相关论文: Largest nearest-neighbour link and connectivity th…

Let $M_n$ be the minimal position at generation $n$, of a real-valued branching random walk in the boundary case. As $n \to \infty$, $M_n- {3 \over 2} \log n$ is tight (see [1][9][2]). We establish here a law of iterated logarithm for the…

We consider the (unoriented) long-range percolation on Z^d in dimensions d\ge1, where distinct sites x,y\in Z^d get connected with probability p_{xy}\in[0,1]. Assuming p_{xy}=|x-y|^{-s+o(1)} as |x-y|\to\infty, where s>0 and |\cdot| is a…

Nearest-neighbor methods are fundamental to classical and modern machine learning, yet their geometric properties are typically analyzed under independent sampling. In this paper, we study the nearest-neighbor radii under dependent…

Given an undirected $n$-vertex graph $G(V,E)$ and an integer $k$, let $T_k(G)$ denote the random vertex induced subgraph of $G$ generated by ordering $V$ according to a random permutation $\pi$ and including in $T_k(G)$ those vertices with…

In this article, we prove Shannon-MacMillan-Breiman Theorem for Wireless Sensor Networks modelled as coloured geometric random graphs. For large $n,$ we show that a Wireless Sensor Network consisting of $n$ sensors in $[0,1]^d$ connected by…

Linear structural equation models postulate noisy linear relationships between variables of interest. Each model corresponds to a path diagram, which is a mixed graph with directed edges that encode the domains of the linear functions and…

In this article, we analyze the limiting eigenvalue distribution (LED) of random geometric graphs (RGGs). The RGG is constructed by uniformly distributing $n$ nodes on the $d$-dimensional torus $\mathbb{T}^d \equiv [0, 1]^d$ and connecting…

We investigate a relationship network of humans located in a metric space where relationships are drawn according to a distance-dependent probability density. The obtained spatial graph allows us to calculate the average separation of…

We show that an intricate relation of cluster properties and optimal bipartitions, which takes place in undirected random graphs, extends to directed and mixed random graphs. In particular, the satisfability threshold coincides with the…

Large real-world networks are typically scale-free. Recent research has shown that such graphs are described best in a geometric space. More precisely, the internet can be mapped to a hyperbolic space such that geometric greedy routing…

$\renewcommand{\Re}{\mathbb{R}}$Given a set $P$ of $n$ points in $\Re^d$, and a parameter $\varepsilon \in (0,1)$, we present a new construction of a directed graph $G$, of size $O(n/\varepsilon^d)$, such that $(1+\varepsilon)$-ANN queries…

We establish maximal trees and graphs for the difference of average distance and proximity proving thus the corresponding conjecture posed in [4]. We also establish maximal trees for the difference of average eccentricity and remoteness and…

In the problem of minimum connected dominating set with routing cost constraint, we are given a graph $G=(V,E)$, and the goal is to find the smallest connected dominating set $D$ of $G$ such that, for any two non-adjacent vertices $u$ and…

Network connectivity is usually addressed for convex domains where a direct line of sight exists between any two transmitting/receiving nodes. Here, we develop a general theory for the network connectivity properties across a small opening,…

We study the connectivity properties of random Bluetooth graphs that model certain "ad hoc" wireless networks. The graphs are obtained as "irrigation subgraphs" of the well-known random geometric graph model. There are two parameters that…

We consider the probability that a dense wireless network confined within a given convex geometry is fully connected. We exploit a recently reported theory to develop a systematic methodology for analytically characterizing the connectivity…

A wide variety of complex networks (social, biological, information etc.) exhibit local clustering with substantial variation in the clustering coefficient (the probability of neighbors being connected). Existing models of large graphs…

We find precise asymptotic estimates for the number of planar maps and graphs with a condition on the minimum degree, and properties of random graphs from these classes. In particular we show that the size of the largest tree attached to…

Let $G$ be a connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity and remoteness of $G$ are defined as the minimum and maximum,…

In this paper, we propose a network whose nodes are labeled by the composite numbers and two nodes are connected by an undirected link if they are relatively prime to each other. As the size of the network increases, the network will be…