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Threshold Functions for Random Graphs on a Line Segment
 Combinatorics, Probability and Computing
, 2001
"... We look at a model of random graphs suggested by Gilbert: given an integer n and δ > 0, scatter n vertices independently and uniformly on a metric space, and then add edges connecting pairs of vertices of distance less than δ apart. We consider the asymptotics... ..."
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We look at a model of random graphs suggested by Gilbert: given an integer n and δ > 0, scatter n vertices independently and uniformly on a metric space, and then add edges connecting pairs of vertices of distance less than δ apart. We consider the asymptotics...
A Geometric Assignment Problem for Robotic Networks
"... Summary. In this chapter we look at a geometric target assignment problem consisting of an equal number of mobile robotic agents and distinct target locations. Each agent has a fixed communication range, a maximum speed, and knowledge of every target’s position. The problem is to devise a distribute ..."
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Summary. In this chapter we look at a geometric target assignment problem consisting of an equal number of mobile robotic agents and distinct target locations. Each agent has a fixed communication range, a maximum speed, and knowledge of every target’s position. The problem is to devise a distributed algorithm that allows the agents to divide the target locations among themselves and, simultaneously, leads each agent to its unique target. We summarize two algorithms for this problem; one designed for “sparse ” environments, in which communication between robots is sparse, and one for “dense ” environments, where communication is more prevalent. We characterize the asymptotic performance of these algorithms as the number of agents increases and the environment grows to accommodate them. 1
Cycles and Components in Geometric Graphs: Adjacency Operator Approach
, 2009
"... Nilpotent and idempotent adjacency operator methods are applied to the study of random geometric graphs in a discretized, ddimensional unit cube [0, 1] d. Cycles are enumerated, sizes of maximal connected components are computed, and closed formulas are obtained for graph circumference and girth. E ..."
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Nilpotent and idempotent adjacency operator methods are applied to the study of random geometric graphs in a discretized, ddimensional unit cube [0, 1] d. Cycles are enumerated, sizes of maximal connected components are computed, and closed formulas are obtained for graph circumference and girth. Expected numbers of kcycles, expected sizes of maximal components, and expected circumference and girth are also computed by considering powers of adjacency operators. 1
Efficient PeertoPeer Lookup in Multihop Wireless Networks
, 2009
"... In recent years the popularity of multihop wireless networks has been growing. Its flexible topology and abundant routing path enables many types of applications. However, the lack of a centralized controller often makes it difficult to design a reliable service in multihop wireless networks. Whil ..."
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In recent years the popularity of multihop wireless networks has been growing. Its flexible topology and abundant routing path enables many types of applications. However, the lack of a centralized controller often makes it difficult to design a reliable service in multihop wireless networks. While packet routing has been the center of attention for decades, recent research focuses on data discovery such as file sharing in multihop wireless networks. Although there are many peertopeer lookup (P2Plookup) schemes for wired networks, they have inherent limitations for multihop wireless networks. First, a wired P2Plookup builds a search structure on the overlay network and disregards the underlying topology. Second, the performance guarantee often relies on specific topology models such as random graphs, which do not apply to multihop wireless networks. Past studies on wireless P2Plookup either combined existing solutions with known routing algorithms or proposed treebased routing, which is prone to traffic congestion. In this paper, we present two wireless P2Plookup schemes that strictly build a topologydependent structure. We first propose the Ring Interval Graph Search (RIGS) that constructs a DHT only through direct connections between the
Performance Testing of RNSC and MCL Algorithms on Random Geometric Graphs
"... The exploration of quality clusters in complex networks is an important issue in many disciplines, which still remains a challenging task. Many graph clustering algorithms came into the field in the recent past but they were not giving satisfactory performance on the basis of robustness, optimality, ..."
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The exploration of quality clusters in complex networks is an important issue in many disciplines, which still remains a challenging task. Many graph clustering algorithms came into the field in the recent past but they were not giving satisfactory performance on the basis of robustness, optimality, etc. So, it is most difficult task to decide which one is giving more beneficial clustering results compared to others in case of real–world problems. In this paper, performance of RNSC (Restricted Neighbourhood Search Clustering) and MCL (Markov Clustering) algorithms are evaluated on a random geometric graph (RGG). RNSC uses stochastic local search method for clustering of a graph. RNSC algorithm tries to achieve optimal cost clustering by assigning some cost functions to the set of clusterings of a graph. Another standard clustering algorithm MCL is based on stochastic flow simulation model. RGG has conventionally been associated with areas such as statistical physics and hypothesis testing but have achieved new relevance with the advent of wireless adhoc and sensor networks. In this study, the performance testing of these methods is conducted on the basis of cost of clustering, cluster size, modularity index of clustering results and normalized mutual information (NMI) using both real and synthetic RGG.
