Preferential attachment is a key process governing the dynamics of many economic, social and biological process. It is the “The rich get richer” mechanism by which a quantity is distributed among individuals according to how much they already have. It also happens in social networks and the ones that have more social connectivity (the “hubs”) receive more new connections than the poorly connected. In a famous paper, Laszlo Barabási and Reka Albert encoded this mechanics in the so called Barabasi-Albert model to generate random scale free-networks.
The model is dynamical since new nodes and edges are created in time. Although the model is analytical tractable and many properties of the networks produced are known, I show you here a video illustrating the temporal network of the model (the details of how this video was done are in this other post)
In my implementation of the model a node is created at regular time intervals and it connects to a (random) number of existing nodes which are chosen randomly but proportionally to their number of existing links. Colors correspond to the “age” of the node in the simulation. Darker means older nodes. Size is log-proportional to the node’s connectivity. The result is a heterogeneous network in which some nodes (most of the times the initial nodes) are highly connected while the rest have a small connectivity.
The video (and model) shows you that under this preferential attachment mechanism the real important thing is to be there first: darker/older nodes are more connected. That’s life experience, or as John Tuld says in the Margin Call movie: “There are three ways to make a living in this business: be first, be smarter, or cheat.”