How to implement an algorithm randomly generating an ergodic Markov chain?

I understand that the graph topology and dynamics of an ergodic Markov chain can be described as irreducible and aperiodic. How can one formulate an algorithm to construct some random directed graph with these properties? Does such an algorithm already exist?

So far I have tried a function with size and density parameters creating a random adjacency matrix, then removing any cycles. However I realize there is a difference between aperiodic states of a Markov chain and acyclic digraphs, so I don't think this is the right way to start.