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Found 41 papers, 15 papers with code
Improving the Noise Estimation of Latent Neural Stochastic Differential Equations
Latent neural stochastic differential equations (SDEs) have recently emerged as a promising approach for learning generative models from stochastic time series data.
Topological Trajectory Classification and Landmark Inference on Simplicial Complexes
Specifically, given a set of labelled trajectories, which we interpret as edge-flows on the underlying simplicial complex, we search for 2-simplices whose deletion results in an optimal separation of the trajectory labels according to the corresponding spectral embedding of the trajectories into the harmonic space.
Residual Connections and Normalization Can Provably Prevent Oversmoothing in GNNs
This results in the convergence of node representations to the top-$k$ eigenspace of the message-passing operator; (c) moreover, we show that the centering step of a normalization layer -- which can be understood as a projection -- alters the graph signal in message-passing in such a way that relevant information can become harder to extract.
Graph Neural Networks Do Not Always Oversmooth
By generalizing methods from conventional deep neural networks (DNNs), we can describe the distribution of features at the output layer of deep GCNs in terms of a GP: as expected, we find that typical parameter choices from the literature lead to oversmoothing.
Node-Level Topological Representation Learning on Point Clouds
Topological Data Analysis (TDA) allows us to extract powerful topological and higher-order information on the global shape of a data set or point cloud.
Learning From Simplicial Data Based on Random Walks and 1D Convolutions
Triggered by limitations of graph-based deep learning methods in terms of computational expressivity and model flexibility, recent years have seen a surge of interest in computational models that operate on higher-order topological domains such as hypergraphs and simplicial complexes.
Position: Topological Deep Learning is the New Frontier for Relational Learning
At the same time, this paper serves as an invitation to the scientific community to actively participate in TDL research to unlock the potential of this emerging field.
TopoX: A Suite of Python Packages for Machine Learning on Topological Domains
We introduce TopoX, a Python software suite that provides reliable and user-friendly building blocks for computing and machine learning on topological domains that extend graphs: hypergraphs, simplicial, cellular, path and combinatorial complexes.
Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphs
In this context, cell complexes are often seen as a subclass of hypergraphs with additional algebraic structure that can be exploited, e. g., to develop a spectral theory.
Disentangling the Spectral Properties of the Hodge Laplacian: Not All Small Eigenvalues Are Equal
The rich spectral information of the graph Laplacian has been instrumental in graph theory, machine learning, and graph signal processing for applications such as graph classification, clustering, or eigenmode analysis.
Non-isotropic Persistent Homology: Leveraging the Metric Dependency of PH
Persistent Homology is a widely used topological data analysis tool that creates a concise description of the topological properties of a point cloud based on a specified filtration.
Optimal transport distances for directed, weighted graphs: a case study with cell-cell communication networks
Comparing graphs by means of optimal transport has recently gained significant attention, as the distances induced by optimal transport provide both a principled metric between graphs as well as an interpretable description of the associated changes between graphs in terms of a transport plan.
Representing Edge Flows on Graphs via Sparse Cell Complexes
In this paper, we generalize this approach to cellular complexes and introduce the flow representation learning problem, i. e., the problem of augmenting the observed graph by a set of cells, such that the eigenvectors of the associated Hodge Laplacian provide a sparse, interpretable representation of the observed edge flows on the graph.
On Performance Discrepancies Across Local Homophily Levels in Graph Neural Networks
We ground the practical implications of this work through granular analysis on five real-world datasets with varying global homophily levels, demonstrating that (a) GNNs can fail to generalize to test nodes that deviate from the global homophily of a graph, and (b) high local homophily does not necessarily confer high performance for a node.
Learning the effective order of a hypergraph dynamical system
Dynamical systems on hypergraphs can display a rich set of behaviours not observable for systems with pairwise interactions.
Topological Point Cloud Clustering
TPCC synthesizes desirable features from spectral clustering and topological data analysis and is based on considering the spectral properties of a simplicial complex associated to the considered point cloud.
Signal Processing on Product Spaces
We establish a framework for signal processing on product spaces of simplicial and cellular complexes.
Dirac signal processing of higher-order topological signals
We discuss in detail the properties of the Dirac operator including its spectrum and the chirality of its eigenvectors and we adopt this operator to formulate Dirac signal processing that can filter noisy signals defined on nodes, links and triangles of simplicial complexes.
On Graph Neural Network Fairness in the Presence of Heterophilous Neighborhoods
We study the task of node classification for graph neural networks (GNNs) and establish a connection between group fairness, as measured by statistical parity and equal opportunity, and local assortativity, i. e., the tendency of linked nodes to have similar attributes.
