Related papers: Comparing Graphs via Persistence Distortion
To perform visual data exploration, many dimensionality reduction methods have been developed. These tools allow data analysts to represent multidimensional data in a 2D or 3D space, while preserving as much relevant information as…
The comparison of graphs is a vitally important, yet difficult task which arises across a number of diverse research areas including biological and social networks. There have been a number of approaches to define graph distance however…
Topological Data Analysis (TDA) is an approach to handle with big data by studying its shape. A main tool of TDA is the persistence diagram, and one can use it to compare data sets. One approach to learn on the similarity between two…
Quantifying the similarity between two graphs is a fundamental algorithmic problem at the heart of many data analysis tasks for graph-based data. In this paper, we study the computational complexity of a family of similarity measures based…
Despite the obvious similarities between the metrics used in topological data analysis and those of optimal transport, an optimal-transport based formalism to study persistence diagrams and similar topological descriptors has yet to come.…
Determining whether two graphs are structurally identical is a fundamental problem with applications spanning mathematics, computer science, chemistry, and network science. Despite decades of study, graph isomorphism remains a challenging…
Steinerberger proposed a notion of curvature on graphs involving the graph distance matrix (J. Graph Theory, 2023). We show that nonnegative curvature is almost preserved under three graph operations. We characterize the distance matrix and…
The computation of distance measures between nodes in graphs is inefficient and does not scale to large graphs. We explore dense vector representations as an effective way to approximate the same information: we introduce a simple yet…
In this paper we propose and study a new structural invariant for graphs, called distance-unbalanced\-ness, as a measure of how much a graph is (un)balanced in terms of distances. Explicit formulas are presented for several classes of…
We introduce the tree distance, a new distance measure on graphs. The tree distance can be computed in polynomial time with standard methods from convex optimization. It is based on the notion of fractional isomorphism, a characterization…
Visualization in the emerging field of topological data analysis has progressed from persistence barcodes and persistence diagrams to display of two-parameter persistent homology. Although persistence barcodes and diagrams have permitted…
Persistent homology allows us to create topological summaries of complex data. In order to analyse these statistically, we need to choose a topological summary and a relevant metric space in which this topological summary exists. While…
We study two popular ways to sketch the shortest path distances of an input graph. The first is distance preservers, which are sparse subgraphs that agree with the distances of the original graph on a given set of demand pairs. Prior work…
Tolerance graphs model interval relations in such a way that intervals can tolerate a certain amount of overlap without being in conflict. In one of the most natural generalizations of tolerance graphs with direct applications in the…
Graph distance metric learning serves as the foundation for many graph learning problems, e.g., graph clustering, graph classification and graph matching. Existing research works on graph distance metric (or graph kernels) learning fail to…
We develop a theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes. Our approach generalizes the usual (co)variance to the setting of…
Discrete curvatures are quantities associated to the nodes and edges of a graph that reflect the local geometry around them. These curvatures have a rich mathematical theory and they have recently found success as a tool to analyze networks…
How does one generalize differential geometric constructs such as curvature of a manifold to the discrete world of graphs and other combinatorial structures? This problem carries significant importance for analyzing models of discrete…
Mapper graphs are widely used tools in topological data analysis and visualization. They can be understood as discrete approximations of Reeb graphs, providing insight into the shape and connectivity of complex data. Given a…
The \emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the…