Is this enough information to fully define a neural network?
Yes.

The neural network mentioned in this paper is simple and straightforward: it is a fully connected network with 2 hidden layers. 
 The authors specified the network structure, the number of neurons in each layer, and the activation functions in each layer and the final loss function.



Given those information, the network is fully specified. For instance, you can create such a network using 

sklearn.neural_network.MLPRegressor(hidden_layer_sizes=(50,7), activation=‘logistic’)