DeFoG: Discrete Flow Matching for Graph Generation

Graph generation is fundamental in diverse scientific applications, due to its ability to reveal the underlying distribution of complex data, and eventually generate new, realistic data points. Despite the success of diffusion models in this domain, those face limitations in sampling efficiency and flexibility, stemming from the tight coupling between the training and sampling stages. To address this, we propose DeFoG, a novel framework using discrete flow matching for graph generation. DeFoG employs a flow-based approach that features an efficient linear interpolation noising process and a flexible denoising process based on a continuous-time Markov chain formulation. We leverage an expressive graph transformer and ensure desirable node permutation properties to respect graph symmetry. Crucially, our framework enables a disentangled design of the training and sampling stages, enabling more effective and efficient optimization of model performance. We navigate this design space by introducing several algorithmic improvements that boost the model performance, consistently surpassing existing diffusion models. We also theoretically demonstrate that, for general discrete data, discrete flow models can faithfully replicate the ground truth distribution - a result that naturally extends to graph data and reinforces DeFoG's foundations. Extensive experiments show that DeFoG achieves state-of-the-art results on synthetic and molecular datasets, improving both training and sampling efficiency over diffusion models, and excels in conditional generation on a digital pathology dataset.

Generative Modelling of Structurally Constrained Graphs

Graph diffusion models have emerged as state-of-the-art techniques in graph generation, yet integrating domain knowledge into these models remains challenging. Domain knowledge is particularly important in real-world scenarios, where invalid generated graphs hinder deployment in practical applications. Unconstrained and conditioned graph generative models fail to guarantee such domain-specific structural properties. We present ConStruct, a novel framework that allows for hard-constraining graph diffusion models to incorporate specific properties, such as planarity or acyclicity. Our approach ensures that the sampled graphs remain within the domain of graphs that verify the specified property throughout the entire trajectory in both the forward and reverse processes. This is achieved by introducing a specific edge-absorbing noise model and a new projector operator. ConStruct demonstrates versatility across several structural and edge-deletion invariant constraints and achieves state-of-the-art performance for both synthetic benchmarks and attributed real-world datasets. For example, by leveraging planarity in digital pathology graph datasets, the proposed method outperforms existing baselines and enhances generated data validity by up to 71.1 percentage points.

Tertiary Lymphoid Structures Generation through Graph-based Diffusion

Graph-based representation approaches have been proven to be successful in the analysis of biomedical data, due to their capability of capturing intricate dependencies between biological entities, such as the spatial organization of different cell types in a tumor tissue. However, to further enhance our understanding of the underlying governing biological mechanisms, it is important to accurately capture the actual distributions of such complex data. Graph-based deep generative models are specifically tailored to accomplish that. In this work, we leverage state-of-the-art graph-based diffusion models to generate biologically meaningful cell-graphs. In particular, we show that the adopted graph diffusion model is able to accurately learn the distribution of cells in terms of their tertiary lymphoid structures (TLS) content, a well-established biomarker for evaluating the cancer progression in oncology research. Additionally, we further illustrate the utility of the learned generative models for data augmentation in a TLS classification task. To the best of our knowledge, this is the first work that leverages the power of graph diffusion models in generating meaningful biological cell structures.

Reconstruction of Time-varying Graph Signals via Sobolev Smoothness

Graph Signal Processing (GSP) is an emerging research field that extends the concepts of digital signal processing to graphs. GSP has numerous applications in different areas such as sensor networks, machine learning, and image processing. The sampling and reconstruction of static graph signals have played a central role in GSP. However, many real-world graph signals are inherently time-varying and the smoothness of the temporal differences of such graph signals may be used as a prior assumption. In the current work, we assume that the temporal differences of graph signals are smooth, and we introduce a novel algorithm based on the extension of a Sobolev smoothness function for the reconstruction of time-varying graph signals from discrete samples. We explore some theoretical aspects of the convergence rate of our Time-varying Graph signal Reconstruction via Sobolev Smoothness (GraphTRSS) algorithm by studying the condition number of the Hessian associated with our optimization problem. Our algorithm has the advantage of converging faster than other methods that are based on Laplacian operators without requiring expensive eigenvalue decomposition or matrix inversions. The proposed GraphTRSS is evaluated on several datasets including two COVID-19 datasets and it has outperformed many existing state-of-the-art methods for time-varying graph signal reconstruction. GraphTRSS has also shown excellent performance on two environmental datasets for the recovery of particulate matter and sea surface temperature signals.

Interpretable Stability Bounds for Spectral Graph Filters

Graph-structured data arise in a variety of real-world context ranging from sensor and transportation to biological and social networks. As a ubiquitous tool to process graph-structured data, spectral graph filters have been used to solve common tasks such as denoising and anomaly detection, as well as design deep learning architectures such as graph neural networks. Despite being an important tool, there is a lack of theoretical understanding of the stability properties of spectral graph filters, which are important for designing robust machine learning models. In this paper, we study filter stability and provide a novel and interpretable upper bound on the change of filter output, where the bound is expressed in terms of the endpoint degrees of the deleted and newly added edges, as well as the spatial proximity of those edges. This upper bound allows us to reason, in terms of structural properties of the graph, when a spectral graph filter will be stable. We further perform extensive experiments to verify intuition that can be gained from the bound.

