Consider a random surfer who begins at a web page (a node of the web graph) and executes a random walk on the Web as follows. At each time step, the surfer proceeds from his current page A to a randomly chosen web page that A hyperlinks to. What if the current location of the surfer, the node A, has no out-links? To address this we introduce an additional operation for our random surfer: the teleport operation. In the teleport operation the surfer jumps from a node to any other node in the web graph. This could happen because he types an address into the URL bar of his browser. The destination of a teleport operation is modeled as being chosen uniformly at random from all web pages. In other words, if N is the total number of nodes in the web graph, the teleport operation takes the surfer to each node with probability 1/N. The surfer would also teleport to his present position with probability 1/N.
So we have a Markov Chain in which we use the teleport operation in two ways: (1) When at a node with no out-links, the surfer invokes the teleport operation. (2)At any node that has outgoing links, the surfer invokes the teleport operation with probability 0 < α < 1 and the standard random walk (follow an out-link chosen uniformly at random) with probability 1 − α.
As the surfer proceeds in this random walk from node to node, he visits some nodes more often than others; intuitively, these are nodes with many links coming in from other frequently visited nodes.
Let's assume that the Markov Chain just described is ergodic and then it admits steady-states.
If in our model we add the chance that the random surfer at each step clicks the "back button" to jump to the previous state, can this be still modelled as an ergodic Markov Chain? Or does it preclude either irreducibility or aperiodicity?
Abstractly, you have a Markov chain on a finite state space of websites. As it stands, it's not irreducible if you have loops that don't allow for "teleportation." Adding a back-button would likely make the state space more irreducible ( have less irreducible components) because then one can get out of one-way accessible subsets. If you allow the user to go to any website at a given time, uniform or otherwise, then the system will be irreducible.
Anyways, for ergodicity in this case, you just need a finite number of states (which you have) and irreducability, which means that you can get from any state to any other state in some sequence of steps. If you get stuck in one irreducible component, then it will be ergodic there.
It will very likely be aperiodic as periodicity is easily broken.