Dynamic Geometric Graph Processes: Adjacency Operator Approach
, 2009
"... Abstract. The ddimensional unit cube [0, 1] d is discretized to create a collection V of vertices used to define geometric graphs. Each subset of V is uniquely associated with a geometric graph. Defining a dynamic random walk on the subsets of V induces a walk on the collection of geometric graphs ..."
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Abstract. The ddimensional unit cube [0, 1] d is discretized to create a collection V of vertices used to define geometric graphs. Each subset of V is uniquely associated with a geometric graph. Defining a dynamic random walk on the subsets of V induces a walk on the collection of geometric graphs in the discretized cube. These walks naturally model additiondeletion networks and can be visualized as walks on hypercubes with loops. Adjacency operators are constructed using subalgebras of Clifford algebras and are used to recover information about the cycle structure and connected components of the n graph of a sequence. 1.
Clique number of random geometric graphs
, 2013
"... The clique number C of a graph is the largest clique size in the graph. For a random geometric graph of n vertices, taken uniformly at random, including an edge beween two vertices if their distance, taken with the uniform norm, is less than a parameter r on a torus T d a, we find the asymptotic beh ..."
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The clique number C of a graph is the largest clique size in the graph. For a random geometric graph of n vertices, taken uniformly at random, including an edge beween two vertices if their distance, taken with the uniform norm, is less than a parameter r on a torus T d a, we find the asymptotic behaviour of the clique number. Setting θ = ( r a)d, in the subcritical regime where θ = o ( 1), we exhibit the intervals of θ where C n takes the same value asymptotically almost surely. In the critical regime, θ ∼ 1, we show that C is growing slightly slower than ln n asymptotically n almost surely. Finally, in the supercritical regime, 1 = o(θ), we prove n that C grows as nθ asymptotically almost surely. We also investigate the behaviour of related graph characteristics: the chromatic number, the maximum vertex degree, and the independence number. 1
Distributions of Eigenvalues of Large Euclidean Matrices Generated from Three Manifolds
"... Let x1, · · · , xn be points randomly chosen from a set G ⊂ R p and f(x) be a function. A special Euclidean random matrix is given by Mn = (f(∥xi − xj ∥ 2))n×n. When p is fixed and n → ∞ we prove that ˆµ(Mn), the empirical distribution of the eigenvalues of Mn, converges to δ0 for a big class of ..."
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Let x1, · · · , xn be points randomly chosen from a set G ⊂ R p and f(x) be a function. A special Euclidean random matrix is given by Mn = (f(∥xi − xj ∥ 2))n×n. When p is fixed and n → ∞ we prove that ˆµ(Mn), the empirical distribution of the eigenvalues of Mn, converges to δ0 for a big class of functions of f(x). Assuming both p and n go to infinity with n/p → y ∈ (0, ∞), we obtain the explicit limit of ˆµ(Mn) when G is the unit sphere S p−1 or the unit ball Bp(0, 1) and the explicit limit of ˆµ((Mn − apIn)/bp) for G = [0, 1] p, where ap and bp are constants. As corollaries, we obtain the limit of ˆµ(An) with An = (d(xi, xj))n×n and d being the geodesic distance on S p−1. We also obtain the limit of ˆµ(An) for the Euclidean distance matrix An = (∥xi − xj∥)n×n as G is S p−1 or Bp(0, 1). The limits are the law of a+bV where a and b are explicit constants and V follows the MarčenkoPastur law. The same are also obtained for other examples including (exp(−λ 2 ∥xi − xj ∥ γ))n×n and (exp(−λ 2 d(xi, xj) γ))n×n.
Empirical Distributions of Laplacian Matrices of Large Dilute Random Graphs
"... We study the spectral properties of the Laplacian matrices and the normalized Laplacian matrices of the ErdösRényi random graph G(n, pn) for large n. Although the graph is simple, we discover some interesting behaviors of the two Laplacian matrices. In fact, under the dilute case, that is, pn ∈ (0 ..."
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We study the spectral properties of the Laplacian matrices and the normalized Laplacian matrices of the ErdösRényi random graph G(n, pn) for large n. Although the graph is simple, we discover some interesting behaviors of the two Laplacian matrices. In fact, under the dilute case, that is, pn ∈ (0, 1) and npn → ∞, we prove that the empirical distribution of the eigenvalues of the Laplacian matrix converges to a deterministic distribution, which is the free convolution of the semicircle law and N(0, 1). However, for its normalized version, we prove that the empirical distribution converges to the semicircle law.