Related papers: Stability of Persistent Path Diagrams
Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of…
We develop the theoretical foundations of a generalized Gromov-Hausdorff distance between functions on networks that has recently been applied to various subfields of topological data analysis and optimal transport. These functional…
The irreducible complexity of natural phenomena has led Graph Neural Networks to be employed as a standard model to perform representation learning tasks on graph-structured data. While their capacity to capture local and global patterns is…
To better understand the structure and function of complex systems, researchers often represent direct interactions between components in complex systems with networks, assuming that indirect influence between distant components can be…
A weighted graph is a graph in which every edge is assigned a non-negative real number. In a weighted graph, the weight of a path is the sum of the weights of its edges, and the weighed degree of a vertex is the sum of the weights of the…
Topological correctness plays a critical role in many image segmentation tasks, yet most networks are trained using pixel-wise loss functions, such as Dice, neglecting topological accuracy. Existing topology-aware methods often lack robust…
Directed graphs arise in many applications where computing persistent homology helps to encode the shape and structure of the input information. However, there are only a few ways to turn the directed graph information into an undirected…
The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space.…
Node2vec is a graph embedding method that learns a vector representation for each node of a weighted graph while seeking to preserve relative proximity and global structure. Numerical experiments suggest Node2vec struggles to recreate the…
A multicomplex structure is defined from an ordered lattice of multigraphs. This structure will help us to observe the features of Persistent Homology in this context, its interaction with the ordering and the repercussions of the process…
A hypergraph is a set V of vertices and a set of non-empty subsets of V, called hyperedges. Unlike graphs, hypergraphs can capture higher-order interactions in social and communication networks that go beyond a simple union of pairwise…
One of the defining features of complex networks is the connectivity properties that we observe emerging from local interactions. Recently, hypergraphs have emerged as a versatile tool to model networks with non-dyadic, higher-order…
We review the main tools which allow for the statistical characterization of weighted networks. We then present two case studies, the airline connection network and the scientific collaboration network, which are representative of critical…
Topological data analysis is a powerful tool for describing topological signatures in real world data. An important challenge in topological data analysis is matching significant topological signals across distinct systems. In geometry and…
A graphical model provides a compact and efficient representation of the association structure of a multivariate distribution by means of a graph. Relevant features of the distribution are represented by vertices, edges and other…
Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs) which are 2D multisets of points. Their variable size makes them, however, difficult to combine…
Complex networks are universal, arising in fields as disparate as sociology, physics, and biology. In the past decade, extensive research into the properties and behaviors of complex systems has uncovered surprising commonalities among the…
In this paper we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the creation and destruction…
The representation of complex systems as networks is inappropriate for the study of certain problems. We show several examples of social, biological, ecological and technological systems where the use of complex networks gives very limited…
Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set…