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I've come across a problem which at first appeared to be a markov process however the transition matrix of the graph is non-stochastic. That is, the probabilities among edges leaving a node do not sum to 1. However,the probabilities on edges entering the node do sum to 1. Therefore the transpose of the transition matrix IS stochastic.

I've built up a model in excel and noticed a few properties of this "transpose markovian process":

  1. all nodes reach the same steady state value.
  2. the steady state *is* dependent on the initial state (unlike a markov process).
  3. the steady state value can be computed as S0 x MT (the initial state vector times the transpose of the markovian steady state vector... obtained by transposing the transition matrix and solving as a markov process).

My question is this: Is there a name for such a process? In searching for it I've found it to be beyond obscure, so my hope is some well-informed human can catalyze me with some search terms.

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    $\begingroup$ $P(\text{staying}) = 1 - P(\text{leaving})$. A Markov process can stay in the same state for many iterations. $\endgroup$
    – Hunaphu
    Commented May 26, 2015 at 16:44

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There is no new name for the processes you mentioned. Markov processes can be defined with row-stochastic or column-stochastic matrices. The choice is only a matter of convention.

If the entries of each row sum up to 1, then the stationary state S0 satisfies $S_0=S_0.M$, otherwise it satisfies $M.S_0=S_0$ .

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