Simplicial complexes provide a powerful framework for modeling higher-order interactions in structured data, making them particularly suitable for applications such as trajectory prediction and mesh processing. However, existing simplicial neural networks (SNNs), whether convolutional or attention-based, rely primarily on discrete filtering techniques, which can be restrictive. In contrast, partial differential equations (PDEs) on simplicial complexes offer a principled approach to capture continuous dynamics in such structures. In this work, we introduce continuous simplicial neural network (COSIMO), a novel SNN architecture derived from PDEs on simplicial complexes. We provide theoretical and experimental justifications of COSIMO's stability under simplicial perturbations. Furthermore, we investigate the over-smoothing phenomenon--a common issue in geometric deep learning--demonstrating that COSIMO offers better control over this effect than discrete SNNs. Our experiments on real-world datasets demonstrate that COSIMO achieves competitive performance compared to state-of-the-art SNNs in complex and noisy environments.

In this paper, we propose HiPoNet, an end-to-end differentiable neural network for regression, classification, and representation learning on high-dimensional point clouds. Our work is motivated by single-cell data which can have very high-dimensionality - exceeding the capabilities of existing methods for point clouds which are mostly tailored for 3D data. Moreover, modern single-cell and spatial experiments now yield entire cohorts of datasets (i.e., one data set for every patient), necessitating models that can process large, high-dimensional point-clouds at scale. Most current approaches build a single nearest-neighbor graph, discarding important geometric and topological information. In contrast, HiPoNet models the point-cloud as a set of higher-order simplicial complexes, with each particular complex being created using a reweighting of features. This method thus generates multiple constructs corresponding to different views of high-dimensional data, which in biology offers the possibility of disentangling distinct cellular processes. It then employs simplicial wavelet transforms to extract multiscale features, capturing both local and global topology from each view. We show that geometric and topological information is preserved in this framework both theoretically and empirically.

Graph-based recommender systems have achieved remarkable effectiveness by modeling high-order interactions between users and items. However, such approaches are significantly undermined by popularity bias, which distorts the interaction graph's structure--referred to as topology bias. This leads to overrepresentation of popular items, thereby reinforcing biases and fairness issues through the user-system feedback loop. Despite attempts to study this effect, most prior work focuses on the embedding or gradient level bias, overlooking how topology bias fundamentally distorts the message passing process itself. We bridge this gap by providing an empirical and theoretical analysis from a Dirichlet energy perspective, revealing that graph message passing inherently amplifies topology bias and consistently benefits highly connected nodes. To address these limitations, we propose Test-time Simplicial Propagation (TSP), which extends message passing to higher-order simplicial complexes. By incorporating richer structures beyond pairwise connections, TSP mitigates harmful topology bias and substantially improves the representation and recommendation of long-tail items during inference. Extensive experiments across five real-world datasets demonstrate the superiority of our approach in mitigating topology bias and enhancing recommendation quality. The implementation code is available at https://github.com/sotaagi/TSP.

Signed networks can reflect more complex connections through positive and negative edges, and cost-effective signed network sketching can significantly benefit an important link sign prediction task in the era of big data. Existing signed network embedding algorithms mainly learn node representation in the Graph Neural Network (GNN) framework with the balance theory. However, the node-wise representation learning methods either limit the representational power because they primarily rely on node pairwise relationship in the network, or suffer from severe efficiency issues. Recent research has explored simplicial complexes to capture higher-order interactions and integrated them into GNN frameworks. Motivated by that, we propose EdgeSketch+, a simple and effective edge embedding algorithm beyond traditional node-centric modeling that directly represents edges as low-dimensional vectors without transitioning from node embeddings. The proposed approach maintains a good balance between accuracy and efficiency by exploiting the Locality Sensitive Hashing (LSH) technique to swiftly capture the higher-order information derived from the simplicial complex in a manner of no learning processes. Experiments show that EdgeSketch+ matches state-of-the-art accuracy while significantly reducing runtime, achieving speedups of up to $546.07\times$ compared to GNN-based methods.

Graph neural tangent kernels give a principled infinite-width theory for graph neural networks, but inherit a basic limitation of graph models: they see only pairwise structure. Many relational systems contain higher-order interactions that are more naturally represented by simplicial complexes. We introduce the Topological Neural Tangent Kernel (TopoNTK), an infinite-width kernel for simplicial message passing on edge features. TopoNTK combines lower Hodge interactions, capturing graph-like coupling through shared vertices, with upper Hodge interactions, capturing coupling through filled simplices. This makes the kernel sensitive to topology invisible to graph kernels, allowing complexes with the same graph but different filled simplices to induce different kernels. Beyond expressivity, the Hodge structure gives the kernel an interpretable learning geometry. Edge signals decompose into gradient-like, harmonic, and local circulation components, and the spectrum of the TopoNTK determines how quickly each component is learned. This yields a topological form of spectral bias: components aligned with large-eigenvalue modes are learned quickly, while global harmonic modes, retained through the residual channel, often lie at smaller eigenvalues and are learned more slowly. We prove expressivity, Hodge-alignment, spectral learning, and stability properties, and validate them on synthetic simplicial tasks and DBLP higher-order link prediction. The results show that topology is not merely extra structure; it can provide coordinates that make relational learning more faithful, interpretable, and effective.

Graph Neural Networks (GNNs) are limited in their expressive power, struggle with long-range interactions and lack a principled way to model higher-order structures. These problems can be attributed to the strong coupling between the computational graph and the input graph structure. The recently proposed Message Passing Simplicial Networks naturally decouple these elements by performing message passing on the clique complex of the graph. Nevertheless, these models can be severely constrained by the rigid combinatorial structure of Simplicial Complexes (SCs). In this work, we extend recent theoretical results on SCs to regular Cell Complexes, topological objects that flexibly subsume SCs and graphs.

Persistent homology (PH) encodes global information, such as cycles, and is thus increasingly integrated into graph neural networks (GNNs). PH methods in GNNs typically traverse an increasing sequence of subgraphs. In this work, we first expose limitations of this inclusion procedure. To remedy these shortcomings, we analyze contractions as a principled topological operation, in particular, for graph representation learning. We study the persistence of contraction sequences, which we call Contraction Homology (CH). We establish that forward PH and CH differ in expressivity. We then introduce Hourglass Persistence, a class of topological descriptors that interleave a sequence of inclusions and contractions to boost expressivity, learnability, and stability. We also study related families parametrized by two paradigms. We also discuss how our framework extends to simplicial and cellular networks. We further design efficient algorithms that are pluggable into end-to-end differentiable GNN pipelines, enabling consistent empirical improvements over many PH methods across standard real-world graph datasets. Code is available at \href{https://github.com/Aalto-QuML/Hourglass}{this https URL}.