The cover time of random geometric graphs

We study the cover time of random geometric graphs. Let $I(d)=[0,1]^{d}$ denote the unit torus in d dimensions. Let $D(x,r)$ denote the ball (disc) of radius r. Let $\Upsilon_d$ be the volume of the unit ball $D(0,1)$ in d dimensions. A random geometric graph $G=G(d,r,n)$ in d dimensions is defined as follows: Sample n points V independently and uniformly at random from $I(d)$ . For each point x draw a ball $D(x,r)$ of radius r about x. The vertex set $V(G)=V$ and the edge set $E(G)=\{\{v,w\}: w\ne v,\,w\in D(v,r)\}$ . Let $G(d,r,n),\,d\geq 3$ be a random geometric graph. Let $C_G$ denote the cover time of a simple random walk on G. Let $c>1$ be constant, and let $r=(c\log n/(\Upsilon_dn))^{1/d}$ . Then whp the cover time satisfies © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 324–349, 2011     

几何随机图大连通分支覆盖面积及其在传感器网络中的应用

随机网络中的大连通分支能体现一个网络的连通情况,是几何随机图研究的-个热点,具有重要的理论意义和应用价值.本文利用渗流理论,研究了几何随机图大连通分支覆盖面积所具有的性质,并将理论结果应用到大型无线传感器网络中,研究了无线传感器网络覆盖的性质.研究结果表明,对于节点服从泊松分布的大型无线传感器网络,其大连通分支覆盖区域大小与总区域大小的比值趋于-个常数,且并估计出了2维空间中没有被大连通分支所覆盖的连通区域(本文称为空洞)的大小.这些结果为衡量无线传感器网络性能提供了理论基础,对实际布网和网络优化等具有一定的指导意义.     

Topological properties of the one dimensional exponential random geometric graph

In this article we study the one‐dimensional random geometric (random interval) graph when the location of the nodes are independent and exponentially distributed. We derive exact results and limit theorems for the connectivity and other properties associated with this random graph. We show that the asymptotic properties of a graph with a truncated exponential distribution can be obtained using the exponential random geometric graph. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Limit laws in the generalized random graphs with random vertex weights

In this note we investigate the number of edges and the vertex degree in the generalized random graphs with vertex weights, which are independent and identically distributed random variables.     

Approximation algorithms for minimum broadcast schedule problem in wireless sensor networks

A wireless sensor network usually consists of a large number of sensor nodes deployed in a field. One of the major communication operations is to broadcast a message from one node to the rest of the others. In this paper, we adopt the conflict-free communication model and study how to compute a transmission schedule that determines when and where a node should forward the message so that all nodes could receive the message in minimum time. We give two approximation algorithms for this NP-hard problem that have better theoretically guaranteed performances than the existing algorithms. The proposed approach could be applied to some other similar problems.     

On properties of geometric random problems in the plane

In this paper, we present results dealing with properties of well-known geometric random problems in the plane, together with their motivations. The paper specifically concentrates on the traveling salesman and minimum spanning tree problems, even though most of the results apply to other problems such as the Steiner tree problem and the minimum weight matching problem.     

Small subgraphs in random graphs and the power of multiple choices

The standard paradigm for online power of two choices problems in random graphs is the Achlioptas process. Here we consider the following natural generalization: Starting with G₀ as the empty graph on n vertices, in every step a set of r edges is drawn uniformly at random from all edges that have not been drawn in previous steps. From these, one edge has to be selected, and the remaining r−1 edges are discarded. Thus after N steps, we have seen rN edges, and selected exactly N out of these to create a graph GN.In a recent paper by Krivelevich, Loh, and Sudakov (2009) [11], the problem of avoiding a copy of some fixed graph F in GN for as long as possible is considered, and a threshold result is derived for some special cases. Moreover, the authors conjecture a general threshold formula for arbitrary graphs F. In this work we disprove this conjecture and give the complete solution of the problem by deriving explicit threshold functions N₀(F,r,n) for arbitrary graphs F and any fixed integer r. That is, we propose an edge selection strategy that a.a.s. (asymptotically almost surely, i.e. with probability 1−o(1) as n→∞) avoids creating a copy of F for as long as N=o(N₀), and prove that any online strategy will a.a.s. create such a copy once N=ω(N₀).     

Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference

From the results of convergence by sampling in linear principal component analysis (of a random function in a separable Hilbert space), the limiting distribution is given for the principal values and the principal factors. These results can be explicitly written in the normal case. Some applications to statistical inference are investigated.     

Geometric structure of the joint N‐voM distribution manifold and its applications to sensor networks

In this paper, the geometric structure for normal distribution manifold, von Mises distribution manifold and their joint distribution manifold are firstly given by the metric, curvature, and divergence, respectively. Furthermore, the active detection with sensor networks is presented by a classical measurement model based on metric manifold, and the information resolution is presented for the range and angle measurements sensor networks. The preliminary analysis results introduced in this paper indicate that our approach is able to offer consistent and more comprehensive means to understand and solve sensor network problems containing sensors management and target detection, which are not easy to be handled by conventional analysis methods.     

