That is a heavily-filtered network: if he didn't drop most of the weak connections he'd end up with everything connected to everything else. On the other hand, if you don't use a sophisticated way to define what a "noisy" edge is, you'll end up with some curious cases like the one you point out. He might have used a naive global threshold -- it's the easiest way to go about it: drop all connections with weight lower than x. But it's also very wrong most of the times :-) Something like the disparity filter [1] works usually well, although you have sure that its null model hypothesis is aligned with what you think is the generative process of your network. The field is "network backboning" and it's a nice one from network science.
[1] https://en.wikipedia.org/wiki/Disparity_filter_algorithm_of_...