To be precise, it should be noted that there are in reality two large families of networks: “small-worlds” characterized by groupings and short paths, and “scale-free” characterized by the presence of hubs. In practice, most networks are * both * small-world and scale-free and therefore exhibit all three properties at the same time as I describe in the video.
The mechanism which produces these graphs will be different for “scale-free” and for “small-world”. If the new points preferentially connect to the big ones, this will give a “scale-free” (and therefore hubs). To produce small-worlds, you need a preferential attachment to neighbors and some random links. For simplicity, I describe in the video a unique mechanism which produces “scale-free” and “small-world”, but they are often described separately in the scientific literature.
Finally on the law of short paths in random networks [Paul Erdös model], the length of the path between two points hardly depends on the size of the network (as indicated at 3:00), but there is however a strong dependence on the number of links that we have placed in the network. In a random network with few social links, the length of the path between two points can be relatively long.