confirm that two-channel P-DivGNNs outperform their one-channel counterparts, and the learned mixing coefficients often emphasize the HP channel on heterophilic tasks, affirming the utility of diversification (Luan et al., 2022, Luan et al., 2020).

4. Layer-Diverse Negative Sampling Extensions

An alternative instantiation of P-DivGNN leverages negative sampling in the message-passing protocol to mitigate over-smoothing and over-squashing. At each layer, for each node, embeddings from a set of negative (non-neighbor) nodes are subtracted, with the negative set chosen to optimize quality and diversity via determinantal point process (DPP) sampling. To enforce diversity across layers, a space-squeezing technique modifies the sample space between layers, ensuring that negative sets are distinct at each level.

The P-DivGNN message update is:

$\alpha_L, \alpha_H \in [0,1]$6

where $\alpha_L, \alpha_H \in [0,1]$7 is a learned negative strength. The approach enhances embedding separation (elevated MAD statistics), increases robustness to depth, widens propagation bottlenecks, and improves expressivity, as evidenced by improved classification metrics across various datasets and GNN backbones (Duan et al., 2024).

5. Physics-Informed P-DivGNN: Divergence-Constrained Message-Passing

In computational mechanics, P-DivGNN refers to a physics-informed GNN incorporating divergence constraints for local field reconstruction in microstructure mechanics. Here, a periodic finite-element mesh is represented as a graph, and the GNN prediction for nodal stresses is regularized by a physics-based loss enforcing local equilibrium (discrete divergence constraint):

$\alpha_L, \alpha_H \in [0,1]$8

This hybrid data-physics loss yields stress reconstructions with accuracy and equilibrium error on par with finite-element solutions but at orders-of-magnitude lower computational cost, especially salient in the nonlinear finite-strain hyperelastic regime. The approach utilizes encode–message-passing–decode GNNs with features and edge augmentations specifically designed to handle periodicity and physical boundary conditions (Garban et al., 5 Jul 2025).

6. Empirical Performance and Application Domains

The two-channel P-DivGNN (FB-GNN) provides consistent accuracy improvements (1–12 percentage points) over baselines including GCN, GAT, GraphSAGE, Geom-GCN, and deep multi-scale variants on node classification tasks—most notably on heterophilic benchmarks (Cornell, Wisconsin, Texas, Actor, Chameleon, Squirrel). Gains persist across shallow and deep networks (Luan et al., 2022, Luan et al., 2020). In layer-diverse negative sampling P-DivGNNs, semi-supervised node classification, bottleneck graph tests, and graph-level classification benchmarks show substantial robustness to over-smoothing, with positive results on datasets such as Citeseer, Cora, PubMed, Coauthor-CS, Amazon-Computers, MUTAG, and Proteins (Duan et al., 2024).

In computational mechanics, P-DivGNN’s divergence-regularized message-passing architecture achieves divergence errors and normalized MSE competitive with FE solutions and enables fast, high-fidelity prediction for both linear and hyperelastic materials with substantial computational speed-ups (Garban et al., 5 Jul 2025).

Variant	Mechanism	Representative Results
Two-channel FB-GNN	LP + HP (diversification) filterbank	1–12 pts ↑ accuracy, all tasks
Layer-diverse Negative	DPP-based negative sampling + space-squeezing	5–20% ↑ accuracy, ↑ MAD
Physics-Informed	Divergence-penalty loss, periodic graph construction	FE-level stress, 500x speedup

7. Context, Limitations, and Extensions

P-DivGNN’s dual-channel and negative-sampling schemes are general, plugin accelerators for a wide variety of GNNs. The HP/diversification channel is crucial in regimes where graph homophily is low, or distinguishing power is bottlenecked by topology and aggregation. In negative sampling P-DivGNNs, computational scaling with graph size and DPP eigen-decompositions can be significant but can be mitigated by sampling on a subset of nodes.

Physics-informed P-DivGNNs’ limitations stem from the discrete divergence operator’s ill-conditioning at boundaries and the need for manual tuning of physics weight $\alpha_L, \alpha_H \in [0,1]$9. Extensions under investigation include integration with inelasticity, temporal dynamics, higher-order PDE constraints, and full-scale multiscale FE$U_L = \text{ReLU}(H^{\ell-1} W_L^{\ell-1})$0 loops (Garban et al., 5 Jul 2025).

P-DivGNN thus serves as a unifying conceptual and methodological framework for enhancing GNN expressivity, discrimination, physical consistency, and scalability in both machine learning and scientific computing applications.