What kind of graph theory problems do you need to solve?

Not only the maximum, but also the medium one, because the maximum will probably not be reached so it is not the most important to know (I suppose a probalistic distribution of member choices, can give an idea of extremes and medium possibilities of a WoT).

Yes the object is fully defined by the uCoin protocol. You will find them in the uCoin official site (probably in the FAQ, or the github, somewhere…).