Subspace Projection Methods for Fast Spectral Embeddings of Evolving Graphs

A crucial assumption in graph signal processing (GSP) is the existence of an underlying graph that captures the pairwise similarities between nodes, allowing filters to be designed based on this graph for tasks such as denoising. For spatial-temporal data in which node-to-node similarities evolve over time, a static spatial graph is insufficient. In this paper, to represent slowly time-varying pairwise relationships, we model the graph changes in two consecutive adjacency matrices $P = W^{(2)} - W^{(1)}$ across time as a low-rank matrix. % Specifically, given an initial adjacency matrix $W^{(1)}$ at time $t=1$, we jointly interpolate a signal $x_2$ and estimate $W^{(2)}$ at $t=2$ using both a graph signal smoothness prior for $x_2$ and a low-rank prior on $¶$. We alternate optimization steps. With $W^{(2)}$ fixed, $x_2$ is interpolated by solving a linear system. Alternatively, holding $x_2$ fixed, $W^{(2)}$ is updated via proximal gradient descent (PGD). The proximal mapping of the rank term $Gamma(W^{(2)} - W^{(1)})$ is approximated in linear time using a fast orthogonal matching pursuit (OMP) algorithm that selects a sparse combination of atoms from a dictionary $cR$ formed by the outer products of $W^{(1)}$'s eigenvectors. We unroll iterations of our algorithm into layers to build a lightweight neural network for limited data-driven parameter tuning. Experiments show that our joint optimization achieves better signal interpolation compared to existing time-varying graph models.

Eigenvalues are a workhorse of graph machine learning: spectral clustering, spectral graph neural networks, and graph kernels all summarize a network by the eigenvalues of a matrix. These eigenvalues are almost always treated as a static list of numbers. When a graph changes, however, its eigenvalues move, and when the matrix is not symmetric, they move in the complex plane, where they can wind around each other. The pattern of this winding is a braid, a classical object from topology. We propose spectral braids, a methodology for dynamic directed networks: each snapshot is aggregated into a community-level transition matrix, its complex eigenvalues are tracked over time, and the braid they trace is read off as a word. The word is then mapped to a matrix by the Burau representation, which gives a fixed-size feature vector for any classifier. The braid has a concrete meaning: each eigenvalue describes how a group of communities behaves, and each crossing records one group overtaking another. We evaluate the method on dynamic network classification, using synthetic networks of online communities whose labels describe how the network evolved between the same starting and ending states. Our experimental results show that the proposed features reach 97.1% accuracy on four classes, against 44.5% for the best feature set that does not track eigenvalues, and that the Burau features also separate networks containing the same events in a different order, which simple counts of crossings cannot do. To the best of our knowledge, this is the first study that defines braids and braid monodromy on graph data and uses them as features for learning on dynamic networks.

Graph Neural Networks (GNNs) are widely used for representation learning on graphs, but most methods assume static topologies, making them inefficient on evolving networks where edges change over time. Existing dynamic approaches either model graph evolution through temporal GNN architectures without focusing on efficient dynamic maintenance, or are restricted to linear propagation models based on Personalized PageRank. In this work, we study how to efficiently maintain node representations for non-linear GNN propagation under edge insertions and deletions. The propagation has no learned parameters, and only a classifier applied afterward is trained. For a broad class of standard activation functions, we develop a residual-based dynamic algorithm that selectively propagates local errors via push operations, maintaining an approximation to the evolving fixed point without full recomputation. We prove that our method achieves amortized $O(1/ε)$ update time per graph change under a degree-normalized error guarantee. Our approach uses a potential-based analysis in a degree-scaled norm and, in contrast to prior work on the linear case, requires no randomness assumptions on either the update sequence or the input vector. For the linear special case, we additionally provide an exact dynamic algorithm via low-rank matrix inverse updates. Experiments on benchmark datasets show that incorporating non-linearity improves accuracy while preserving efficient update performance, yielding a scalable and theoretically grounded method for maintaining this propagation on dynamic graphs.