Graph Structure Learning

Temporal graph learning is crucial for dynamic networks where nodes and edges evolve over time and new nodes continuously join the system. Inductive representation learning in such settings faces two major challenges: effectively representing unseen nodes and mitigating noisy or redundant graph information. We propose GTGIB, a versatile framework that integrates Graph Structure Learning (GSL) with Temporal Graph Information Bottleneck (TGIB). We design a novel two-step GSL-based structural enhancer to enrich and optimize node neighborhoods and demonstrate its effectiveness and efficiency through theoretical proofs and experiments. The TGIB refines the optimized graph by extending the information bottleneck principle to temporal graphs, regularizing both edges and features based on our derived tractable TGIB objective function via variational approximation, enabling stable and efficient optimization. GTGIB-based models are evaluated to predict links on four real-world datasets; they outperform existing methods in all datasets under the inductive setting, with significant and consistent improvement in the transductive setting.

We study neural architectures in which each hidden layer is defined by the stationary state of a dissipative Schrödinger-type dynamics on a learned latent graph. On stable branches, the local stationary problem defines a differentiable implicit graph layer. To learn the graph itself, we optimize over the stratified moduli space of weighted graphs and equip each stratum with a non-degenerate Kähler-Hessian metric that keeps natural-gradient descent and face crossing well posed. We then show that a multilayer stationary network is equivalent to an exact global stationary problem on a supra-graph, and that it admits a penalized global relaxation whose stationary states converge to the exact one as the penalty parameter tends to infinity. Reverse-mode differentiation is recovered as the adjoint of the exact global system, and the penalized adjoint converges to it in the same limit. Finally, under finite-dimensional strong-monotonicity and admissible-lift assumptions, the corresponding represented hypothesis classes coincide among resolvent feed-forward networks, graph-stationary networks, supra-graph stationary systems, and sheaf-based architectures with unitary connection. The resulting structural identifications yield complexity bounds controlled by sparse graph or supra-graph geometry rather than dense ambient connectivity.

Signed graphs are equipped with both positive and negative edge weights, encoding pairwise correlations as well as anti-correlations in data. A balanced signed graph is a signed graph with no cycles containing an odd number of negative edges. Laplacian of a balanced signed graph has eigenvectors that map via a simple linear transform to ones in a corresponding positive graph Laplacian, thus enabling reuse of spectral filtering tools designed for positive graphs. We propose an efficient computation method to learn a balanced signed graph Laplacian directly from data. Specifically, extending a previous linear programming (LP) based sparse inverse covariance estimation method called CLIME, we formulate a new LP problem for each Laplacian column $i$, where the linear constraints restrict weight signs of edges stemming from node $i$, so that nodes of same / different polarities are connected by positive / negative edges. We derive a feasible CLIME parameter $ρ_i$ for each sign-constrained column problem. We solve the LP problem efficiently by tailoring a sparse LP method based on ADMM. We theoretically prove that the row / column updates produce a non-increasing objective sequence, and show that the iterations are terminated in a finite number of steps. Extensive experimental results on synthetic and real-world datasets show that our balanced graph learning method outperforms competing methods and enables reuse of spectral filters, wavelets, and graph neural nets (GNN) constructed for positive graphs.

We propose a novel framework for learning time-varying graphs from spatiotemporal measurements. Given an appropriate prior on the temporal behavior of signals, our proposed method can estimate time-varying graphs from a small number of available measurements. To achieve this, we introduce three regularization terms in convex optimization problems that constrain the sparseness of temporal variations of the time-varying networks. Moreover, a computationally scalable algorithm is introduced to solve the optimization problem efficiently. The experimental results with synthetic and real datasets (point cloud, temperature, and EEG data) demonstrate that our proposed method outperforms state-of-the-art methods.

Reconstructing complex network topologies from data is a fundamental challenge in cybernetics and graph signal processing, with applications in neuroscience, sensor, and social networks. In practice, target-domain samples are scarce while heterogeneous source-domain data are abundant. Fusing these sources is challenging: Euclidean averaging works for homogeneous sources but degrades sharply as inter-source divergence grows, collapsing distinct geometries into an inflated, biased consensus. We exploit the Wasserstein metric's distribution-preserving properties to counter heterogeneity while preserving each source's intrinsic geometry. We propose MS-WDRO, a multi-source Wasserstein distributionally robust graph learning framework that fuses heterogeneous sources via their weighted Wasserstein barycenter, a geometrically principled nominal distribution, then builds an ambiguity ball around it to hedge residual uncertainty. Minimizing worst-case risk yields a tractable regularized Laplacian estimator solved efficiently via a provably convergent ADMM scheme. We establish non-asymptotic guarantees: a finite-sample concentration bound for the empirical barycenter, a pooling bias lower bound proving naive aggregation is suboptimal, and an out-of-sample excess risk bound decaying at a parametric rate with only logarithmic dependence on source count. To calibrate hyperparameters governing robustness, sparsity, and source fusion, we unroll the solver into a differentiable architecture trained end-to-end, achieving data-adaptive calibration beyond cross-validation while retaining interpretability. Experiments on synthetic benchmarks and the multi-site ABIDE~I neuroimaging dataset show MS-WDRO consistently outperforms seven baselines in graph recovery, sample efficiency, and downstream diagnostic utility, with the largest gains in the sample-scarce regime.