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Title: Random walks, effective resistance and neighbourhood statistics in binomial random graphs
Author: Sylvester, John A.
ISNI:       0000 0004 7425 5946
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2017
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The binomial random graph model G(n; p), along with its near-twin sibling G(n; m), were the starting point for the entire study of random graphs and even probabilistic combinatorics as a whole. The key properties of these models are woven into the fabric of the field and their behaviour serves as a benchmark to compare any other model of random structure. In this thesis we contribute to the already rich literature on G(n; p) in a number of directions. Firstly, vertex to vertex hitting times of random walks in G(n; p) are considered via their interpretation as potential differences in an electrical network. In particular we show that in a graph satisfying certain connectivity properties the effective resistance between two vertices is typically determined, up to lower order terms, by the degrees of these vertices. We apply this to obtain the expected values of hitting times and several related indices in G(n; p), and to prove that these values are concentrated around their mean. We then study the statistics of the size of the r-neighbourhood of a vertex in G(n; p). We show that the sizes of these neighbourhoods satisfy a central limit theorem. We also bound the probability a vertex in G(n; p) has an r-neighbourhood of size k from above and below by functions of n; p and k which match up to constants. Finally, in the last chapter the extreme values of the r-degree sequence are studied. We prove a novel neighbourhood growth estimate which states that with high probability the size of a vertex's r neighbourhood is determined, up to lower order terms, by the size of its first neighbourhood. We use this growth estimate to bound the number of vertices attaining a smallest r-neighbourhood.
Supervisor: Not available Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics