1. Two-state Markov chain
Use a column-stochastic matrix $A=\begin{bmatrix}a&b\\1-a&1-b\end{bmatrix}$.
2. Iteration plot
The plot shows the first component of $x_k$ over time. Convergence appears as flattening.
3. PageRank damping
For a three-page web, form $G=\alpha P+(1-\alpha)\frac{1}{3}\mathbf 1\mathbf 1^T$.
4. Web graph picture
5. Leslie population model
Use $L=\begin{bmatrix}f_1&f_2\\s_1&0\end{bmatrix}$.
6. Population plot
7. Independent-study tasks with answers
Task A. Use $A=\begin{bmatrix}0.9&0.2\\0.1&0.8\end{bmatrix}$. Find the stationary distribution.
Answer: $\pi=(2/3,1/3)^T$.
Answer: $\pi=(2/3,1/3)^T$.
Task B. Use $A=\begin{bmatrix}0&1\\1&0\end{bmatrix}$. Explain the long-term behavior.
Answer: The state alternates, so it does not converge from $x_0=(1,0)^T$.
Answer: The state alternates, so it does not converge from $x_0=(1,0)^T$.
Task C. Increase the PageRank damping teleportation term by lowering $\alpha$. What happens?
Answer: The ranking becomes closer to uniform because random jumps become more important.
Answer: The ranking becomes closer to uniform because random jumps become more important.
Task D. In the Leslie model, what does the dominant eigenvalue represent?
Answer: The asymptotic growth factor per time step.
Answer: The asymptotic growth factor per time step.