Asymptotic properties of Euclidean shortest-path trees in random geometric graphs

2015 ◽  
Vol 107 ◽  
pp. 122-130 ◽  
Author(s):  
Christian Hirsch ◽  
David Neuhäuser ◽  
Catherine Gloaguen ◽  
Volker Schmidt
Keyword(s):  
Shortest Path ◽  
Asymptotic Properties ◽  
Geometric Graphs ◽  
Random Geometric Graphs ◽  
Euclidean Shortest Path ◽  
Shortest Path Trees

First Passage Percolation on Random Geometric Graphs and an Application to Shortest-Path Trees

2015 ◽  
Vol 47 (02) ◽  
pp. 328-354
Author(s):  
C. Hirsch ◽  
D. Neuhäuser ◽  
C. Gloaguen ◽  
V. Schmidt
Keyword(s):  
Shortest Path ◽  
Linear Growth ◽  
Telecommunication Networks ◽  
Geometric Graphs ◽  
First Passage Percolation ◽  
First Passage ◽  
Growth Property ◽  
Random Geometric Graphs ◽  
Shape Theorem ◽  
Path Lengths

We consider Euclidean first passage percolation on a large family of connected random geometric graphs in the d-dimensional Euclidean space encompassing various well-known models from stochastic geometry. In particular, we establish a strong linear growth property for shortest-path lengths on random geometric graphs which are generated by point processes. We consider the event that the growth of shortest-path lengths between two (end) points of the path does not admit a linear upper bound. Our linear growth property implies that the probability of this event tends to zero sub-exponentially fast if the direct (Euclidean) distance between the endpoints tends to infinity. Besides, for a wide class of stationary and isotropic random geometric graphs, our linear growth property implies a shape theorem for the Euclidean first passage model defined by such random geometric graphs. Finally, this shape theorem can be used to investigate a problem which is considered in structural analysis of fixed-access telecommunication networks, where we determine the limiting distribution of the length of the longest branch in the shortest-path tree extracted from a typical segment system if the intensity of network stations converges to 0.

First Passage Percolation on Random Geometric Graphs and an Application to Shortest-Path Trees

2015 ◽  
Vol 47 (2) ◽  
pp. 328-354 ◽  
Author(s):  
C. Hirsch ◽  
D. Neuhäuser ◽  
C. Gloaguen ◽  
V. Schmidt
Keyword(s):  
Shortest Path ◽  
Linear Growth ◽  
Telecommunication Networks ◽  
Geometric Graphs ◽  
First Passage Percolation ◽  
First Passage ◽  
Growth Property ◽  
Random Geometric Graphs ◽  
Shape Theorem ◽  
Path Lengths

We consider Euclidean first passage percolation on a large family of connected random geometric graphs in the d-dimensional Euclidean space encompassing various well-known models from stochastic geometry. In particular, we establish a strong linear growth property for shortest-path lengths on random geometric graphs which are generated by point processes. We consider the event that the growth of shortest-path lengths between two (end) points of the path does not admit a linear upper bound. Our linear growth property implies that the probability of this event tends to zero sub-exponentially fast if the direct (Euclidean) distance between the endpoints tends to infinity. Besides, for a wide class of stationary and isotropic random geometric graphs, our linear growth property implies a shape theorem for the Euclidean first passage model defined by such random geometric graphs. Finally, this shape theorem can be used to investigate a problem which is considered in structural analysis of fixed-access telecommunication networks, where we determine the limiting distribution of the length of the longest branch in the shortest-path tree extracted from a typical segment system if the intensity of network stations converges to 0.

