The topological structure of a Markov chain can be represented as a digraph and denoted by a pair (V,E ) with V being the vertices and E being the edges of the graph. The graph of a chain refers to the structure of the process that does not depend on the exact values of the transition probabilities but only on which of them are positive. We say that a state i leads to state j if, starting from i the chain will visit j at some finite time.
We denote this by i ~ j and it is equivalent to stating the probability for some final state j after n iterations starting from i is greater than zero. This relation is reflexive. In fact for all the states in the event space, we can have probabilities between pairs to be greater than zero.
This relation is also transitive. If we can reach state j from i and k from j then it implies that there is a chain possible from i to k .
If it's a symmetric relationship which means that there is a chain from i to j and j to i then we can say that i communicates with j.
The communication relationship is an equivalence relation which means it is symmetric, reflexive and transitive. Equivalence classes also called as communicating classes partition the space. The communicating class corresponding to state i is, by definition the set of all states that communicate with i.
Two communicating classes are either identical or completely disjoint.
A communicating class is said to be closed when all the states belong to the communicating class. Closed communicating classes are particularly important because they decompose the chain into smaller more manageable parts.The single element state i is essential if it belongs to closed communicating class. Otherwise the state is inessential. If all the states communicate with all others, then the chain is considered irreducible.