Related papers: A note on discrete Holonomy through directed edges…
We establish tight lower and upper bounds on the number of edges in traceable graphs in several classes of dense graphs. A graph is traceable if it has a Hamiltonian path. We show that the bound is: - quadratic for the class of graphs of…
In phylogenetic networks, it is desirable to estimate edge lengths in substitutions per site or calendar time. Yet, there is a lack of scalable methods that provide such estimates. Here we consider the problem of obtaining edge length…
These lecture notes attempt to invite the reader towards the theory of singular foliations, both smooth and holomorphic. In addition to a systematic review of the foundations, and an attempt to put in order examples and several elementary…
Liu et al recently proposed a minimum number of driver nodes needed to obtain full structural controllability over a directed network. Driver nodes are unmatched nodes, from which there are directed paths to all matched nodes. Their most…
Many of the properties of sectional category, topological complexity and homotopic distance are in fact derived from a small number of basic properties, which, once established, lead to all the others without further recourse to topology.…
Constructing a discretization of a given set is a major problem in various theoretical and applied disciplines. An offset discretization of a set $X$ is obtained by taking the integer points inside a closed neighborhood of $X$ of a certain…
We introduce a new type of distinct distances result: a lower bound on the number of distances between points on a line and points on a two-dimensional strip. This can be seen as a generalization of the well-studied problems of distances…
We consider an extension of the triangular-distance Delaunay graphs (TD-Delaunay) on a set $P$ of points in the plane. In TD-Delaunay, the convex distance is defined by a fixed-oriented equilateral triangle $\triangledown$, and there is an…
We define strict confluent drawing, a form of confluent drawing in which the existence of an edge is indicated by the presence of a smooth path through a system of arcs and junctions (without crossings), and in which such a path, if it…
A dimension group is an ordered abelian group that is an inductive limit of a sequence of simplicial groups, and a stationary dimension group is such an inductive limit in which the homomorphism is the same at every stage. If a simple…
We introduce new broadcast and receive communicability indices that can be used as global measures of how effectively information is spread in a directed network. Furthermore, we describe fast and effective criteria for the selection of…
Homophily is a graph property describing the tendency of edges to connect similar nodes. There are several measures used for assessing homophily but all are known to have certain drawbacks: in particular, they cannot be reliably used for…
Role is a fundamental concept in the analysis of the behavior and function of interacting entities represented by network data. Role discovery is the task of uncovering hidden roles. Node roles are commonly defined in terms of equivalence…
We study the directed maximum common edge subgraph problem (DMCES) for the class of directed graphs that are finite, weakly connected, oriented, and simple. We use DMCES to define a metric on partially ordered sets that can be represented…
The holomorph of a discrete group $G$ is the universal semi-direct product of $G$. In chapter 1 we describe why it is an interesting object and state main results. In chapter 2 we recall the classical definition of the holomorph as well as…
Complex networks are made up of vertices and edges. The latter connect the vertices. There are several ways to measure the importance of the vertices, e.g., by counting the number of edges that start or end at each vertex, or by using the…
In this short note, we argue that directed homotopy can be given the structure of generalized modules, over particular monoids. This is part of a general attempt for refoundation of directed topology.
Discrete models have a long tradition in engineering, including finite state machines, Boolean networks, Petri nets, and agent-based models. Of particular importance is the question of how the model structure constrains its dynamics. This…
In weighted graphs the shortest path between two nodes is often reached through an indirect path, out of all possible connections, leading to structural redundancies which play key roles in the dynamics and evolution of complex networks. We…
Modularity is a widely used measure for evaluating community structure in networks. The definition of modularity involves a comparison of within-community edges in the observed network and that number in an equivalent randomized network.…