Damping Factor Analysis for PageRank

Abstract: The purpose of this thesis is to present research related to the damping factor in relation to the PageRank algorithm where a method of symbolic calculations is used to calculate eigenvalues, eigenvectors corresponding to the Google matrix in relation to both directed and undirected graphs. These graphs given comprise all the directed graphs up to four vertices and in addition the undirected graphs of five vertices are given in this thesis. A central research question has been to determine how $d$ behaves in relation to effecting the result of the dominant eigenvector for corresponding graphs such as to determine how the PageRank is directly influenced. A few selected graphs along with their calculations were extracted and analyzed in terms of the parameter $d$. For the calculations in this thesis probability matrices were constructed for all graphs and calculations were made using Matlab where eigenvalues, eigenvectors corresponding to the Google matrix were returned along with the input probability matrix and the Google matrix. In addition, the thesis contains a theoretical portion related to the theory behind PageRank along with relevant proofs, theorems and definitions which are used throughout the thesis. Some brief mention of the historical background and applications of the PageRank are also given. \bigskip A discussion of the results is provided involving the interaction of the damping factor with the dominant PageRank eigenvector. Lastly, a conclusion is given and future prospects relating to the topic of research is discussed. The work in this thesis is inspired by a previous work done by Silvestrov et al. in $2008$ where we have here placed further emphasis on the damping factor.