Small-world network models and their average path length

Taha, Samah M. Osman (2014-12)

Thesis (MSc)--Stellenbosch University, 2014.

Thesis

ENGLISH ABSTRACT: Socially-based networks are of particular interest amongst the variety of communication 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 theoretical 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 dimensional lattice, and rewire links locally and probabilistically in the former or probabilistically adding extra links in the latter. These models are investigated and compared with real-world networks. We consider a model in which randomness is provided by the Erdos-Rényi random network models superposed 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 smallworld 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 εn (where ε 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.

AFRIKAANSE OPSOMMING: Sosiaal-baseerde netwerke is van besondere belang onder die verskeidenheid kommunikasie netwerke. Hulle word onderskei deur ’n klein gemiddelde skeidingsafstand en hoë samedrommingskoëffisiënt, en is voorbeelde van kleinwêreld 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 Strogatz (1998) en Newman en Watts (1999a), voeg randvoorwaardes by tot eendimensionele roosters, en herbedraad nedwerkskakels gebaseer op lokale kennis 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 gestruktureerde 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, verbeter 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 εn (waar εn n konstante groter as nul is) ewekansige skakels bygevoeg moet word by ’n een-dimensionele rooster om ‘n gemiddelde padlengte van orde log n te verseker.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/95834
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