Topological Deep Learning: Going Beyond Graph Data
Topological deep learning is a rapidly growing field that pertains to the development of deep learning models for data supported on topological domains such as simplicial complexes, cell complexes, and hypergraphs, which generalize many domains encountered in scientific computations.
Simplicial Convolutional Filters
We study linear filters for processing signals supported on abstract topological spaces modeled as simplicial complexes, which may be interpreted as generalizations of graphs that account for nodes, edges, triangular faces etc.
Outlier Detection for Trajectories via Flow-embeddings
Here we consider trajectories as edge-flow vectors defined on a simplicial complex, a higher-order generalization of graphs, and use the Hodge 1-Laplacian of the simplicial complex to derive embeddings of these edge-flows.
Signal Processing on Cell Complexes
The processing of signals supported on non-Euclidean domains has attracted large interest recently.
Hodgelets: Localized Spectral Representations of Flows on Simplicial Complexes
We first show that the Hodge Laplacian can be used in lieu of the graph Laplacian to construct a family of wavelets for higher-order signals on simplicial complexes.
How does Heterophily Impact the Robustness of Graph Neural Networks? Theoretical Connections and Practical Implications
We bridge two research directions on graph neural networks (GNNs), by formalizing the relation between heterophily of node labels (i. e., connected nodes tend to have dissimilar labels) and the robustness of GNNs to adversarial attacks.
Signal processing on simplicial complexes
Higher-order networks have so far been considered primarily in the context of studying the structure of complex systems, i. e., the higher-order or multi-way relations connecting the constituent entities.
Local, global and scale-dependent node roles
This paper re-examines the concept of node equivalences like structural equivalence or automorphic equivalence, which have originally emerged in social network analysis to characterize the role an actor plays within a social system, but have since then been of independent interest for graph-based learning tasks.
Finite Impulse Response Filters for Simplicial Complexes
In this paper, we study linear filters to process signals defined on simplicial complexes, i. e., signals defined on nodes, edges, triangles, etc.
Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond
In the context of simplicial complexes, we specifically focus on signal processing using the Hodge Laplacian matrix, a multi-relational operator that leverages the special structure of simplicial complexes and generalizes desirable properties of the Laplacian matrix in graph signal processing.
Modularity maximisation for graphons
The detection of communities or other meso-scale structures is a prominent topic in network science as it allows the identification of functional building blocks in complex systems.
Detectability of hierarchical communities in networks
We study the problem of recovering a planted hierarchy of partitions in a network.
Hierarchical community structure in networks
A great deal of effort has gone into trying to detect and study these structures.
Blind identification of stochastic block models from dynamical observations
We consider a blind identification problem in which we aim to recover a statistical model of a network without knowledge of the network's edges, but based solely on nodal observations of a certain process.
Graph-based Semi-Supervised & Active Learning for Edge Flows
The first strategy selects edges to minimize the reconstruction error bound and works well on flows that are approximately divergence-free.
Spectral partitioning of time-varying networks with unobserved edges
We discuss a variant of `blind' community detection, in which we aim to partition an unobserved network from the observation of a (dynamical) graph signal defined on the network.
Random Walks on Simplicial Complexes and the normalized Hodge Laplacian
Simplicial complexes, a mathematical object common in topological data analysis, have emerged as a model for multi-nodal interactions that occur in several complex systems; for example, biological interactions occur between a set of molecules rather than just two, and communication systems can have group messages and not just person-to-person messages.
Social and Information Networks Discrete Mathematics Algebraic Topology Physics and Society
Multiscale dynamical embeddings of complex networks
Complex systems and relational data are often abstracted as dynamical processes on networks.
Social and Information Networks Systems and Control Physics and Society
Simplicial Closure and higher-order link prediction
Networks provide a powerful formalism for modeling complex systems by using a model of pairwise interactions.
Centrality measures for graphons: Accounting for uncertainty in networks
Using the theory of linear integral operators, we define degree, eigenvector, Katz and PageRank centrality functions for graphons and establish concentration inequalities demonstrating that graphon centrality functions arise naturally as limits of their counterparts defined on sequences of graphs of increasing size.
Random Multi-Hopper Model. Super-Fast Random Walks on Graphs
Stated differently, for small parameter values the multi-hopper explores a general graph as fast as possible when compared to a random walker on a full graph.
Physics and Society Statistical Mechanics Social and Information Networks Mathematical Physics Mathematical Physics Probability
The many facets of community detection in complex networks
Community detection, the decomposition of a graph into essential building blocks, has been a core research topic in network science over the past years.
Social and Information Networks Data Analysis, Statistics and Probability Physics and Society
