Graphs and Markov chains

Learning Objectives

• Express a graph as a sparse matrix.
• Identify the performance benefits of a sparse matrix.

Graphs

Undirected Graphs:

The following is an example of an undirected graph:

The adjacency matrix, $\mathbf{A}$, for undirected graphs is always symmetric and is defined as:

where $a_{ij}$ is the $(i,j)$ element of $\mathbf{A}$. The adjacency matrix which describes the example graph above is:

Directed Graphs:

The following is an example of a directed graph:

The adjacency matrix, $\mathbf{A}$, for directed graphs is defined as:

where $a_{ij}$ is the $(i,j)$ element of $\mathbf{A}$. The adjacency matrix which describes the example graph above is:

Weighted Directed Graphs:

The following is an example of a weighted directed graph:

The adjacency matrix, $\mathbf{A}$, for weighted directed graphs is defined as:

where $a_{ij}$ is the $(i,j)$ element of $\mathbf{A}$, and $w_{ij}$ is the link weight associated with edge connecting nodes $i$ and $j$. The adjacency matrix which describes the example graph above is:

Typically, when we discuss weighted directed graphs it is in the context of transition matrices for Markov chains where the link weights across each column sum to $1$.

Markov Chain

A Markov chain is a stochastic model where the probability of future (next) state depends only on the most recent (current) state. This memoryless property of a stochastic process is called Markov property. From a probability perspective, the Markov property implies that the conditional probability distribution of the future state (conditioned on both past and current states) depends only on the current state.

Markov Matrix

A Markov/Transition/Stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a non-negative real number representing a probability. Based on Markov property, next state vector ${\mathbf{x}}_{k+1}$ is obtained by left-multiplying the Markov matrix $\mathbf{M}$ with the current state vector ${\mathbf{x}}_k$.

In this course, unless specifically stated otherwise, we define the transition matrix $\mathbf{M}$ as a left Markov matrix where each column sums to $1$.

Note: Alternative definitions in outside resources may present $\mathbf{M}$ as a right markov matrix where each row of $\mathbf{M}$ sums to $1$ and the next state is obtained by right-multiplying by $\mathbf{M}$, i.e. ${\mathbf{x}}_{k+1}^T={\mathbf{x}}_k^T\mathbf{M}$.

A steady state vector ${\mathbf{x}}^{\ast }$ is a probability vector (entries are non-negative and sum to $1$) that is unchanged by the operation with the Markov matrix $M$, i.e.

Therefore, the steady state vector ${\mathbf{x}}^{\ast }$ is an eigenvector corresponding to the eigenvalue $\lambda =1$ of matrix $\mathbf{M}$. If there is more than one eigenvector with $\lambda =1$, then a weighted sum of the corresponding steady state vectors will also be a steady state vector. Therefore, the steady state vector of a Markov chain may not be unique and could depend on the initial state vector.

Markov Chain Example

Suppose we want to build a Markov Chain model for weather predicting in UIUC during summer. We observed that:

• a sunny day is $60\mathrm{\%}$ likely to be followed by another sunny day, $10\mathrm{\%}$ likely followed by a rainy day and $30\mathrm{\%}$ likely followed by a cloudy day;
• a rainy day is $40\mathrm{\%}$ likely to be followed by another rainy day, $20\mathrm{\%}$ likely followed by a sunny day and $40\mathrm{\%}$ likely followed by a cloudy day;
• a cloudy day is $40\mathrm{\%}$ likely to be followed by another cloudy day, $30\mathrm{\%}$ likely followed by a rainy day and $30\mathrm{\%}$ likely followed by a sunny day.

The state diagram is shown below:

The Markov matrix is

If the weather on day $0$ is known to be rainy, then

and we can determine the probability vector for day $1$ by

The probability distribution for the weather on day $n$ is given by

Review Questions

• See this review link

ChangeLog

• 2018-04-01 Erin Carrier ecarrie2@illinois.edu: Minor reorg and formatting changes
• 2018-03-25 Yu Meng yumeng5@illinois.edu: adds Markov chains
• 2018-03-01 Erin Carrier ecarrie2@illinois.edu: adds more review questions
• 2018-01-14 Erin Carrier ecarrie2@illinois.edu: removes demo links
• 2017-11-02 Erin Carrier ecarrie2@illinois.edu: adds changelog, fix COO row index error
• 2017-10-25 Erin Carrier ecarrie2@illinois.edu: adds review questions, minor fixes and formatting changes
• 2017-10-25 Arun Lakshmanan lakshma2@illinois.edu: first complete draft
• 2017-10-16 Luke Olson lukeo@illinois.edu: outline