I have a network (London Underground) with the entries and exits for a day and year. This obviously gives me a weighted mode problem.
However I want to look at centrality of the network, which means I need weighted edges - is there a way of doing this other than manually working though the whole network to work out how many people are still on the train?
Thank you for your help
With some additional assumptions: yes. In general: no.
For example, if you have $+100$ in node A, $-100$ in node Z and $0$ everywhere else, the you can't tell how the traffic was routed if there is more than one way.
On the other hand, you could assume some capacities on the edges and ask for a flow that minimizes congestion or more generally some given norm over that flow (most common are $l_1$, $l_2$ and $l_\infty$).
There is yet one more important thing: you say you have the number of people going in and out, and in reality one can assume that people who do enter are not exiting immediately after. On the other hand plain network flows will just do the opposite (because it's the least costly way of matching given data). Thus, you would need to tell the algorithm in which direction did they go. If you have no such data, a first approximation could be to specify some kind of "center area" and route all the non-center entrances to center exits and center entrances to non-center exists. If you have more accurate data available you might be interested in fractional multi-commodity flows (fractional, because you can solve them using linear programming and some already available solvers).
I hope this helps $\ddot\smile$