Is there any way to calculate the equilibrium (stationary) distribution for a **weighted directed acyclic** graph?
Some references emphasized adjacency matrix to be symmetric.
<https://arxiv.org/abs/1012.1211#content>
[Graph Example](https://i.stack.imgur.com/2lrPd.jpg)



Stationary distribution of a weighted directed acyclic graph
If the graph is acyclic, then either you have sinks (nodes with no edge out), or your graph is infinite.


If you have sinks, then this would indicate that you have a loop of probability $1$ at this node (you have to do something of your probability). In such a configuration, only the sink nodes can have a non-zero stationary probability. And any probability distribution whose support is among the sink nodes is stationnary.


If you don't have sinks, then you have infinite paths, so infinite (discrete) number of nodes. This means discrete probability techniques won't work (no unique node can have a non-zero probability). Then, maybe the results you could have there depend on the $\Sigma$-algebra you use, but my guess is that in most interesting cases you won't have any stationary distribution.


