From the internet to airport trips, many networks are characterized by a few nodes with many connections, and many nodes with very few connections. These networks are called heavy-tailed because, when a histogram of the node degrees is drawn, the high-connectivity nodes form a tail.
There are many ways to generate heavy-tailed networks, but one of the most widely-used is the Barabási-Albert preferential attachment model (Albert & Barabási, 1999). The preferential attachment model mimics processes where the rich get richer. Every time a node is added, it is randomly connected to existing nodes, with high-degree nodes being more likely.
In NetworkX, the barabasi_albert_graph() function generates preferential attachment networks. The following code shows an example of such a network with 35 nodes:
G_preferential_35 ...