Related papers: Negatively Curved Graphs

In the present work, we study the properties of biological networks by applying analogous notions of fundamental concepts in Riemannian geometry and optimal mass transport to discrete networks described by weighted graphs. Specifically, we…

Graphs can model real-world, complex systems by representing entities and their interactions in terms of nodes and edges. To better exploit the graph structure, graph neural networks have been developed, which learn entity and edge…

Let $N$ be a closed enlargeable manifold in the sense of Gromov-Lawson and $M$ a closed spin manifold of equal dimension, a famous theorem of Gromov-Lawson states that the connected sum $M\# N$ admits no metric of positive scalar curvature.…

A spanning tree of a network or graph is a subgraph that connects all nodes with the least number or weight of edges. The spanning tree is one of the most straightforward techniques for network simplification and sampling, and for…

Graphs can represent relational information among entities and graph structures are widely used in many intelligent tasks such as search, recommendation, and question answering. However, most of the graph-structured data in practice suffers…

Graph Neural Networks achieve remarkable results on problems with structured data but come as black-box predictors. Transferring existing explanation techniques, such as occlusion, fails as even removing a single node or edge can lead to…

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…

We examine data-processing of Markov chains through the lens of information geometry. We first establish a theory of congruent Markov morphisms within the framework of stochastic matrices. Specifically, we introduce and justify the concept…

Convolutional networks are ubiquitous in deep learning. They are particularly useful for images, as they reduce the number of parameters, reduce training time, and increase accuracy. However, as a model of the brain they are seriously…

Building upon [1], this study aims to introduce fractal geometry into graph theory, and to establish a potential theoretical foundation for complex networks. Specifically, we employ the method of substitution to create and explore…

A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalisation of the Leibnitz rules of commutative geometry and uses the bimodule structure of $\Omega^1$. A special role is played…

In this paper we study the inverse eigenvector centrality problem on directed graphs: given a prescribed node centrality profile, we seek edge weights that realize it. Since this inverse problem generally admits infinitely many solutions,…

Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed…

Network or graph structures are ubiquitous in the study of complex systems. Often, we are interested in complexity trends of these system as it evolves under some dynamic. An example might be looking at the complexity of a food web as…

The highly non-linear nature of deep neural networks causes them to be susceptible to adversarial examples and have unstable gradients which hinders interpretability. However, existing methods to solve these issues, such as adversarial…

A graphical model is a statistical model that is associated to a graph whose nodes correspond to variables of interest. The edges of the graph reflect allowed conditional dependencies among the variables. Graphical models admit…

We address the problem of efficiently gathering correlated data from a wired or a wireless sensor network, with the aim of designing algorithms with provable optimality guarantees, and understanding how close we can get to the known…

We define notions of local topological convergence and local geometric convergence for embedded graphs in $\mathbb{R}^n,$ and study their properties. The former is related to Benjamini-Schramm convergence, and the latter to weak convergence…

Data describing the three-dimensional structure of physical networks is increasingly available, leading to a surge of interest in network science to explore the relationship between the shape and connectivity of physical networks. We…

In an increasingly interconnected world, understanding and summarizing the structure of these networks becomes increasingly relevant. However, this task is nontrivial; proposed summary statistics are as diverse as the networks they…