On the Stability of Graph Convolutional Neural Networks under Edge Rewiring

Graph neural networks are experiencing a surge of popularity within the machine learning community due to their ability to adapt to non-Euclidean domains and instil inductive biases. Despite this, their stability, i.e., their robustness to small perturbations in the input, is not yet well understood. Although there exists some results showing the stability of graph neural networks, most take the form of an upper bound on the magnitude of change due to a perturbation in the graph topology. However, these existing bounds tend to be expressed in terms of uninterpretable variables, limiting our understanding of the model robustness properties. In this work, we develop an interpretable upper bound elucidating that graph neural networks are stable to rewiring between high degree nodes. This bound and further research in bounds of similar type provide further understanding of the stability properties of graph neural networks.

Graph signal processing for machine learning: A review and new perspectives

The effective representation, processing, analysis, and visualization of large-scale structured data, especially those related to complex domains such as networks and graphs, are one of the key questions in modern machine learning. Graph signal processing (GSP), a vibrant branch of signal processing models and algorithms that aims at handling data supported on graphs, opens new paths of research to address this challenge. In this article, we review a few important contributions made by GSP concepts and tools, such as graph filters and transforms, to the development of novel machine learning algorithms. In particular, our discussion focuses on the following three aspects: exploiting data structure and relational priors, improving data and computational efficiency, and enhancing model interpretability. Furthermore, we provide new perspectives on future development of GSP techniques that may serve as a bridge between applied mathematics and signal processing on one side, and machine learning and network science on the other. Cross-fertilization across these different disciplines may help unlock the numerous challenges of complex data analysis in the modern age.

node2coords: Graph Representation Learning with Wasserstein Barycenters

In order to perform network analysis tasks, representations that capture the most relevant information in the graph structure are needed. However, existing methods do not learn representations that can be interpreted in a straightforward way and that are robust to perturbations to the graph structure. In this work, we address these two limitations by proposing node2coords, a representation learning algorithm for graphs, which learns simultaneously a low-dimensional space and coordinates for the nodes in that space. The patterns that span the low dimensional space reveal the graph's most important structural information. The coordinates of the nodes reveal the proximity of their local structure to the graph structural patterns. In order to measure this proximity by taking into account the underlying graph, we propose to use Wasserstein distances. We introduce an autoencoder that employs a linear layer in the encoder and a novel Wasserstein barycentric layer at the decoder. Node connectivity descriptors, that capture the local structure of the nodes, are passed through the encoder to learn the small set of graph structural patterns. In the decoder, the node connectivity descriptors are reconstructed as Wasserstein barycenters of the graph structural patterns. The optimal weights for the barycenter representation of a node's connectivity descriptor correspond to the coordinates of that node in the low-dimensional space. Experimental results demonstrate that the representations learned with node2coords are interpretable, lead to node embeddings that are stable to perturbations of the graph structure and achieve competitive or superior results compared to state-of-the-art methods in node classification.

Mask Combination of Multi-layer Graphs for Global Structure Inference

Structure inference is an important task for network data processing and analysis in data science. In recent years, quite a few approaches have been developed to learn the graph structure underlying a set of observations captured in a data space. Although real world data is often acquired in settings where relationships are influenced by a priori known rules, this domain knowledge is still not well exploited in structure inference problems. In this paper, we identify the structure of signals defined in a data space whose inner relationships are encoded by multi-layer graphs. We aim at properly exploiting the information originating from each layer to infer the global structure underlying the signals. We thus present a novel method for combining the multiple graphs into a global graph using mask matrices, which are estimated through an optimization problem that accommodates the multi-layer graph information and a signal representation model. The proposed mask combination method also estimates the contribution of each graph layer in the structure of signals. The experiments conducted both on synthetic and real world data suggest that integrating the multi-layer graph representation of the data in the structure inference framework enhances the learning procedure considerably by adapting to the quality and the quantity of the input data

Combining Anatomical and Functional Networks for Neuropathology Identification: A Case Study on Autism Spectrum Disorder

While the prevalence of Autism Spectrum Disorder (ASD) is increasing, research towards the definition of a common etiology is still ongoing. In this regard, modern machine learning and network science pave the way for a better understanding of the pathology and the development of diagnosis aid systems. At the same time, the culture of data sharing heads favorably in that direction, with the availability of large datasets such as the Autism Brain Imaging Data Exchange (ABIDE) one. The present work addresses the classification of neurotypical and ASD subjects by combining knowledge about both the anatomy and the functional activity of the brain. In particular, we model the brain structure as a graph, and the time-varying resting-state functional MRI (rs-fMRI) signals as values that live on the nodes of that graph. We then borrow tools from the emerging field of Graph Signal Processing (GSP) to build features related to the frequency content of these signals. In order to make these features highly discriminative, we apply an extension of the Fukunaga-Koontz transform. Finally, we use these new markers to train a decision tree, an interpretable classification scheme, which results in a final diagnosis aid model. Interestingly, the resulting decision tree outperforms state-of-the-art methods on the ABIDE dataset. Moreover, the analysis of the predictive markers reveals the influence of the frontal and temporal lobes in the diagnosis of the disorder, which is in line with previous findings in the literature of neuroscience. Our results indicate that exploiting jointly structural and functional information of the brain can reveal important information about the complexity of the neuropathology.