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### Highly connected random geometric graphs

Paul Balister, Béla Bollobás, Amites Sarkar, Mark Walters
2009 Discrete Applied Mathematics
We construct a random geometric graph G n,k by joining each point of P to its k nearest neighbours.  ...  For many applications it is desirable that G n,k is highly connected, that is, it remains connected even after the removal of a small number of its vertices.  ...  We define the random geometric graph G n,k by joining each point of P to its k nearest neighbours. Here, and throughout this paper, distance is measured using the Euclidean l 2 norm.  ...

### Probabilistic Properties of Highly Connected Random Geometric Graphs [chapter]

Bodo Manthey, Victor M. J. J. Reijnders
2018 Lecture Notes in Computer Science
We study reliability in terms of k-edge-connectivity in graphs.  ...  These results are then extended to optimal k-edge-connected power assignment graphs, where we assign transmit power to vertices, and two vertices are connected if they both have sufficient transmit power  ...  Conclusions and Open Problems In this paper, we have looked at fault tolerant networks in terms of k-edge-connected graphs. We studied both a standard (wired) model and a model for wireless networks.  ...

### From Graph Theory to Network Science: The Natural Emergence of Hyperbolicity (Tutorial)

Tobias Friedrich, Michael Wagner
2019 Symposium on Theoretical Aspects of Computer Science
Hyperbolic random graphs are obtained by choosing random points in the hyperbolic plane and connecting pairs of points that are geometrically close.  ...  The vertices of a network can be mapped to points in this geometry, such that geometric distances are similar to graph distances.  ...  An example of such models are geometric random graphs [38] (also called random unit disk graphs).  ...

### Random geometric graphs

Jesper Dall, Michael Christensen
2002 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
We analyse graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present.  ...  We derive an analytical expression for the cluster coefficient which shows that the graphs are distinctly different from standard random graphs, even for infinite dimensionality.  ...  A random geometric graph (RGG) is a random graph with a metric.  ...

### On the Connectivity of Key-Distribution Strategies in Wireless Sensor Networks

2009 GLOBECOM 2009 - 2009 IEEE Global Telecommunications Conference
In this paper, we study the connectivity of key-distribution mechanisms in secured topologies of wireless sensor networks.  ...  We explore the effect of the radio range on the connectivity of the network and provide a lower bound on the radio range under which the cover time of the underlying topology decreases significantly.  ...  Random Geometric Graph with radio range r 2 (n).  ...

### Multiscale saliency detection for 3D meshes using random walk

Se-Won Jeong, Jae-Young Sim
2014 Signal and Information Processing Association Annual Summit and Conference (APSIPA), 2014 Asia-Pacific
We perform random walk on the graph and find the stationary distribution of random walker, which is used as an initial saliency distribution.  ...  We construct a weighted undirected graph on an input 3D mesh model, by taking the vertices and edges in the mesh as the nodes and links of the graph.  ...  Costa [2] regarded saliency as visiting frequencies of random walker to the nodes of a fully connected graph defined on an image. Kim et al.  ...

### Modeling interactome: scale-free or geometric?

N. Przulj, D. G. Corneil, I. Jurisica
2004 Bioinformatics
We show that the structure of PPI networks is better modeled by a geometric random graph than by a scale-free model.  ...  We examine the fit of four different network models, including Erdös-Rényi, scalefree, and geometric random network models, to these PPI networks with respect to the measures of local and global network  ...  Geometric Random Graphs A geometric graph ¥ § ¦ © with radius is a graph with node set¨of points in a metric space and edge set !  ...

### Spectral statistics of random geometric graphs

C. P. Dettmann, O. Georgiou, G. Knight
2017 Europhysics letters
We use random matrix theory to study the spectrum of random geometric graphs, a fundamental model of spatial networks.  ...  That is the spectral statistics of spatial random geometric graphs fits the universality of random matrix theory found in other models such as Erdos-Renyi, Barabasi-Albert and Watts-Strogatz random graph  ...  Many types of random graph 25 models have been analysed, however, the ubiquitous and 26 fundamental class of geometric graphs which are the sim-27 Fig. 1: A random geometric graph.  ...

