Would an optimum combination of weights for a given topology necessarily be the the optimum for a different topology ?
No, an optimum combination of weights for a given topology not necessarily be the the optimum for a different topology.
Weight of a given neural network depends on the structure of the network and training data which vary network to network (or problem to problem).
Always the weights of a network are adjusted to match our desire output to achieve our goal.
Network topology also have a critical role in selection an optimum combination of weight.
Network with one hidden layer:
Network withe two hidden layer:
In above figures shows classification of a linearly separable data on two different network topology.
Here, thick dash line shows strong connection(higher weight) and thin dash line shows poor connection(lower weight).
You can visualize how weights get changes when we fed same data to two different network.
I recommend you to go to http://playground.tensorflow.org/ and play around different setting and observe the changes.