On a random graph with immigrating vertices: Emergence of the giant component

A randomly evolving graph, with vertices immigrating at rate n and each possible edge appearing at rate 1/n, is studied. The detailed picture of emergence of giant components with O(n^2/3) vertices is shown to be the same as in the Erdős–Rényi graph process with the number of vertices fixed at n at the start. A major difference is that now the transition occurs about a time t=π/2, rather than t=1. The proof has three ingredients. The size of the largest component in the subcritical phase is bounded by comparison with a certain multitype branching process. With this bound at hand, the growth of the sum‐of‐squares and sum‐of‐cubes of component sizes is shown, via martingale methods, to follow closely a solution of the Smoluchowsky‐type equations. The approximation allows us to apply results of Aldous [Brownian excursions, critical random graphs and the multiplicative coalescent, Ann Probab 25 (1997), 812–854] on emergence of giant components in the multiplicative coalescent, i.e., a nonuniform random graph process. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 79–102, 2000     

Geometry of the vacant set left by random walk on random graphs,Wright's constants,and critical random graphs with prescribed degrees

We provide an explicit algorithm for sampling a uniform simple connected random graph with a given degree sequence. By products of this central result include: (1) continuum scaling limits of uniform simple connected graphs with given degree sequence and asymptotics for the number of simple connected graphs with given degree sequence under some regularity conditions, and (2) scaling limits for the metric space structure of the maximal components in the critical regime of both the configuration model and the uniform simple random graph model with prescribed degree sequence under finite third moment assumption on the degree sequence. As a substantive application we answer a question raised by ?erný and Teixeira study by obtaining the metric space scaling limit of maximal components in the vacant set left by random walks on random regular graphs.     

Diffusion approximation for the components in critical inhomogeneous random graphs of rank 1.

Consider the random graph on n vertices 1,…,n. Each vertex i is assigned a type x_i with x₁,…,x_n being independent identically distributed as a nonnegative random variable X. We assume that EX³< ∞. Given types of all vertices, an edge exists between vertices i and j independent of anything else and with probability \begin{align*}\min \{1, \frac{x_ix_j}{n}\left(1+\frac{a}{n^{1/3}} \right) \}\end{align*}. We study the critical phase, which is known to take place when EX² = 1. We prove that normalized by n^‐2/3the asymptotic joint distributions of component sizes of the graph equals the joint distribution of the excursions of a reflecting Brownian motion with diffusion coefficient \begin{align*}\sqrt{{\textbf{ E}}X{\textbf{ E}}X^3}\end{align*}and drift \begin{align*}a-\frac{{\textbf{ E}}X^3}{{\textbf{ E}}X}s\end{align*}. In particular, we conclude that the size of the largest connected component is of order n^2/3. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 43, 486–539, 2013     

Variable neighborhood search for extremal graphs. 16. Some conjectures related to the largest eigenvalue of a graph

We consider four conjectures related to the largest eigenvalue of (the adjacency matrix of) a graph (i.e., to the index of the graph). Three of them have been formulated after some experiments with the programming system AutoGraphiX, designed for finding extremal graphs with respect to given properties by the use of variable neighborhood search. The conjectures are related to the maximal value of the irregularity and spectral spread in n-vertex graphs, to a Nordhaus–Gaddum type upper bound for the index, and to the maximal value of the index for graphs with given numbers of vertices and edges. None of the conjectures has been resolved so far. We present partial results and provide some indications that the conjectures are very hard.     

Some properties for a class of interchange graphs

The Wiener number is the sum of distances between all pairs of vertices of a connected graph. In this paper, we give an explicit algebraic formula for the Wiener number of a class of interchange graphs. Moreover, distance-related properties and cliques of this class of interchange graphs are investigated.     

Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over

We consider ergodic random Schrödinger operators on the metric graph with random potentials and random boundary conditions taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable. This limit, the integrated density of states, can be expressed by a closed Shubin–Pastur type trace formula. It supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. Among other examples we discuss random magnetic fields and percolation models.     

The use of random weights for the training of multilayer networks of neurons with Heaviside characteristics

Artificial neural networks have, in recent years, been very successfully applied in a wide range of areas. A major reason for this success has been the existence of a training algorithm called backpropagation. This algorithm relies upon the neural units in a network having input/output characteristics that are continuously differentiable. Such units are significantly less easy to implement in silicon than are neural units with Heaviside (step-function) characteristics. In this paper, we show how a training algorithm similar to backpropagation can be developed for 2-layer networks of Heaviside units by treating the network weights (i.e., interconnection strengths) as random variables. This is then used as a basis for the development of a training algorithm for networks with any number of layers by drawing upon the idea of internal representations. Some examples are given to illustrate the performance of these learning algorithms.     

Braess's paradox for the spectral gap in random graphs and delocalization of eigenvectors

We study how the spectral gap of the normalized Laplacian of a random graph changes when an edge is added to or removed from the graph. There are known examples of graphs where, perhaps counter‐intuitively, adding an edge can decrease the spectral gap, a phenomenon that is analogous to Braess's paradox in traffic networks. We show that this is often the case in random graphs in a strong sense. More precisely, we show that for typical instances of Erd?s‐Rényi random graphs G (n, p ) with constant edge density , the addition of a random edge will decrease the spectral gap with positive probability, strictly bounded away from zero. To do this, we prove a new delocalization result for eigenvectors of the Laplacian of G (n, p ), which might be of independent interest. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 584–611, 2017     

On the combination of kernel principal component analysis and neural networks for process indirect control

ABSTRACT

A new adaptive kernel principal component analysis (KPCA) for non-linear discrete system control is proposed. The proposed approach can be treated as a new proposition for data pre-processing techniques. Indeed, the input vector of neural network controller is pre-processed by the KPCA method. Then, the obtained reduced neural network controller is applied in the indirect adaptive control. The influence of the input data pre-processing on the accuracy of neural network controller results is discussed by using numerical examples of the cases of time-varying parameters of single-input single-output non-linear discrete system and multi-input multi-output system. It is concluded that, using the KPCA method, a significant reduction in the control error and the identification error is obtained. The lowest mean squared error and mean absolute error are shown that the KPCA neural network with the sigmoid kernel function is the best.