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Random graphs on the hyperbolic plane

Bode, Michel (2016)
Ph.D. thesis, University of Birmingham.

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In this thesis, we study a recently proposed model of random graphs that exhibit properties which are present in a wide range of networks arising in real world settings. The model creates random geometric graphs on the hyperbolic plane, where vertices are connected if they are within a certain threshold distance. We study typical properties of these graphs.

We identify two critical values for one of the parameters that act as sharp thresholds. The three resulting intervals of the parameters that correspond to three possible phases of the random structure: A.a.s., the graph is connected; A.a.s., the graph is not connected, yet there is a giant component; A.a.s., every component is of sublinear size. Furthermore, we determine the behaviour at the critical values.

We also consider typical distances between vertices and show that the ultra-small world phenomenon is present. Our results imply that most pairs of vertices that belong to the giant component are within doubly logarithmic distance.

Type of Work:Ph.D. thesis.
Supervisor(s):Fountoulakis, Nikolaos
School/Faculty:Colleges (2008 onwards) > College of Engineering & Physical Sciences
Department:School of Mathematics
Subjects:QA Mathematics
Institution:University of Birmingham
ID Code:6526
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
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