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Transient and slim versus recurrent and fat: Random walks and the trees they grow

Part of: Graph theory Special processes

Published online by Cambridge University Press:  01 October 2019

Giulio Iacobelli ,
Daniel R. Figueiredo  and
Giovanni Neglia
Giulio Iacobelli*
Affiliation:
Federal University of Rio de Janeiro
Daniel R. Figueiredo*
Affiliation:
Federal University of Rio de Janeiro
Giovanni Neglia*
Affiliation:
Inria, Université Côte d’Azur
*
* Postal address: Instituto de Matemática, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil. Email address giulio@im.ufrj.br
** Postal address: Department of Computer and System Engineering, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil. Email address daniel@cos.ufrj.br
*** Postal address: NEO Team, Inria, Sophia-Antipolis, France.Email address giovanni.neglia@inria.fr
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Abstract

The no restart random walk (NRRW) is a random network growth model driven by a random walk that builds the graph while moving on it, adding and connecting a new leaf node to the current position of the walker every s steps. We show a fundamental dichotomy in NRRW with respect to the parity of s: for ${s}=1$ we prove that the random walk is transient and non-leaf nodes have degrees bounded above by an exponential distribution; for s even we prove that the random walk is recurrent and non-leaf nodes have degrees bounded below by a power law distribution. These theoretical findings highlight and confirm the diverse and rich behaviour of NRRW observed empirically.

Keywords

Random walkrandom graphnetwork growth model

MSC classification

Primary: 60K37: Processes in random environments
Secondary: 05C80: Random graphs 05C81: Random walks on graphs
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 3 , September 2019 , pp. 769 - 786
Copyright
© Applied Probability Trust 2019 

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