Lecture L15: Markov Chain Convergence and PageRank

What does the term $\pi^{(t)}$ mean in a Markov chain?

It represents the state probabilities (visiting probabilities) at time $t$.

What is $\pi^{(0)}$ called?

The initial state (or initial distribution) of the Markov chain.

How do you mathematically compute the state $\pi^{(t)}$ for any given time $t$?

Using the equation $\pi^{(t)}=\pi^{(0)}\cdot T^{t}$.

What does an initial distribution represented by a unit vector mean?

It means that the random walk starts from one specific node.

What does an initial distribution represented by a uniform vector mean?

It indicates that the probability of being at any node in the network during the random walk is equal.

What is the result of advancing the distribution by $k$ steps forward?

It is equivalent to a single multiplication by $T^{k}$, which is the $k$-th power of the transition matrix.

What do the elements of the matrix $T^{k}$ represent?

They capture the transition probabilities after exactly $k$ steps.

Does $\pi^{(t)}$ describe the exact deterministic path of the random walk?

No, these are probability distributions

What does the mathematical condition$\delta (\pi^{(t)}, \pi^{(t+1)}) \rightarrow 0$as$t \rightarrow \infty$signify?

It signifies that the visitation probabilities have stopped changing, meaning the Markov chain has reached its stationary state.How is the stationary state$\pi$mathematically defined using limits?