I am modelling something with a flow network.
- There are a number of layers to the network, each layer has a different number of nodes.
- Each node receives some flow from all the nodes in the previous layer, and splits its output into nodes in the next layer.
- The final layer output loops back into the first layer.
I believe (although I currently do not have rigorous proof) that such a steady state exists for such a network... when all the flows are positive.
But actually, this model allows any node to "suck" flow backwards -- the flow rate between two such nodes is a negative number.
Does a steady state exist for such a network?
(The flow through each node in a layer can be calculated by constructing a suitable matrix and multiplying the previous layer by that matrix. I believe this means the problem is equivalent to finding an eigenvector; assuming that flows have been properly normalised, the eigenvalue will be 1. In which case I need to prove that this solution exists, but now I'm stuck.)