Related papers: Hierarchy Structure of Graphs and Weighted Condens…
The hierarchy associated to clusters in the HDBSCAN algorithm has layers, which are defined by cardinality. The layers define a layer subposet of the HDBSCAN hierarchy, which is a strong deformation retract and admits a stability analysis.…
Graph embeddings have emerged as a powerful tool for representing complex network structures in a low-dimensional space, enabling the use of efficient methods that employ the metric structure in the embedding space as a proxy for the…
We present an exact formula for the ordinary generating series of the simple paths between any two vertices of a graph. Our formula involves the adjacency matrix of the connected induced subgraphs and remains valid on weighted and directed…
We call a topological ordering of a weighted directed acyclic graph non-negative if the sum of weights on the vertices in any prefix of the ordering is non-negative. We investigate two processes for constructing non-negative topological…
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…
In order to facilitate the accesses of general users to knowledge graphs, an increasing effort is being exerted to construct graph-structured queries of given natural language questions. At the core of the construction is to deduce the…
We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and…
A rooted arborescence of a directed graph is a spanning tree directed towards a particular vertex. A recent work of Chepuri et al. showed that the arborescences of a covering graph of a directed graph G are closely related to the…
Graph symmetries intervene in diverse applications, from enumeration, to graph structure compression, to the discovery of graph dynamics (e.g., node arrival order inference). Whereas Erd\H{o}s-R\'enyi graphs are typically asymmetric, real…
We introduce the weighted path homology on the category of weigh\-ted directed hypergraphs and describe conditions of homotopy invariance of weighted path homology groups. We give several examples that explain the nontriviality of the…
An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. We define the adjacency, incidence and Laplacian matrices of an oriented hypergraph and study each of them. We extend several matrix…
Hypergraph is a topological model for networks. In order to study the topology of hypergraphs, the homology of the associated simplicial complexes and the embedded homology have been invented. In this paper, we give some algorithms to…
Arithmetical structures on a graph were introduced by Lorenzini as some intersection matrices that arise in the study of degenerating curves in algebraic geometry. In this article we study these arithmetical structures, in particular we are…
A highly successful approach to route planning in networks (particularly road networks) is to identify a hierarchy in the network that allows faster queries after some preprocessing that basically inserts additional "shortcut"-edges into a…
Many real-world complex systems are characterized by non-pairwise -- higher-order -- interactions among system's units, and can be effectively modeled as hypergraphs. Directed hypergraphs distinguish between source and target sets within…
We consider evolution algebras and their related substructures: evolution ideals and evolution subalgebras. After exposing some of the concepts related to them in the literature, we explore the order structures that arise in the sets of…
In various subjects including mathematics, one can hope to use mathematical thinking well when the right kinds of algebraic structure to consider can be discovered or spotted. Therefore, it would help to understand kinds of algebraic…
Recent studies have revealed that neural networks learn interpretable algorithms for many simple problems. However, little is known about how these algorithms emerge during training. In this article, I study the training dynamics of a small…
A directed hypergraph (dihypergraph) consists of a set of vertices and a set of hyperarcs, where each hyperarc is partitioned into a head and a tail. Directed hypergraphs are useful in many applications, including the study of chemical…
Recently, the deep learning community has given growing attention to neural architectures engineered to learn problems in relational domains. Convolutional Neural Networks employ parameter sharing over the image domain, tying the weights of…