  Markov chains are a class of probabilistic models that have achieved
widespread application in the quantitative sciences. This is in part due to
their versatility, but is compounded by the ease with which they can be probed
analytically. This tutorial provides an in-depth introduction to Markov chains,
and explores their connection to graphs and random walks. We utilize tools from
linear algebra and graph theory to describe the transition matrices of
different types of Markov chains, with a particular focus on exploring
properties of the eigenvalues and eigenvectors corresponding to these matrices.
The results presented are relevant to a number of methods in machine learning
and data mining, which we describe at various stages. Rather than being a novel
academic study in its own right, this text presents a collection of known
results, together with some new concepts. Moreover, the tutorial focuses on
offering intuition to readers rather than formal understanding, and only
assumes basic exposure to concepts from linear algebra and probability theory.
It is therefore accessible to students and researchers from a wide variety of
disciplines.

Machine Learning