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Small-World Network Models and Their Average Path Length Small-World Network Models and Their Average Path Length by Samah M.Osman Taha Thesis presented in fulfilment of the requirements for the degree of Master of Science in Mathematics at Stellenbosch University Department of Mathematical Sciences, Faculty of Science, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa. Supervisors: Prof. J.W. Sanders Dr. D. Ralaivaosaona Stellenbosch University http://scholar.sun.ac.za Declaration By submitting this thesis electronically, I declare that the entirety of the work contained therein is my own, original work, that I am the owner of the copy- right thereof (unless to the extent explicitly otherwise stated) and that I have not previously in its entirety or in part submitted it for obtaining any qualifi- cation. Signature: . Samah Taha 2014/4/8 Date: . Copyright © 2014 Stellenbosch University All rights reserved. ii Stellenbosch University http://scholar.sun.ac.za Abstract Small-World Network Models and Their Average Path Length Samah Taha Department of Mathematical Sciences, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa. Thesis: April 2014 Socially-based networks are of particular interest amongst the variety of com- munication networks arising in reality. They are distinguished by having small average path length and high clustering coefficient, and so are examples of small-world networks. This thesis studies both real examples and theoreti- cal models of small-world networks, with particular attention to average path length. Existing models of small-world networks, due to Watts and Strogatz (1998) and Newman and Watts (1999a), impose boundary conditions on a one di- mensional lattice, and rewire links locally and probabilistically in the former or probabilistically adding extra links in the latter. These models are in- vestigated and compared with real-world networks. We consider a model in which randomness is provided by the Erdős-Rényi random network models su- perposed on a deterministic one dimensional structured network. We reason about this model using tools and results from random graph theory. Given a disordered network C(n; p) formed by adding links randomly with probability p to a one dimensional network C(n). We improve the analytical result regarding the average path length by showing that the onset of small- world behaviour occurs if pn is bounded away from zero. Furthermore, we show that when pn tends to zero, C(n; p) is no longer small-world. We display that the average path length in this case approaches infinity with the network order. We deduce that at least "n (where " is a constant bigger than zero) random links should be added to a one dimensional lattice to ensure average path length of order log n. iii Stellenbosch University http://scholar.sun.ac.za Uittreksel Small-World Network Models and Their Average Path Length Samah Taha Departement Wiskundige Wetenskappe, Universiteit van Stellenbosch, Privaatsak X1, Matieland 7602, Suid Afrika. Tesis: April 2014 Sosiaal-baseerde netwerke is van besondere belang onder die verskeidenheid kommunikasie netwerke. Hulle word onderskei deur ’n klein gemiddelde skei- dingsafstand en hoë samedrommingskoëffisiënt, en is voorbeelde van klein- wêreld netwerke. Hierdie verhandeling bestudeer beide werklike voorbeelde en teoretiese modelle van klein-wêreld netwerke, met besondere aandag op die gemiddelde padlengte. Bestaande modelle van klein-wêreld netwerke, te danke aan Watts en Stro- gatz (1998) en Newman en Watts (1999a), voeg randvoorwaardes by tot een- dimensionele roosters, en herbedraad nedwerkskakels gebaseer op lokale ken- nis in die eerste geval en voeg willekeurig ekstra netwerkskakels in die tweede. Hierdie modelle word ondersoek en vergelyk met werklike-wêreld netwerke. Ons oorweeg ’n prosedure waarin willekeurigheid verskaf word deur die Erdös- Renyi toevalsnetwerk modelle wat op ’n een-dimensionele deterministiese gest- ruktureerde netwerk geimposeer word. Ons redeneer oor hierdie modelle deur gebruik te maak van gereedskap en resultate toevalsgrafieke teorie. Gegewe ’n wanordelike netwerk wat gevorm word deur skakels willekeurig met waarskynlikheid p tot ‘n een-dimensionele netwerk C(n) toe te voeg, ver- beter ons die analitiese resultaat ten opsigte van die gemiddelde padlengte deur te wys dat die aanvang van klein-wêreld gedrag voorkom wanneer pn weg van nul begrens is. Verder toon ons dat, wanneer pn neig na nul, C(n; p) nie meer klein-wêreld is nie. Ons toon dat die gemiddelde padlengte in hierdie geval na oneindigheid streef saam met die netwerk groote. Ons lei af dat ten minste "n (waar "n n konstante groter as nul is) ewekansige skakels bygevoeg moet word iv Stellenbosch University http://scholar.sun.ac.za UITTREKSEL v by ’n een-dimensionele rooster om ‘n gemiddelde padlengte van orde log n te verseker. Stellenbosch University http://scholar.sun.ac.za Acknowledgements I would sincerely like to express my sincere gratitude to my supervisors Prof. J.W. Sanders