Related papers: Vertex Degree of Random Geometric Graph on Exponen…

A random geometric graph $G_n$ is given by picking $n$ vertices in $\mathbb{R}^d$ independently under a common bounded probability distribution, with two vertices adjacent if and only if their $l^p$-distance is at most $r_n$. We investigate…

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The total…

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…

In this paper we study the one dimensional random geometric graph when the location of the nodes are independent and exponentially distributed. We derive exact results and the limit theorems for the connectivity and other properties…

Let $G=(V,E)$ be a finite, connected graph. We consider a greedy selection of vertices: given a list of vertices $x_1, \dots, x_k$, take $x_{k+1}$ to be any vertex maximizing the sum of distances to the existing vertices and iterate: we…

Let $G$ be a graph with a vertex set $V$. The graph $G$ is path-proximinal if there are a semimetric $d \colon V \times V \to [0, \infty[$ and disjoint proximinal subsets of the semimetric space $(V, d)$ such that $V = A \cup B$, and…

Consider a random multigraph G* with given vertex degrees d_1,...,d_n, contructed by the configuration model. We show that, asymptotically for a sequence of such multigraphs with the number of edges (d_1+...+d_n)/2 tending to infinity, the…

Random graph matching refers to recovering the underlying vertex correspondence between two random graphs with correlated edges; a prominent example is when the two random graphs are given by Erd\H{o}s-R\'{e}nyi graphs $G(n,\frac{d}{n})$.…

We propose the following model of a random graph on n vertices. Let F be a distribution in R_+^{n(n-1)/2} with a coordinate for every pair i$ with 1 \le i,j \le n. Then G_{F,p} is the distribution on graphs with n vertices obtained by…

Given a graphical degree sequence ${\bf d}=(d_1,\ldots, d_n)$, let $G(n, {\bf d})$ denote a uniformly random graph on vertex set $[n]$ where vertex $ i$ has degree $d_i$ for every $1\le i\le n$. We give upper and lower bounds on the joint…

A classical result of Koml\'os, S\'ark\"ozy and Szemer\'edi states that every $n$-vertex graph with minimum degree at least $(1/2+ o(1))n$ contains every $n$-vertex tree with maximum degree $O(n/\log{n})$ as a subgraph, and the bounds on…

In this paper, we give an analytic solution for graphs with n nodes and E edges for which the probability of obtaining a given graph G is specified in terms of the degree sequence of G. We describe how this model naturally appears in the…

We find an asymptotic enumeration formula for the number of simple $r$-uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations.…

We revisit the problem of designing sublinear algorithms for estimating the average degree of an $n$-vertex graph. The standard access model for graphs allows for the following queries: sampling a uniform random vertex, the degree of a…

Let $F$ be a probability distribution with support on the non-negative integers. Two algorithms are described for generating a stationary random graph, with vertex set $\mathbb{Z}$, so that the degrees of the vertices are i.i.d.\ random…

Consider~\(n\) nodes~\(\{X_i\}_{1 \leq i \leq n}\) independently distributed in the unit square~\(S,\) each according to a distribution~\(f.\) Nodes~\(X_i\) and~\(X_j\) are joined by an edge if the Euclidean distance~\(d(X_i,X_j)\) is less…

We consider a random graph G(n,p) whose vertex set V has been randomly embedded in the unit square and whose edges are given weight equal to the geometric distance between their end vertices. Then each pair {u,v} of vertices have a distance…

We propose a consistent approach to the statistics of the shortest paths in random graphs with a given degree distribution. This approach goes further than a usual tree ansatz and rigorously accounts for loops in a network. We calculate the…

We provide optimal rates of convergence to the asymptotic distribution of the (properly scaled) degree of a fixed vertex in two preferential attachment random graph models. Our approach is to show that these distributions are unique fixed…

Given an $n\times n$ symmetric matrix $W\in [0,1]^{[n]\times [n]}$, let $\mathcal{G}(n,W)$ be the random graph obtained by independently including each edge $jk$ with probability $W_{jk}$. Given a degree sequence ${\bf d}=(d_1,\ldots,…