Generating realistic graphs faces challenges in estimating accurate distribution of graphs in an embedding space while preserving structural characteristics. However, existing graph generation methods primarily focus on approximating the joint distribution of nodes and edges, often overlooking topological properties such as connected components and loops, hindering accurate representation of global structures. To address this issue, we propose a Topology-Aware diffusion-based Graph Generation (TAGG), which aims to sample synthetic graphs that closely resemble the structural characteristics of the original graph based on persistent homology. Specifically, we suggest two core components: 1) Persistence Diagram Matching (PDM) loss which ensures high topological fidelity of generated graphs, and 2) Topology-aware Attention Module (TAM) which induces the denoising network to capture the homological characteristics of the original graphs. Extensive experiments on conventional graph benchmarks demonstrate the effectiveness of our approach indicating high generation performance across various metrics, while achieving closer alignment with the distribution of topological features observed in the original graphs.

In semi-supervised segmentation, capturing meaningful semantic structures from unlabeled data is essential. This is particularly challenging in histopathology image analysis, where objects are densely distributed. To address this issue, we propose a semi-supervised segmentation framework designed to robustly identify and preserve relevant topological features. Our method leverages multiple perturbed predictions obtained through stochastic dropouts and temporal training snapshots, enforcing topological consistency across these varied outputs. This consistency mechanism helps distinguish biologically meaningful structures from transient and noisy artifacts. A key challenge in this process is to accurately match the corresponding topological features across the predictions in the absence of ground truth. To overcome this, we introduce a novel matching strategy that integrates spatial overlap with global structural alignment, minimizing discrepancies among predictions. Extensive experiments demonstrate that our approach effectively reduces topological errors, resulting in more robust and accurate segmentations essential for reliable downstream analysis. Code is available at https://github.com/MelonXu/MATCH.

Topological descriptors have been increasingly utilized for capturing multiscale structural information in relational data. In this work, we consider various filtrations on the (box) product of graphs and the effect on their outputs on the topological descriptors - the Euler characteristic (EC) and persistent homology (PH). In particular, we establish a complete characterization of the expressive power of EC on general color-based filtrations. We also show that the PH descriptors of (virtual) graph products contain strictly more information than the computation on individual graphs, whereas EC does not. Additionally, we provide algorithms to compute the PH diagrams of the product of vertex-and edge-level filtrations on the graph product. We also substantiate our theoretical analysis with empirical investigations on runtime analysis, expressivity, and graph classification performance. Overall, this work paves way for powerful graph persistent descriptors via product filtrations.

Molecular dynamics (MD) simulations generate trajectories in a high-dimensional configuration space whose analysis critically depends on molecular descriptors, typically handcrafted observables or learned kinetic embeddings. Designing descriptors that are both expressive and broadly applicable, however, remains challenging. We study persistent homology (PH) as a general-purpose representation for MD and introduce the masked Flood complex, a protein-tailored modification of a recently introduced simplicial complex construction that emphasizes inter-residue structure at low computational cost. Vectorized persistence diagrams then provide information-rich, geometry-aware summaries of protein conformations, which we evaluate on protein class prediction, frame-level observable regression, and Markov state model (MSM) estimation from learned low-dimensional coordinates in a single shared representation space. Results on the mdCATH dataset show that PH-based descriptors are competitive across tasks, with masked Flood PH yielding the most consistent overall performance. Further, when using topologically-informed MSMs as a drop-in replacement within the recent MarS-FM framework for generative modeling of protein conformations, we obtain consistently better ensemble statistics than MSMs based on physical observables. Finally, we explore the transferability of the generative model to qualitatively different, fast folding, proteins.

Persistence diagrams are common representations in topological data analysis, but they do not naturally live in a vector space, and the statistical tools developed for comparing them have largely evolved separately from those used for downstream prediction. We introduce STRAND (Survival Topological Representation ANalysis of Diagrams), which treats (collections of) PDs as survival data: each topological feature with persistence value $p = d - b$ is a fully observed time-to-event, and the persistence survival function $S(t) = \mathbb{P}(p > t)$ is the central object for comparing diagrams. From this single representation we derive (i) a non-parametric two-sample test with calibrated Type I error and high power from a small number of diagrams; (ii) interpretable effect sizes; and (iii) a 1-Wasserstein-stable feature vector for downstream machine learning. We validate calibration and power on synthetic manifolds with controlled topology, demonstrate competitive vectorisation across 14 graph and 3D point cloud benchmarks, and apply the method to study functional brain connectivity in fMRI/neuroscience data. To our knowledge, STRAND is the first method to provide hypothesis testing and vectorisation for persistence diagrams from a single coherent and interpretable representation.

We introduce the \emph{Topological Stability Index} (TSI), a variance-based scalar measure for persistence barcodes that quantifies the dispersion of persistence lifetimes. Unlike persistent entropy, which depends only on normalized weights, the TSI captures absolute variability and is sensitive to heterogeneous feature scales. We establish fundamental properties of the TSI, including its scaling behavior, invariance under lifetime translation and explicit update formulas under insertion and deletion of bars. We also consider a complementary first-moment-type quantity, the Topological Signal Index (TSigI), which captures the typical scale of persistence lifetimes and provides additional interpretability alongside the TSI. We further introduce a normalized version, $cv\text{TSI}$, which is scale invariant and admits an explicit algebraic relation to the Rényi entropy of order two. In particular, $cv\text{TSI}$ is an affine function of the collision probability $\sum_i p_i^2$, and therefore a monotone reparametrization of the Rényi entropy, providing a direct link between variance-based and entropy-based summaries in topological data analysis. Numerical experiments on synthetic data and stochastic time series demonstrate that the TSI captures structural variability complementary to entropy: it is relatively insensitive to deterministic trends, while responding strongly to stochastic fluctuations and variations in persistence magnitude.

We present a unified pipeline for univariate time series classification via complex networks and persistent homology. A time series is mapped to a graph through one of five constructions across three families--visibility (natural and horizontal visibility graphs), transition, and proximity--and the graph is converted to a dissimilarity matrix from which a Vietoris-Rips filtration yields persistence diagrams. These diagrams are vectorized into fixed-length features through persistence landscapes and topological summary statistics. By standardizing the downstream processing, differences in classification performance are attributable to the network construction and distance metric alone. Experiments on twelve UCR benchmarks show that (i) no single construction dominates: the optimal graph type depends on the signal's discriminative structure; (ii) the graph distance metric is a first-order design choice, with diffusion distance uniformly outperforming shortest-path alternatives; and (iii) persistence-based features degrade gracefully under noise, consistent with the classical stability theorem of persistent homology.

Machine learning for point clouds has been attracting much attention, with many applications in various fields, such as shape recognition and material science. For enhancing the accuracy of such machine learning methods, it is often effective to incorporate global topological features, which are typically extracted by persistent homology. In the calculation of persistent homology for a point cloud, we choose a filtration for the point cloud, an increasing sequence of spaces. Since the performance of machine learning methods combined with persistent homology is highly affected by the choice of a filtration, we need to tune it depending on data and tasks. In this paper, we propose a framework that learns a filtration adaptively with the use of neural networks. In order to make the resulting persistent homology isometry-invariant, we develop a neural network architecture with such invariance. Additionally, we show a theoretical result on a finite-dimensional approximation of filtration functions, which justifies the proposed network architecture. Experimental results demonstrated the efficacy of our framework in several classification tasks.