Can this gaming problem be represented as an electric circuit?
You can't. There is something missing.

You can look at this as an electrical problem, where each node is a capacitor charged at a specific voltage. When connecting them simultaneously trough equal resisntances a current will strat to flow trough each edge(resistor) to equalize the voltage. Am I right?

You can also think of it as a hydrostatic problem. Each ot the nodes is cylindrical tank filled with water to a certain height. All tanks are connected at their bottoms by pipes with equal cross-section. The water will flow till all the tanks equal their water levels.

The thing you are missing is (a) the capacitance of the capacitor (electrically) or (b) the diameter ot the water tank. Actually the same thing.

In either way the curent or the water will flow for a certain time until voltage/height is equal. The current flowing during this period will not be constant - it will decrease with the voltages getting close to each other. The process is transient and is hard to calculate for more than 2 nodes. That's why all electric simulators use numerical methods of prediction, trying different values and etc. till they solve all variables for a given moment of time, then move to next moment of time... an so on...

UPDATE:
Say you've got only 2 tanks - with 10m of water in the first tank and 6m in the second. In the end both will have equal level and the interesting thing is that the level may not be 8m !!! It will depend on both tanks diameters. Only if they are equal the level will be 8m. Same is electrical - depends on the capacitance. 

The other thing - if you assume they are equal you can easily get the final level. But to know the exact ammount of water flowed from one tank to another except height you must know the diameter(cross-section) to calculate the volume. Its the same with electricity - except voltage delta you must know the capacitance to calculate the ammount of energy in Joules that moved from one cap. to another.