who has been one of the most patient and helpful guides one can ask for. Working with him in this thesis was a great learning experience for me and I am grateful to him for giving me this opportunity. Your advice on both research as well as on my career have been priceless. Also, I would like to express my special appreciation and thanks to Dr. D. Ralaivaosaona. Having him as a co-supervisor has been a privilege. I would like to thank him for being a tremendous mentor for me. Without their endless supports and thoughts I could not have successfully completed my thesis. Thanks to Dr. Paul Taylor, who took time out of his busy schedules for careful reading my work. Thanks to my office-mate Mr. Ahmed Hassan, who was always ready to help when I needed help. Finally, my debt in gratitude to all the people -family, friends, teachers, colleagues, and others - who have helped me reach this point. The support I had is greater than I could ever possibly repay. vi Stellenbosch University http://scholar.sun.ac.za Dedications To My parents: M.Osman and Ibtesam vii Stellenbosch University http://scholar.sun.ac.za Contents Declaration ii Abstract iii Uittreksel iv Acknowledgements vi Dedications vii Contents viii List of Figures x List of Tables xii Nomenclature xiii 1 Introduction 1 1.1 Small-World Network . 1 1.2 Research question . 1 1.3 Related Work . 2 1.4 Contribution . 5 1.5 Thesis Organization . 6 2 Foundation 7 2.1 Introducing Graphs . 7 2.2 Complex Networks Properties . 8 2.3 Random Network . 10 2.4 Touching upon Real-World Networks . 14 2.5 Conclusion . 22 3 Small-World Networks 24 3.1 Structure and Definition . 24 3.2 Small-World Network Models . 26 3.3 Small-World Network Properties . 29 viii Stellenbosch University http://scholar.sun.ac.za CONTENTS ix 3.4 Conclusion . 40 4 Average Path Length in Small-World Networks 42 4.1 Small-world Networks . 42 4.2 The diameter of small-world networks . 43 4.3 The Average Path Length . 45 4.4 Random Network Structure . 45 4.5 Random Process . 52 4.6 Early Evolution of The Random Tree Process . 56 4.7 The result . 59 4.8 Conclusion . 61 5 Conclusion and Future work 62 5.1 Conclusion . 62 5.2 Future Work . 63 Bibliography 65 Stellenbosch University http://scholar.sun.ac.za List of Figures 2.1 The global clustering coefficient. 10 2.2 The degree distribution of a scientific collaboration done by (New- man, 2001c). .............................. 15 2.3 The degree distribution of Astro physics collaboration network in the period from January 1993 to April 2003. 16 2.4 The degree distribution of the high energy physics citation network in the period of 1993-2003. 17 (a) In-degree distribution . 17 (b) Out-degree distribution . 17 2.5 Backstrom et al. (2012)’ result regarding the path length distribu- tion and the average path length of Facebook network 2012. 19 (a) The distribution of the path length. 19 (b) The average path length. 19 2.6 The clustering coefficient and the degree distribution of Facebook network (Ugander et al., 2011). 20 (a) Clustering . 20 (b) Degree distribution . 20 2.7 The degree distribution of the Facebook data collected from MyNet- work application in 2013. The data represents one active user friendship. 21 2.8 Comparison between Random networks and real-world networks. 22 (a) Collaboration vs. Random network . 22 (b) Citation in-degree vs. Random network . 22 (c) Citation out-degree vs. Random network . 22 3.1 The rewiring model of Watts and Strogatz (1998) . 27 (a) p = 0 .............................. 27 (b) 0 < p < 1 ............................ 27 (c) p = 1 .............................. 27 3.2 Newman and Watts (1999a) small-world network model . 28 (a) p = 0 .............................. 28 (b) 0 < p < 1 ............................ 28 (c) p = 1 .............................. 28 x Stellenbosch University http://scholar.sun.ac.za LIST OF FIGURES xi 3.3 The degree distribution of Watts and Strogatz (1998) and Newman and Watts (1999a) model . 30 (a) p = 2−3 ............................. 30 (b) p = 2−1 ............................. 30 (c) p = 1 .............................. 30 (d) p = 2−3 ............................. 30 (e) p = 2−1 ............................. 30 (f) p = 1 .............................. 30 3.4 average path length l(p) verses the clustering coefficient cc(p) in Watts and Strogatz (1998) and Newman and Watts (1999a) models of small-world network. 32 (a) Without log scale. 32 (b) Log scale. 32 (c) Without log scale. 32 (d) With log scale. 32 3.5 The characteristic path length in small-world models. 33 (a) Watts-Strogatz model. 33 (b) Newman-Watts model. 33 3.6 The average path length distribution in small-world models . 33 (a) p = 2−9 ............................. 33 (b) p = 2−7 ............................. 33 (c) p = 2−4 ............................. 33 (d) p = 2−9 ............................. 33 (e) p = 2−7 ............................. 33 (f) p = 2−4 ............................. 33 3.7 Cross-over picture of small-world model. 34 (a) Watts-Strogatz model .
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