On the distribution of typical shortest-path lengths in connected random geometric graphs

2012 ◽  
Vol 71 (1-2) ◽  
pp. 199-220 ◽  
Author(s):  
D. Neuhäuser ◽  
C. Hirsch ◽  
C. Gloaguen ◽  
V. Schmidt
Keyword(s):  
Shortest Path ◽  
Geometric Graphs ◽  
Random Geometric Graphs ◽  
Path Lengths

scholarly journals Normalized Sombor Indices as Complexity Measures of Random Networks

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 976
Author(s):  
R. Aguilar-Sánchez ◽  
J. Méndez-Bermúdez ◽  
José Rodríguez ◽  
José Sigarreta
Keyword(s):  
Random Matrix ◽  
Adjacency Matrix ◽  
Matrix Theory ◽  
Computational Study ◽  
Random Networks ◽  
Geometric Graphs ◽  
Theory Approach ◽  
Complexity Measures ◽  
Random Geometric Graphs ◽  
Highly Correlated

We perform a detailed computational study of the recently introduced Sombor indices on random networks. Specifically, we apply Sombor indices on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the network, scale with the average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random networks and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the adjacency matrix.

scholarly journals Connectivity in one-dimensional soft random geometric graphs

2020 ◽  
Vol 102 (6) ◽  
Author(s):  
Michael Wilsher ◽  
Carl P. Dettmann ◽  
Ayalvadi Ganesh
Keyword(s):  
Geometric Graphs ◽  
Random Geometric Graphs ◽  
One Dimensional

scholarly journals Convergence Theorems for Some Layout Measures on Random Lattice and Random Geometric Graphs

2000 ◽  
Vol 9 (6) ◽  
pp. 489-511 ◽  
Author(s):  
JOSEP DÍAZ ◽  
MATHEW D. PENROSE ◽  
JORDI PETIT ◽  
MARÍA SERNA
Keyword(s):  
Constant Factor ◽  
Geometric Graphs ◽  
Convergence Theorems ◽  
Linear Arrangement ◽  
Random Lattice ◽  
Random Geometric Graphs ◽  
Unit Square ◽  
Subcritical Regime ◽  
Probabilistic Setting ◽  
Lattice Graphs

This work deals with convergence theorems and bounds on the cost of several layout measures for lattice graphs, random lattice graphs and sparse random geometric graphs. Specifically, we consider the following problems: Minimum Linear Arrangement, Cutwidth, Sum Cut, Vertex Separation, Edge Bisection and Vertex Bisection. For full square lattices, we give optimal layouts for the problems still open. For arbitrary lattice graphs, we present best possible bounds disregarding a constant factor. We apply percolation theory to the study of lattice graphs in a probabilistic setting. In particular, we deal with the subcritical regime that this class of graphs exhibits and characterize the behaviour of several layout measures in this space of probability. We extend the results on random lattice graphs to random geometric graphs, which are graphs whose nodes are spread at random in the unit square and whose edges connect pairs of points which are within a given distance. We also characterize the behaviour of several layout measures on random geometric graphs in their subcritical regime. Our main results are convergence theorems that can be viewed as an analogue of the Beardwood, Halton and Hammersley theorem for the Euclidean TSP on random points in the unit square.

scholarly journals Symmetric motifs in random geometric graphs

2017 ◽  
Vol 6 (1) ◽  
pp. 95-105 ◽  
Author(s):  
Carl P Dettmann ◽  
Georgie Knight
Keyword(s):  
Geometric Graphs ◽  
Random Geometric Graphs

Dynamically Maintaining Shortest Path Trees under Batches of Updates

2013 ◽  
pp. 286-297 ◽  
Author(s):  
Annalisa D’Andrea ◽  
Mattia D’Emidio ◽  
Daniele Frigioni ◽  
Stefano Leucci ◽  
Guido Proietti
Keyword(s):  
Shortest Path ◽  
Shortest Path Trees

scholarly journals High-Dimensional Random Geometric Graphs and their Clique Number

2011 ◽  
Vol 16 (0) ◽  
pp. 2481-2508 ◽  
Author(s):  
Luc Devroye ◽  
András György ◽  
Gábor Lugosi ◽  
Frederic Udina
Keyword(s):  
Clique Number ◽  
High Dimensional ◽  
Geometric Graphs ◽  
Random Geometric Graphs
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