Figure 11

Maximum path length against the log of the number of paths per pair of source and target ports (left) as well as the log of the average number of paths (right, main plot) and including the amount of variation measured by one standard deviation (right, inset plot). As the maximum path length between a pair increases, so does the average number of paths between pairs with that maximum path length. An intuitive explanation for this trend is that longer minimum-route paths are concatenations of shorter minimum-route paths between intermediate source and target pairs (e.g. the loop on line 12 of Algorithm 1) that are all interchangeable. This implies that the number of paths between a pair with large maximum path length is a function of the product of the number of minimum-route paths at smaller lengths, and so the number of paths grows much more quickly than the maximum path length. However, at the highest maximum path lengths this trend does not hold. We attribute this to the fact that the very longest paths are likely to appear between ports that are not well-connected, thus there are few (perhaps only 1) viable minimum-route paths between some of these pairs, dragging the average down. Put another way: very long paths usually occur between poorly connected nodes, meaning they are more likely to be unique or few in number