We study a broad class of graphical models whose independencies correspond to vertex separation in mixed graphs with directed, undirected, and bidirected edges, that are capable of encoding independence structures arising from feedback, latent and selection mechanisms. In particular, we introduce separable graphs, in which each missing edge implies the existence of a separating set for its endpoints, and essentially separable graphs, those graphs separation equivalent to a separable graph. We show that these models include many existing graph families used to define graphical models an provide several characterizations of separable graphs and essentially separable graphs. We also provide multiple characterizations of separation equivalence for separable graphs. One is a graphical characterization in terms of ordinary graph properties, extending earlier results for specific subfamilies Another is a separational characterization depending only on graph separation properties. Finally, we provide a canonical representation for the equivalence classes of essentially separable graphs and develop an algorithm that, under suitable assumptions, identifies the equivalence class of any essentially separable graph.

A growing number of techniques leverage the spatial structures that underlie many real-world datasets. Despite these advances, the complementary task of estimating spatial structures and understanding their role within these techniques has often been overlooked. In neurophysiological data analysis specifically, numerous methods exist to estimate brain connectivity, but most are not explicitly model-based, dynamic, multivariate, or directed. To address these limitations, we previously introduced noise-driven heat modelling on graphs for neurophysiological connectivity estimation. In this study, we extend this framework by relaxing earlier noise assumptions and adding regularisation to improve robustness. We also develop a simulation procedure to characterise and evaluate our technique in a controlled setting. Finally, we demonstrate that the technique is able to capture meaningful spatial structure across two experiments, each using two real-world datasets. The explicit model formulation of our connectivity estimator has the potential to improve the interpretability of graph-based techniques across a wide range of applications. The code implementing our method is available at https://github.com/sgoerttler/Heat_Connectivity.

We consider sparse multivariate stochastic systems that evolve in continuous time according to a causal mechanism and present methodology to recover the system's time-infinitesimal transition mechanism from mere cross-sectional data. This observational paradigm is motivated by applications such as gene expression analysis, where destructive experimental techniques may only allow recording data once over a cell's lifetime. Precisely, we assume the system follows a time-homogeneous diffusion process that has reached an equilibrium distribution at observation time. Further, we assume the causal mechanism is fully described by the diffusion drift, is acyclic, and its causal structure graph is known. In this setting, we prove that the full causal mechanism, i.e., the drift function, can be non-parametrically identified under a weak non-explosion criterion. We derive a non-parametric kernel estimator for this challenging inverse problem and prove its consistency. Moreover, we propose a cross-validation scheme for hyperparameter tuning, illustrate the behavior of our estimator in simulations, and we discuss connections with irreversible generative diffusion models and low-frequency sampled data.

Inference-time scaling is a promising paradigm to improve generative models, especially when outputs must satisfy structural constraints or optimize downstream rewards. We consider Masked Diffusion Model (MDM) and introduce MDM-VGB, a discrete diffusion sampler that augments unmasking generation with theoretically principled reward-guided remasking. Inspired by the recent success of the classical Jerrum-Sinclair backtracking Markov chain in reward-tilted generation, MDM-VGB extends the backtracking random walk from a fixed prefix tree to a masked-state graph, allowing tokens to be unmasked and remasked at arbitrary positions. The resulting sampler favors unmasking and remasking moves that lead to higher-value partial configurations, enabling both effective high-reward generation and efficient repair of low-reward samples. We prove that MDM-VGB is robust to process-verifier noise and achieves quadratic complexity, while popular test-time heuristics such as best-of-$N$ can incur exponential complexity due to error accumulation. Our theoretical findings are corroborated by strong empirical performance, particularly on popular constraint-satisfaction and scientific benchmarks such as Sudoku and QM9.

Causal representation learning for time series has developed strong identifiability results in discrete-time latent causal models, but identifiability in continuous-time latent stochastic differential equation (SDE) models remains largely open. We address this gap using environment-induced shifts in diffusion covariance. We study additive-noise latent SDEs observed through an unknown nonlinear diffeomorphism, with shared drift but environment-specific diffusion covariance. We show that two diagonal diffusion regimes with pairwise distinct coordinate-wise variance ratios identify the latent coordinates up to permutation and scaling, without any sparsity assumption on the drift. We first prove this result for linear Ornstein--Uhlenbeck systems and then extend it to general additive-noise latent SDEs. Under mild smoothness, the instantaneous drift-Jacobian causal graph is identifiable up to the same permutation. We propose a two-stage estimator for latent disentanglement and optional graph recovery; experiments on synthetic systems confirm the predicted identifiability boundary, and an application to Hardanger Bridge monitoring data illustrates the approach on real sensor trajectories.

