Size of the giant component in a random geometric graph

Ghurumuruhan Ganesan

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 4, page 1130-1140
  • ISSN: 0246-0203

Abstract

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In this paper, we study the size of the giant component C G in the random geometric graph G = G ( n , r n , f ) of n nodes independently distributed each according to a certain density f ( · ) in [ 0 , 1 ] 2 satisfying inf x ∈ [ 0 , 1 ] 2 f ( x ) g t ; 0 . If c 1 n ≤ r n 2 ≤ c 2 log n n for some positive constants c 1 , c 2 and n r n 2 ⟶ ∞ as n → ∞ , we show that the giant component of G contains at least n - o ( n ) nodes with probability at least 1 - e - β n r n 2 for all n and for some positive constant β . We also obtain estimates on the diameter and number of the non-giant components of G .

How to cite

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Ganesan, Ghurumuruhan. "Size of the giant component in a random geometric graph." Annales de l'I.H.P. Probabilités et statistiques 49.4 (2013): 1130-1140. <http://eudml.org/doc/271997>.

@article{Ganesan2013,
abstract = {In this paper, we study the size of the giant component $C_\{G\}$ in the random geometric graph $G=G(n,r_\{n\},f)$ of $n$ nodes independently distributed each according to a certain density $f(\cdot )$ in $[0,1]^\{2\}$ satisfying $\inf _\{x\in [0,1]^\{2\}\}f(x)&gt;$$0$. If $\frac\{c_\{1\}\}\{n\}\le r_\{n\}^\{2\}\le c_\{2\}\frac\{\log \{n\}\}\{n\}$ for some positive constants $c_\{1\}$, $c_\{2\}$ and $nr_\{n\}^\{2\}\longrightarrow \infty $ as $n\rightarrow \infty $, we show that the giant component of $G$ contains at least $n-\mathrm \{o\}(n)$ nodes with probability at least $1-\mathrm \{e\}^\{-\beta nr^\{2\}_\{n\}\}$ for all $n$ and for some positive constant $\beta $. We also obtain estimates on the diameter and number of the non-giant components of $G$.},
author = {Ganesan, Ghurumuruhan},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random geometric graphs; Size of giant component; number of components; random geometric graph; size of giant component; graph diameter},
language = {eng},
number = {4},
pages = {1130-1140},
publisher = {Gauthier-Villars},
title = {Size of the giant component in a random geometric graph},
url = {http://eudml.org/doc/271997},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Ganesan, Ghurumuruhan
TI - Size of the giant component in a random geometric graph
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 4
SP - 1130
EP - 1140
AB - In this paper, we study the size of the giant component $C_{G}$ in the random geometric graph $G=G(n,r_{n},f)$ of $n$ nodes independently distributed each according to a certain density $f(\cdot )$ in $[0,1]^{2}$ satisfying $\inf _{x\in [0,1]^{2}}f(x)&gt;$$0$. If $\frac{c_{1}}{n}\le r_{n}^{2}\le c_{2}\frac{\log {n}}{n}$ for some positive constants $c_{1}$, $c_{2}$ and $nr_{n}^{2}\longrightarrow \infty $ as $n\rightarrow \infty $, we show that the giant component of $G$ contains at least $n-\mathrm {o}(n)$ nodes with probability at least $1-\mathrm {e}^{-\beta nr^{2}_{n}}$ for all $n$ and for some positive constant $\beta $. We also obtain estimates on the diameter and number of the non-giant components of $G$.
LA - eng
KW - random geometric graphs; Size of giant component; number of components; random geometric graph; size of giant component; graph diameter
UR - http://eudml.org/doc/271997
ER -

References

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  3. [3] G. Grimmett. Percolation, 2nd edition. Grundlehren der Mathematischen Wissenschaften 321. Springer, Berlin, 1999. MR1707339
  4. [4] P. Gupta and P. R. Kumar. Critical power for asymptotic connectivity in wireless networks. In Stochastic Analysis, Control, Optimization and Applications 547–566. Systems Control Found. Appl. Birkhäuser, Boston, MA, 1999. Zbl0916.90101MR1702981
  5. [5] S. Muthukrishnan and G. Pandurangan. The bin-covering technique for thresholding random geometric graph properties. In Proc. SODA 989–998. ACM, New York, 2005. Zbl1297.05221MR2298358
  6. [6] M. Penrose. Random Geometric Graphs. Oxford Studies in Probability 5. Oxford Univ. Press, Oxford, 2003. MR1986198

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