A variant of the graph flow problem

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I have the following problem: A complete graph with an unknown capacity-function for edge-flow is assumed. The vertices are assumed to not influence flow. I know the source vertex, the amount of flow put into the network, and can observe a binary $0-1$ variable for each edge if there is any flow over it at time $t$ or not. The problem is: Is there a way to determine if the underlying capacity-function is fixed over time $t$? And is there a way to estimate the capacity-function if it is?