I have а flow network $G$ and for every vertex $v$ I have the flow value $f(v)$ that is going through that vertex, so I have no information about flow on edges.
The problem is that such measurements of the flow may be incorrect (because real network has additional edges).
So the question is, how can I check flow conservation if I'm not given the flow number on edges.
My guess is to write a linear system $Ax = f$, where $A$ is an incidence matrix of $G$, $f$ is a vector of flow values on vertices and $x$ is a flow value on edges. So then I can check if this linear system is not consistent (there is no such flow on edges).
But I have no idea how to prove this (that linear system will be inconsistent).
For example, if I know that real flow network $G'$ has one additional edge $e$ with non-zero flow $f'(e)>0$ on it. The flow $f'$ in $G'$ is correct (satisfies the flow conservation condition in $G'$).
Then I take $f'(v)$ values from $G'$ and write them to $G = G'-e$ (without edge $e$).
So I need to prove that the flow network $G$ has no valid flow $f$ that satisfies the flow conservation condition and has its values on vertices $f(v)$ equal to $f'(v)$.