In this paper, we propose a non-parametric task-agnostic representation learning method for cell complexes (CCs). Specifically, we propose a scattering transform for CCs that extends geometric scattering to CCs. In addition, we introduce scattering transforms for product cell complexes (PCCs), which are a particular type of CCs that can be factored as the Cartesian product of two smaller simplicial complexes (SCs). We utilize the intrinsic product structure of PCCs to develop separable filters across the two constituent factors of PCCs. The proposed scattering transform for PCCs is computationally efficient, as it can be expressed in terms of scattering on the individual SCs. The effectiveness of the proposed model is demonstrated through experiments on various tasks, such as trajectory classification, missing edge feature estimation, and edge feature prediction, on both synthetic and real-world PCCs.

Simplicial neural networks (SNNs) are deep models for higher-order graph representation learning. SNNs learn low-dimensional embeddings of simplices in a simplicial complex by aggregating features of their respective upper, lower, boundary, and coboundary adjacent simplices. The aggregation in SNNs is carried out during training. Since the number of simplices of various orders in a simplicial complex is significantly large, the memory and training-time requirement in SNNs is enormous. In this work, we propose a scalable simplicial-aware neural network (SaNN) model with a constant run-time and memory requirements independent of the size of the simplicial complex and the density of interactions in it. SaNN is based on pre-aggregated simplicial-aware features as inputs to a neural network, so it has a strong simplicial-structural inductive bias. We provide theoretical conditions under which SaNN is provably more powerful than the Weisfeiler-Lehman (WL) graph isomorphism test and as powerful as the simplicial Weisfeiler-Lehman (SWL) test. We also show that SaNN is permutation and orientation equivariant and satisfies simplicial-awareness of the highest order in a simplicial complex. We demonstrate via numerical experiments that despite being computationally economical, the proposed model achieves state-of-the-art performance in predicting trajectories, simplicial closures, and classifying graphs.

Simplicial neural network models are becoming popular for processing and analyzing higher-order graph data, but they suffer from high training complexity and dependence on task-specific labels. To address these challenges, we propose simplicial scattering networks (SSNs), a parameter-free model inspired by scattering transforms designed to extract task-agnostic features from simplicial complex data without labels in a principled manner. Specifically, we propose a simplicial scattering transform based on random walk matrices for various adjacencies underlying a simplicial complex. We then use the simplicial scattering transform to construct a deep filter bank network that captures high-frequency information at multiple scales. The proposed simplicial scattering transform possesses properties such as permutation invariance, robustness to perturbations, and expressivity. We theoretically prove that including higher-order information improves the robustness of SSNs to perturbations. Empirical evaluations demonstrate that SSNs outperform existing simplicial or graph neural models in many tasks like node classification, simplicial closure, graph classification, trajectory prediction, and simplex prediction while being computationally efficient.

In this paper, we focus on learning sparse graphs with a core-periphery structure. We propose a generative model for data associated with core-periphery structured networks to model the dependence of node attributes on core scores of the nodes of a graph through a latent graph structure. Using the proposed model, we jointly infer a sparse graph and nodal core scores that induce dense (sparse) connections in core (respectively, peripheral) parts of the network. Numerical experiments on a variety of real-world data indicate that the proposed method learns a core-periphery structured graph from node attributes alone, while simultaneously learning core score assignments that agree well with existing works that estimate core scores using graph as input and ignoring commonly available node attributes.