Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-29T06:22:59.347Z Has data issue: false hasContentIssue false
  • English
  • Français

On the connectivity and diameter of small-world networks

Part of: Graph theory

Published online by Cambridge University Press:  01 July 2016

Ayalvadi Ganesh  and
Feng Xue
Ayalvadi Ganesh*
Affiliation:
Microsoft Research
Feng Xue*
Affiliation:
University of Illinois at Urbana-Champaign
*
∗ Current address: Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK. Email address: a.ganesh@bristol.ac.uk
∗∗ Current address: Communications Technology Laboratory, Intel Corporation, Santa Clara, CA 95052, USA. Email address: feng.xue1@intel.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider two different models of small-world graphs on nodes whose locations are modelled by a stochastic point process. In the first model each node is connected to a fixed number of its nearest neighbours, while in the second model each node is connected to all nodes located within some fixed distance. In both models, nodes are additionally connected via shortcuts to other nodes chosen uniformly at random. We obtain sufficient conditions for connectivity in the first model, and necessary conditions in the second model. Thereby, we show that connectivity is achieved at a smaller value of total degree (nearest neighbours plus shortcuts) in the first model. We also obtain bounds on the diameter of the graph in this model.

Keywords

Random graphsmall-world modelconnectivity

MSC classification

Primary: 05C80: Random graphs
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 39 , Issue 4 , December 2007 , pp. 853 - 863
Copyright
Copyright © Applied Probability Trust 2007 

References

Balister, P., Bollobas, B., Sarkar, A. and Walters, M. (2005). Connectivity of random k-nearest-neighbour graphs. Adv. Appl. Prob. 37, 1–24.Google Scholar
Barbour, A. D. and Reinert, G. (2001). Small worlds. Random Structures Algorithms 19, 54–74.CrossRefGoogle Scholar
Bollobas, B. (2001). Random Graphs. Cambridge University Press.Google Scholar
Coppersmith, D., Gamarnik, D. and Sviridenko, M. (2002). The diameter of a long-range percolation graph. Random Structures Algorithms 21, 1–13.CrossRefGoogle Scholar
Erdős, P. and Rényi, A. (1960). On the evolution of random graphs. Mat. Kutato Int. Közl. 5, 17–60.Google Scholar
Gupta, P. and Kumar, P. R. (1999). Critical power for asymptotic connectivity in wireless networks. In Stochastic Analysis, Control, Optimization and Applications, eds McEneaney, W. H. et al., Birkhäuser, Boston, MA, pp. 547–566.Google Scholar
Kleinberg, J. (2000). The small world phenomenon: an algorithmic perspective. In Proc. 32nd ACM Symp. Theory Comp. (STOC), ACM, New York, pp. 163–170.Google Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. Dover, New York.Google Scholar
Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.Google Scholar
Penrose, M. (1997). The longest edge of the random minimal spanning tree. Ann. Appl. Prob. 7, 340–361.CrossRefGoogle Scholar
Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.Google Scholar
Watts, D. J. (1999). Small Worlds. Princeton University Press.Google Scholar
Xue, F. and Kumar, P. R. (2004). The number of neighbours needed for connectivity of wireless networks. Wireless Networks 10, 169–181.Google Scholar