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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: [email protected]
∗∗ Current address: Communications Technology Laboratory, Intel Corporation, Santa Clara, CA 95052, USA. Email address: [email protected]
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Abstract

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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 

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