In page rank we are multiplying Transpose Adjacency matrix with current page rank vector. If the result is stationary distribution it means it will converge and all values will stay same. We can think this as incoming probability to a node is equal to outgoing probability.
For detailed balance condition T(x'->x)P(x') = T(x->x')P(x) if the graph is undirected
So total flow of x' times fraction of flow to x T(x'->x) is equal to total flow of x times the fraction of flow from x to x' is equal means incoming and outgoing probability flow in the full duplex edge.
It is possible that, this concept is already originating from real physical phenomena such as fluid flow or air flow.
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