EDT-PA: Evolvable Graph Diffusion & Pattern Alignment

• The paper introduces EDT-PA, a unified framework that combines adaptive graph diffusion, optimal transport, and pattern-specific alignment to handle evolving graph structures.
• It leverages entropy-regularized optimal transport and iterative neural aggregation to selectively align informative substructures in complex networks.
• Empirical results show improved performance in applications like brain connectome analysis and multimodal data fusion, addressing scalability and robustness challenges.

Evolvable Graph Diffusion Optimal Transport with Pattern-Specific Alignment (EDT-PA) synthesizes adaptive graph diffusion processes, optimal transport theory, and pattern-aware alignment methodologies to provide a unified framework for complex graph-based analysis and data integration. EDT-PA targets the modeling, comparison, and alignment of graph-structured data, particularly in settings where the underlying topology or feature distributions evolve and where the alignment of local or global patterns is of fundamental importance. The approach is designed to address both scalability and robustness issues that arise in large-scale problems such as brain connectome modeling, multimodal data fusion, and ensemble-based analysis.

1. Foundational Principles and Conceptual Underpinnings

EDT-PA is motivated by limitations of static or fixed-topology graph models, especially in contexts where graph structure-function relationships are not fixed but subject to nonlinear evolution or reconfiguration. Traditional approaches to graph alignment (e.g., treating structural connectivity as an invariant scaffold for functional connectivity) can introduce distortions when higher-order dependencies and misaligned patterns exist between modalities or measurement conditions (Sheng et al., 16 Sep 2025). EDT-PA instead pursues a dynamic, modular pipeline in which both the underlying graph topology and the alignment between representations are allowed to evolve ("evolvable modeling blocks") through diffusion, optimal transport, and nonlinear aggregation.

The distinguishing feature of EDT-PA is its explicit incorporation of pattern-specific alignment mechanisms, enabling selective fusion or alignment of connectivity structures that correspond to meaningful subnetworks or spatial patterns. This is accomplished by formulating entropy-regularized optimal transport problems—where the transport plan is constrained or biased toward the alignment of informative substructures—rather than performing naive global distribution matching.

2. Evolvable Graph Diffusion Processes

At the core of EDT-PA is an iterative graph diffusion procedure that allows the structural representation of the network to evolve. The process is mathematically expressed as:

$$A^{(t+1)} = \mathcal{T}\Big( \alpha S A^{(t)} S^{\top} + (1-\alpha)A \Big)$$

where $A$ is the adjacency matrix, $S = D^{-1/2}AD^{-1/2}$ is the normalized diffusion operator, $\alpha$ is a parameter controlling the mix between diffused and original topology, and $\mathcal{T}$ is a class-aware transformer facilitating the integration of both local and global information (Sheng et al., 16 Sep 2025). The algorithm may append task-aware soft class tokens to the node representations to introduce latent supervision.

Diffusion and energy minimization are tightly coupled, especially in transformer-based neural architectures where layer-wise evolutionary states are computed as:

$$z_i^{(k+1)} = (1 - \tau \sum_j S_{ij}^{(k)}) z_i^{(k)} + \tau \sum_j S_{ij}^{(k)} z_j^{(k)}$$

with the pairwise diffusivity $S_{ij}^{(k)}$ adaptively computed from energy functional derivatives (Wu et al., 2023). These mechanisms allow the structure to adapt via iterative propagation and nonlinear mixing, producing high-order dependencies and contextually adaptive latent geometries.

3. Pattern-Specific Alignment via Optimal Transport

Pattern-specific alignment in EDT-PA is accomplished by leveraging (entropy-regularized) optimal transport to align distributions of node features or local graph patterns between evolving modalities. Given empirical distributions $u = \sum_i \mu_i \delta_{a_i}$ and $v = \sum_j \nu_j \delta_{s_j}$ for evolved structure and functional similarity, pattern-specific alignment is formulated as:

$$T^* = \arg\min_{T\in \mathbb{R}^{N\times N}} \langle T, C\rangle - \varepsilon Z(T)$$

$$\text{subject to} \quad T\mathbf{1} = \mu, \quad T^\top\mathbf{1} = \nu$$

where $C$ is a pattern-aware cost matrix (e.g., cosine distances), $Z(T)$ is the entropy, and $\varepsilon$ is a smoothing parameter (Sheng et al., 16 Sep 2025). The resultant transport plan $T^*$ is used to refine or fuse node representations:

$H^* = T^* H + H$

focusing alignment on shared or informative patterns rather than global distributions (Sheng et al., 16 Sep 2025).

