Geometric Evolution Graph Convolutional Networks: Enhancing Graph Representation Learning via Ricci Flow

Message passing is a core mechanism in Graph Neural Networks (GNNs), enabling the iterative update of node embeddings by aggregating information from neighboring nodes. Graph Convolutional Networks (GCNs) exemplify this approach by adapting convolutional operations for graph structures, allowing features from adjacent nodes to be combined effectively. However, GCNs encounter challenges with complex or dynamic data. Capturing long-range dependencies often requires deeper layers, which not only increase computational costs but also lead to over-smoothing, where node embeddings become indistinguishable. To overcome these challenges, reservoir computing has been integrated into GNNs, leveraging iterative message-passing dynamics for stable information propagation without extensive parameter tuning. Despite its promise, existing reservoir-based models lack structured convolutional mechanisms, limiting their ability to accurately aggregate multi-hop neighborhood information. To address these limitations, we propose RGC-Net (Reservoir-based Graph Convolutional Network), which integrates reservoir dynamics with structured graph convolution. Key contributions include: (i) a reimagined convolutional framework with fixed-random reservoir weights and a leaky integrator to enhance feature retention; (ii) a robust, adaptable model for graph classification; and (iii) an RGC-Net-powered transformer for graph generation with application to dynamic brain connectivity. Extensive experiments show RGC-Net achieves state-of-the-art performance in classification and generative tasks, including brain graph evolution, with faster convergence and mitigated over-smoothing. Our source code is available at https://github.com/basiralab/RGC-Net.

Learning local geometry enables graph neural networks (GNNs) to adapt how they compare and integrate neighborhood information. However, estimating geometry from aggregated representations can overlook variation among individual messages and dependencies across feature dimensions. We propose GeoF, a recurrent framework that jointly evolves node features and propagation geometry through message-passing feedback. Each node maintains a local symmetric positive-definite geometry, initialized from a structure-aware prototype atlas and parameterized in block log-triangular coordinates. At each step, the geometry determines neighborhood weights, while triangular frame transport maps transformed source messages into the target node's local coordinates before aggregation. Weighted second-order statistics of residuals between aligned messages and the transformed target state capture directional variation and within-block dependencies, yielding a geometric update target. A shared controller learns complementary corrections through task supervision. A bounded log-triangular update combines these corrections, the target, and the previous geometric state while preserving positive definiteness. The geometry governs subsequent propagation, closing the feedback loop. With parameters shared across recurrent steps, task-specific readouts support node classification, link prediction, and graph classification. Experiments on benchmark datasets show that GeoF consistently outperforms state-of-the-art GNN baselines.

Dynamic graph neural networks (DGNNs) that operate on snapshot sequences typically fall into one of two categories. Temporal-first approaches build per-node temporal embeddings and only afterward perform spatial aggregation, whereas Spatial-first approaches invert this order, feeding the output of a graph convolution into a downstream temporal module. In either case, the rigid sequencing forces the second stage to consume an already-compressed summary produced by the first, ruling out joint reasoning over topology and evolution; effectively, the message-passing operator never gets to weight a neighbor's contribution by that neighbor's past trajectory. This paper introduces SiST-GNN (Simultaneous Spatial-Temporal GNN), which fuses the two signals inside a single message-passing operation rather than chaining them. At each snapshot, we maintain a recurrent hidden state per node that summarises its history, pairs it with the node's current feature vector, and treats the pair as two nodes joined by a cross-time edge; running a standard graph convolution on this temporally augmented graph yields the updated representation. We compare against fourteen link-prediction baselines under both the fixed-split and live-update evaluation regimes, and eleven baselines on node classification. Across the public benchmarks, SiST-GNN improves on the strongest prior method in link prediction by 1-18% in the fixed-split setting, and is the leading learned method on five of six datasets in the live-update setting, improving on the strongest prior method by 1-158% there. We additionally derive three dynamic node-classification tasks by discretizing the underlying continuous-time event streams; here SiST-GNN beats the leading discrete-time (DTDG) baseline by 7-23% and matches continuous-time (CTDG) methods that consume the raw events directly.