Related papers: Geometric random graphs on circles
An isometric embedding of a graph into a metric space is an embedding of the vertices such that the smallest number of edges connecting any two vertices equals to the distance in the metric space between the images. In this paper, we study…
We study graphs that are formed by independently-positioned needles (i.e., line segments) in the unit square. To mathematically characterize the graph structure, we derive the probability that two line segments intersect and determine…
We extend the concept of the law of a finite graph to graphings, which are, in general, infinite graphs whose vertices are equipped with the structure of a probability space. By doing this, we obtain a vast array of new unimodular measures.…
We consider the flip-width of geometric graphs, a notion of graph width recently introduced by Toru\'nczyk. We prove that many different types of geometric graphs have unbounded flip-width. These include interval graphs, permutation graphs,…
We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random…
We investigate how the metric dimension of infinite graphs change when we add edges to the graph. Our two main results: (1) there exists a growing sequence of graphs (under the subgraph relation, but without adding vertices) for which the…
For an ordered set $W=\{w_1,w_2,...,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),.,d(v,w_k))$ is called the (metric) representation of $v$ with respect to $W$, where…
The entropy of a digraph is a fundamental measure which relates network coding, information theory, and fixed points of finite dynamical systems. In this paper, we focus on the entropy of undirected graphs. We prove that for any integer $k$…
In this paper, we study rare events in spherical and Gaussian random geometric graphs in high dimensions. In these models, the vertices correspond to points sampled uniformly at random on the $d$ dimensional unit sphere or correspond to $d$…
Twisted hypercubes are generalizations of the Boolean hypercube, obtained by iteratively connecting two instances of a graph by a uniformly random perfect matching. Dudek et al. showed that when the two instances are independent, these…
In this paper we study the threshold model of \emph{geometric inhomogeneous random graphs} (GIRGs); a generative random graph model that is closely related to \emph{hyperbolic random graphs} (HRGs). These models have been observed to…
We consider the geometric random (GR) graph on the $d-$dimensional torus with the $L_\sigma$ distance measure ($1 \leq \sigma \leq \infty$). Our main result is an exact characterization of the probability that a particular labeled cycle…
A geometric graph is a graph embedded in the plane with vertices at points and edges drawn as curves (which are usually straight line segments) between those points. The average transversal complexity of a geometric graph is the number of…
We introduce the notion of a "random basic walk" on an infinite graph, give numerous examples, list potential applications, and provide detailed comparisons between the random basic walk and existing generalizations of simple random walks.…
We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting…
One of the main questions that arise when studying random and quasi-random structures is which properties P are such that any object that satisfies P "behaves" like a truly random one. In the context of graphs, Chung, Graham, and Wilson…
The ring of graph invariants is spanned by the basic graph invariants which calculate the number of subgraphs isomorphic to a given graph in other graphs. These subgraphs counting invariants are not algebraically independent. In our view…
A metric space (X,d) is monotone if there is a linear order < on X and a constant c>0 such that d(x,y) < c d(x,z) for all x<y<z in X. Properties of continuous functions with monotone graph (considered as a planar set) are investigated. It…
Consider a random geometric graph $G$ with a vertex set defined by a Poisson point process with intensity $t>0$ in a convex body. We can generate a drawing of the graph by projecting the construction onto some plane $L$. Choosing different…
For an ordered set $W=\{w_1,w_2,...,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),...,d(v,w_k))$ is called the (metric) representation of $v$ with respect to $W$, where…