I am a final year mathematics and computer science student, and doing my dissertation at the moment. I would appreciate any ideas on the following:

It is well known that Markov Chains which are ergodic (irreducible and aperiodic) have a unique stationary distribution.
Let's view a Markov Chain as a directed graph with labels as probabilities.
For my project, I define a new system that is exactly like a Markov Chain, except that from each state we visit 2 new states. We may reach different '2 states'
with different probability. This process repeats, so that for each of the 2 states we would also define a transition to 2 new states.
We can view this system as a hyper-graph with hyper-edges including 3 states (1 source and 2 target states).

Now, I am looking to see when such a system would have a unique stationary distribution. More specifically:
A vector $\displaystyle $\pi$$ s.t $\displaystyle $\pi_i = \sum_{j, k \in S} \pi_j * \pi_k * P_{i(j, k)}$ $ where $\displaystyle $P_{i(j, k)}$$ is the probability of moving from state i to states j, k.

I would be grateful for any ideas.