Hybrid frameworks, such as those based on Fused Unbalanced Gromov-Wasserstein (FUGW) losses, further extend pattern specificity by introducing explicit trade-offs between feature and structural matching (Mazelet et al., 21 May 2025):

$$L^{(\alpha, \rho)}(G_1, G_2, P) = (1-\alpha) \sum_{ij} \|F_1^i - F_2^j\|^2 P_{ij} + \alpha \sum_{ijkl} |D_1^{ik} - D_2^{jl}|^2 P_{ij} P_{kl} + \rho\,(\text{KL-divergence terms})$$

This enables gradient-based optimization of selective alignment objectives, sensitive to network patterns, feature dissimilarity, or marginal constraints.

4. Neural Aggregation and Nonlinear Representation

Advanced node aggregation modules, such as Kolmogorov–Arnold Networks (KANs), enable node-level updates through generic nonlinear transformations:

$$h^*_i = KAN(h^*_i, \{h^*_j\}) = \Phi_{L-1} \circ \cdots \circ \Phi_0 (h^*_i, \{h^*_j\})$$

Each $\Phi_\ell$ is a layer-specific nonlinear mixing function permitting the modeling of complex cross-node interactions, thereby capturing subtle network-wide dependencies and non-additive effects (Sheng et al., 16 Sep 2025). After iterative aggregation, global embeddings are constructed by pooling refined node features for downstream tasks (e.g., graph classification).

Parametric prototype point clouds and cross-attention mechanisms, as in OT-GNNs and ULOT, further enable the abstraction and comparison of complex local graph patterns (Chen et al., 2020, Mazelet et al., 21 May 2025).

5. Scalability, Implementation, and Optimization

Empirical advances in iterative OT solvers—such as Sinkhorn-Knopp, inexact Newton-Raphson methods, and algebraic multigrid preconditioners—yield favorable computational scaling (e.g., $O(1)$ to $O(M^{0.36})$, where $M$ is edge count) for graph-based OT and diffusion processes (Facca et al., 2020). Deep neural predictors (e.g., ULOT) accelerate computation via direct regression of OT plans, allowing for fast, differentiable, and scalable routines that facilitate "warm starts" for classical solvers and enable end-to-end optimization in evolvable graph environments (Mazelet et al., 21 May 2025).

Evolutionary algorithms provide efficient black-box search for inference-time alignment in diffusion generative models, handling both differentiable and non-differentiable alignment objectives and yielding substantial efficiency gains in memory and runtime (Jajal et al., 30 May 2025).

6. Applications and Empirical Results

ETD-PA is effective in modeling and classifying brain disorders via structural-functional connectome integration. On datasets such as REST-meta-MDD and ADNI, the framework outperforms state-of-the-art models, revealing disorder-specific subnetworks and achieving improvements of up to 5.4% in accuracy and 6.0% in F1 score (Sheng et al., 16 Sep 2025). Statistical analyses confirm that evolvable diffusion produces discriminative connectivity patterns and interpretable disease biomarkers.

Further applications include multimodal data alignment with domain-specific regions (partial OT), ensemble political district mapping (hierarchical OT partitions), scalable node and graph classification in molecular graphs, and cross-modal pattern alignment in image-text retrieval and translation tasks (Abrishami et al., 2019, Duque et al., 2022, Chen et al., 2020, Chen et al., 2020, Mazelet et al., 21 May 2025).

7. Future Directions and Open Problems

Potential research directions for EDT-PA include:

• Finer-grained resolution of structure-function misalignments in connectomic modeling, with adaptation to additional neuroimaging modalities.
• Extension of framework components (e.g., basis selection, filter parameterization) to enhance flexibility and generalizability (Maretic et al., 2021).
• Exploration of thermodynamic constraints in pattern evolution, particularly quantifying trade-offs between dissipation, speed, and pattern accuracy in reaction-diffusion systems on networks (Nagayama et al., 2023).
• Investigation of robustness, interpretability, and scalability for large-scale heterogeneous graph ensembles and evolving relational data.

EDT-PA provides a principled, modular paradigm for integrating adaptive graph diffusion, flexible optimal transport, and pattern-specific alignment, achieving computational tractability, theoretical robustness, and accuracy across a range of high-dimensional graph analysis tasks.