Causal discovery seeks to uncover the causal dependencies among variables. For this purpose, we propose an algorithm called Tensor-based Second-order Causal Discovery (TSCD). Its input is a tensor obtained from the covariance matrices of observational and interventional data. Assuming the causal dependencies follow a linear structural equation model on a directed acyclic graph (DAG), TSCD outputs the DAG and the functions on its edges, requiring only that the noise variables are uncorrelated. We also implement a version of the approach for nonlinear models. Our focus on second-order statistics (via the covariance matrices) is motivated by their statistical and computational efficiency relative to higher-order moments, their identifiability relative to first-order statistics, and that they work regardless of whether the variables are Gaussian. We show that TSCD has identifiable causal order and parameters from a number of interventions that is logarithmic in the number of variables. Experiments show that TSCD is robust to noise, competitive with existing methods, and scales to hundreds of variables.

Causal discovery in the form of a directed acyclic graph (DAG) for dynamic time series data has been widely studied in various applications. In this work, we propose a dynamic DAG discovery algorithm, Meta-D2AG, based on online metalearning. Meta-D2AG is designed to learn dynamic DAG structures from potentially nonlinear and non-stationary time series datasets, accounting for changes in both parameters and graph structures. Unlike most of the existing work focusing on observational, offline, and/or stationary settings, Meta-D2AG explicitly treats data collected at different time points with distribution shifts as distinct domains, which is assumed to occur as a result of external interventions. Moreover, MetaD2AG involves a new online meta-learning framework to take advantage of the temporal transition among existing domains such that it can quickly adapt to new domains with few measurements. A first-order optimization approach is utilized to efficiently solve the meta-learning framework, and theoretical analysis establishes the identifiability conditions and the convergence of the learning process. We demonstrate the promising performance of the proposed meta learning framework through better accuracy on benchmark datasets against state-of-the-art baselines.

Masked Graph Autoencoders (MGAEs) have gained significant attention recently. Their proxy tasks typically involve random corruption of input graphs followed by reconstruction. However, in the molecular domain, two main issues arise: the predetermined mask ratio and reconstruction objectives can lead to suboptimal performance or negative transfer due to overly simplified or complex tasks, and these tasks may deviate from chemical priors. To tackle these challenges, we propose Dynamic and Chemical Constraints (DyCC) for MGAEs. This includes a masking strategy called GIBMS, which preserves essential semantic information during graph masking while adaptively adjusting the mask ratio and content for each molecule. Additionally, we introduce a Soft Label Generator (SLG) that reconstructs masked tokens as learnable prototypes (soft labels) rather than hard labels. These components adhere to chemical constraints and allow dynamic variation of proxy tasks during training. We integrate the model-agnostic DyCC into various MGAEs and conduct comprehensive experiments, demonstrating significant performance improvements. Our code is available at https://github.

Graph few-shot learning has attracted increasing attention due to its ability to rapidly adapt models to new tasks with only limited labeled nodes. Despite the remarkable progress made by existing graph few-shot learning methods, several key limitations remain. First, most current approaches rely on predefined and unified graph filters (e.g., low-pass or high-pass filters) to globally enhance or suppress node frequency signals. Such fixed spectral operations fail to account for the heterogeneity of local topological structures inherent in real-world graphs. Moreover, these methods often assume that the support and query sets are drawn from the same distribution. However, under few-shot conditions, the limited labeled data in the support set may not sufficiently capture the complex distribution of the query set, leading to suboptimal generalization.

Large language models (LLMs) have been used in many zero-shot learning problems, with their strong generalization ability. Recently, adopting LLMs in textattributed graphs (TAGs) has drawn increasing attention. However, the adoption of LLMs faces two major challenges: limited information on graph structure and unreliable responses. LLMs struggle with text attributes isolated from the graph topology. Worse still, they yield unreliable predictions due to both information insufficiency and the inherent weakness of LLMs (e.g., hallucination). Towards this end, this paper proposes a novel method named Dynamic Text Bundling Supervision (DENSE) that queries LLMs with bundles of texts to obtain bundle-level labels and uses these labels to supervise graph neural networks.