Networks of agents having a degree preference

I will discuss a model of   social network of agents  in which individuals can add or delete links connecting them to others, making the network evolve in  time. Each  agent is  typically comfortable with a certain number of contacts, i.e., they have preferred degree, and add links if they have too few, or delete if they have too many.  It is difficult to determine  the steady state of such networks in general. A particular simplifying limit is interesting to study: It is in which  members are either extreme introverts, who do not want any links, or extreme extroverts who want as many as possible. Remarkably, in this case,  the exact probability distribution of different  network configurations in the steady state  can be found, and has an interesting  analytical form. Further, the network  exhibits a phase transition in which the fractional  number of links in the system  jumps discontinuously from a value near zero, to near 1, as we vary a parameter.  I will also discuss variants of the model where the agents also have a preference about which links to add or remove, and these variations show qualitatively new features.