Hittingtime and PageRank

Abstract

In this thesis, we study convergence of finite state, discrete, and time homogeneous Markov chains to a stationary distribution. Expressing the probability of transitioning between states as a matrix allows us to look at the conditions that make the matrix primitive. Using the Perron-Frobenius theorem we find the stationary distribution of a Markov chain to be the left Perron vector of the probability transition matrix. We study a special type of Markov chain &mdash random walks on connected graphs. Using the concept of fundamental matrix and the method of spectral decomposition, we derive a formula that calculates expected hitting times for random walks on finite, undirected, and connected graphs. The mathematical theory behind Google's vaunted search engine is its PageRank algorithm. Google interprets the web as a strongly connected, directed graph and browsing the web as a random walk on this graph. PageRank is the stationary distribution of this random walk. We define a modified random walk called the lazy random walk and define personalized PageRank to be its stationary distribution. Finally, we derive a formula to relate hitting time and personalized PageRank by considering the connected graph as an electrical network, hitting time as voltage potential difference between nodes, and effective resistance as commute time.