What do graphs of signal vs background neural network outputs represent? For example, this image shows the output of a neural network I assume but I am not sure how the output is not symmetric. So if the neural network gives a .6 for signal it should give a .4 for background. 


This is my neural network output that is symmetric. However, I have not seen anyone else get a symmetric output like this. Does this mean that I'm doing something wrong or does it mean that the first graph is not the output of the neural network.
Okay, your output makes sense, given what you wrote in the comments. But if you don't know where the other output comes from, all else is speculation. It could be that it was generated by a neural network with two sigmoid outputs, rather then softmax, and the classification was done based on which output was larger. In that case you are not guaranteed symmetric graphs. So the asymmetric graph does not necessarily imply that it's not generated by a neural network, only that the output does not come from a two-dimensional softmax or a pair of sigmoid/1-sigmoid.