R. Hekmat, P. Van Mieghem
2006 Journal on spesial topics in mobile networks and applications
In this paper we study connectivity in wireless ad-hoc networks by modeling the network as an undirected geometric random graph.  ...  Firstly, they reduce the amount of correlation between links causing the geometric random graph tend to behave like a random graph with uncorrelated links.  ...  Therefore we expect our geometric random graph model to shift between a highly clustered regular lattice (for ξ → 0) and a random graph (for highest values of ξ ).  ...

### A curious gap in one-dimensional geometric random graphs between connectivity and the absence of isolated node [article]

Jun Zhao, Osman Yağan, Virgil Gligor
2015 arXiv   pre-print
c ∈( 1/2, 1), a one-dimensional geometric random graph has no isolated node but is not connected.  ...  This curious gap in one-dimensional geometric random graphs is in sharp contrast to the prevalent phenomenon in many other random graphs such as two-dimensional geometric random graphs, Erdős-Rényi graphs  ...  geometric random graphs.  ...

### Connectivity and Coverage in Hybrid Wireless Sensor Networks using Dynamic Random Geometric Graph Model

Jasmine Norman
2011 International Journal on Applications of Graph Theory In wireless Ad Hoc Networks And sensor Networks
In this paper, I study connectivity and coverage in hybrid WSN based on dynamic random geometric graph.  ...  Random Geometric Graphs have been a very influential and well-studied model of large networks, such as sensor networks, where the network nodes are represented by the vertices of the RGG, and the direct  ...  In [1, 13, 18] the authors studied the toplogy and connectivity properties of random geometric graphs.  ...

### A 360-degree Holographic Display System for Radiotherapy Treatment Planning

Yukying Lu, Shara Weeyee Lee, Kim Khaw, Ruxu Du
2015 Computer-Aided Design and Applications
It would be parameterized to generate graphs with some desirable proprieties, like highly or weakly decomposable graphs, or restricting the generated graph to a specific class of geometric configuration  ...  Each one is solved, separately, and the final solution is obtained by recomposing the solved sub-graphs. To the best of our knowledge, there is no random geometric constraint graph generator so far.  ...  G 6. replace the edge (x, y) by the graph H as follows: 7. pick two random vertices x' and y' from the graph H 8. connect x', to all neighbours of x 9. connect y', to all neighbours of y 10. delete x  ...

### Scale-invariant geometric random graphs

Zheng Xie, Tim Rogers
2016 Physical review. E
We introduce and analyze a class of growing geometric random graphs that are invariant under rescaling of space and time.  ...  Through calculations and numerical simulations we explore the consequences of scale-invariance for geometric random graphs generated this way.  ...  Minkowski space), it becomes possible to discuss dynamical random geometric graphs.  ...

### Introduction to the Special Issue on AI and Networks

Marie Des Jardins, Matthew E. Gaston, Dragomir R. Radev
2008 The AI Magazine
AI Magazine's Special Issueon Networks and AI summarizes the seven articles in thespecial issue by characterizing the nature of thenetworks that are the focus of each of the papers.A short tutorial on graph  ...  A random geometric graph is a special case of a random graph that is generated by randomly placing N agents in the unit square, then connecting pairs of agents if they are within some specified distance  ...  More specifically, two agents, i and j are connected in a random geometric graph if d(i, j) < φ, where φ is a threshold parameter of the model.  ...

### Enhanced Random Walk with Choice: An Empirical Study [article]

John Alexandris, Gregory Karagiorgos 'and' Ioannis Stavrakakis
2010 arXiv   pre-print
Simulation results show a significant improvement in cover time, maximum node load and load balancing, mainly in random geometric graphs.  ...  We compare the random walk with choice with its enhanced counterpart.  ...  The best known cases correspond to dense, highly connected graphs; on the other hand, when connectivity decreases and bottlenecks exist in the graph, the cover time increases.  ...
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