I'm working on a package for analysis of Boolean networks (implementing some of the analysis from this paper), and I'm trying to figure out how to figure something out.

Say I have a state transition graph $G = (V, E)$, where every $v \in V$ represents the configuration of the network, and every $e \in E$ represents a deterministic, directed transition from one state, to the next. Every node has an out-degree of 1, but the in-degrees can vary. All edges are unweighted.

If I am on vertex $v$, which represents being in state $s$ at time $t_{1}$, and $v$ that has an in-degree $k > 0$. How can I create the best probability distribution that describes the probability that, at time $t_{0}$, I was at each node that feeds into $v$?

My initial thought was to have a uniform distribution, where, for the set of possible prior states, the probability is uniformly $1/k$, but that doesn't take into consideration that some prior nodes are more likely to be visited than others, and so on. Perhaps normalizing the probabilities by the Katz Centrality, or the PageRank of the nodes that feed into $v$ would work?

I'm sure there are many possible implementations, but I'm looking for something that is both (1) accurate, and (2) reasonably computationally tractable.


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