The flow rate through the network will most definitely not change only with the area of the pipes.
As water flows through a system, its pressure (often referred to as head) is reduced by friction along the way and changes in elevation.
Pipe friction is a function of pipe diameter, pipe length, fluid characteristics, and pipe wall smoothness. Every bend in a pipe also introduces additional head loss. The head loss due to this directly affects the flow rate through the pipe.
But the key thing that will affect flow through the network is the usage by each end user. I doubt that these pipes just empty into rooms. So the flows in each branch pipe will be determined by the outlet of the pipes. In a non-industrial use we can probably assume that water requirements fluctuate greatly. For industrial use we can determine flows by direct measurement (buckets, etc), or flow measurement devices in the pipes. If we had static pressures throughout the system we might be able to back in to flows, given that we know the total. I feel like this might have to be an iterative process.