Reinforcement and Preferential Attachment Models via Pólya Urns
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Authors
Singh, Somya
Date
Type
thesis
Language
eng
Keyword
Pólya urns , Contagion networks , Dynamical systems with time-delay , Consensus dynamics , Preferential attachment graphs
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Abstract
In this thesis, we devise two different types of discrete-time stochastic models using modified Pólya urn schemes. The first set of models concerns interacting contagion networks constructed using two-colour (red and black) finite memory Pólya urns in which reinforcing balls are removed M time steps after being added (where M is the “memory” of the urn). The urns interact in the sense that the probability of drawing a red ball (which represents an infectious state or an opinion) for a given urn, not only depends on the ratio of red balls in that urn but also on the ratio of red balls in other urns in a network representing the interconnections, hence accounting for the effect of spatial contagion. The finite memory reinforcement provides a diminishing effect of past draws which represents curing of an infection in an epidemic spread model, or lessening influence of a popular opinion in a social network. We examine the stochastic properties of the underlying Markov draw process and construct a class of dynamical systems to approximate the asymptotic marginal distributions. We also design a consensus achieving connected network of agents via two-color finite memory Pólya urns. The interaction between urns is time-varying and is represented via “super-urns” which combine for each node its own urn with its neighbouring urns. We obtain the consensus value in terms of the network’s reinforcement parameters and memory.
In the second part of this thesis, we introduce a novel preferential attachment model using the draw variables of a modified Pólya urn with an expanding number of colors, notably capable of modeling influential opinions (in terms of vertices of high degree) as the graph evolves. Unlike the Barabási-Albert model, the color-coded vertices in conjunction with the time-varying reinforcing parameter in our model allows for the vertices added (born) later in the process to potentially attain a high degree in a way that is not captured by the former. We study the degree count of the vertices in the graphs generated via our model by analyzing the draw vectors of the underlying stochastic process. Furthermore, we compare our model with the Barabási-Albert model via simulations
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Attribution 4.0 International
Queen's University's Thesis/Dissertation Non-Exclusive License for Deposit to QSpace and Library and Archives Canada
Proquest PhD and Master's Theses International Dissemination Agreement
Intellectual Property Guidelines at Queen's University
Copying and Preserving Your Thesis
This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owne
Queen's University's Thesis/Dissertation Non-Exclusive License for Deposit to QSpace and Library and Archives Canada
Proquest PhD and Master's Theses International Dissemination Agreement
Intellectual Property Guidelines at Queen's University
Copying and Preserving Your Thesis
This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owne