Why do neural networks seem to perform better with restrictions placed on their topology?
Fewer nodes/edges (or edges with fixed weights) means that there are fewer parameters whose values need to be found, and this typically reduces the time to learn. Also, when there are fewer parameters, the space that can be expressed by the neural network has fewer dimensions, so the neural network can only express more general models. It is thus is less capable of over-fitting the data, and hence the models will seem more general.