Discussion
From these results, we can infer a few ideas. Centrality is a important sign that separates more effective networks from less effective ones. We see in both Betweenness and PageRank a tendency for most nodes to have a low centrality, and a few nodes to have a slightly higher centrality as opposed to the other networks where the mean PageRank centrality and the variance in the betweenness centrality is larger. The exact nature of centrality and its role in effective networks is an area for future study and it would be good to see if there is a way to separate the correlation between PageRank and betweenness centrality before we can purpose a particular course of action for network designers. This leads us back to evidence found in the literature about the role of bottlenecks. We saw that effective networks do not bottleneck a large amount flow through a few nodes, however they do select a few nodes to be more "important" than the rest. While this is not what we were expecting, it still is consistent with what is shown in Radford et al. \cite{Radford2017} and Guimar et al. \cite{Guimar1995} where there is assignment "main predictors and correction factors."
In addition to this, we saw through the maximum flow path of the networks, that a network narrows, as it goes from source to sink naturally. This was a surprising but welcomed result as it serves to confirm some of the intuition applied to neural network design. We see that instead of leveraging the capacity provided later in the networks, the networks continue to condense. This gives designer a metric to work with, which is to make the hidden layer before the output layer has enough capacity to encode the representation of the input data and make the earlier hidden layers wider so that the network may condense to a larger and more precise encoding. The narrowing of the network also explains the larger varience in betweeness centrality and the larger mean and mode of PageRank.
Combined, these results show that perhaps there are qualities about the topology of effective networks that distinguish them from less effective counter parts and a promising platform for future work.