Geometry-Aware Graph Construction via Adaptive Spectral Bandwidth Control

Kernelized graph methods - spectral clustering, diffusion maps, and sparse kernel -regression graphs - that use Gaussian kernels depend on the choice of Gaussian bandwidth sigma, which governs the spectral character of the local kernel operator. When sigma is too small, the kernel overestimates local complexity and treats each sample as an independent direction; when sigma is too large, the kernel collapses multiple directions together, the condition number diverges, and all geometric discrimination is lost. We propose a choice of scale to make the spectral complexity of the kernel consistent with the intrinsic complexity of the underlying manifold. We propose a per-node bandwidth criterion that operationalizes this principle by jointly matching the kernel's effective rank to the local intrinsic dimension estimated via minimum spanning tree, anchoring the search in the manifold-consistent log-log scaling regime. We evaluate SSL embeddings from six encoders on CIFAR-100, showing that adaptive bandwidth consistently improves leave-one-out (LOO) classification and label propagation (LP) accuracy over fixed-bandwidth methods and competing adaptive methods.

Spectral Embeddings of Degree-$α$ Laplacians in Random Dot Product Graphs

Spectral clustering methods for network data are commonly based on a few matrix representations, such as the adjacency matrix and the symmetric Laplacian. We study a continuum of degree-normalized spectral embeddings that includes these commonly used choices as special cases. Under a random dot product graph model, we establish a row-wise central limit theorem for this family of embeddings. The result provides an explicit description of how degree normalization affects both population geometry and the local uncertainty of embedded nodes. We use the limiting distributions to compare different normalizations in two-community stochastic block models through a projected-Gaussian Bayes-error diagnostic. These comparisons show that no single normalization is uniformly preferred. Instead, the favored normalization depends on network density, community imbalance, and block-probability structure. Typically, stronger normalization is favored in lower-density or more imbalanced settings. These results provide a unified distributional understanding of when and why alternative normalizations may improve spectral clustering.

Learning and Clustering on Temporal Graphs: Principles, Primitives, and Pooling

This work focuses on the problem of learning on temporal graphs, with particular emphasis on the task of clustering: obtaining coarse-grained representations by aggregating information from nodes, edges, and temporal dynamics - a task related to pooling in machine learning on graphs, or community detection in network science. Although graph neural networks reach state-of-the-art performance across many downstream graph tasks, their advantage over established descriptive and inferential clustering algorithms is far less settled, especially under demands of efficiency and recovery accuracy. We frame this tension through three linked perspectives: principles, connecting graph learning and community detection through shared spectral foundations and detectability thresholds in stochastic block model regimes; primitives, making spectral clustering and multislice modularity optimization tractable through GPU-accelerated temporal backends; and pooling, viewing principled community detection as a theory-grounded coarse-graining operator for temporal graphs. Our results indicate that algorithmic methods remain the appropriate tool where attributes are absent or weak - scalability rather than accuracy being the binding obstacle - while neural models are most compelling when structural, temporal, and attribute signals align. By making temporal clustering scalable, GPU-accelerated primitives suggest a route toward theory-grounded pooling, while raising a central question: when does community-based coarse-graining preserve the dynamics needed for downstream learning tasks?

Accelerating Dynamic Graph Clustering on GPU Architectures with cuGraph

This work addresses community detection in temporal networks through GPU-accelerated extensions of spectral clustering and modularity-based algorithms originally designed for static graphs. Built on the NVIDIA RAPIDS ecosystem, the framework enables the characterization and tracking of communities in snapshot-based dynamic graphs, either by Leiden greedy optimization with multi-GPU support via Dask-based workload distribution, or eigendecomposition of a symmetric Bethe-Hessian operator. Our multislice modularity backend achieves up to roughly three orders of magnitude speedup over the CPU reference under an equal-work budget, depending on graph density and snapshot count, while preserving compatibility with existing graph analytics pipelines. We demonstrate its applicability on real-world and synthetic datasets, facilitating exploratory analysis of structural network properties over time. Such capabilities are relevant across several application domains, such as epidemic spreading, financial systems, cybersecurity, and trajectory and mobility analysis. We release our implementation as free and open-source software, including Python bindings through the NetworkX-Temporal library for ease of use and zero-code acceleration with existing codebases.