Studies of adaptive networks with preferred degree

We study a simple model involving adaptive networks in which the nodes add or cut links to other nodes according to a set preferred degree,k This behavior seems more natural for human beings as they form a circle of a preferred number of friends or contacts. In the simplest model, a node with degree k will add (cut) a link with probability w+ (k) (1– w+ (k)). Several forms of w+ are considered, e.g., a step function that drops abruptly from unity to zero as k increases beyond k. Using simulations, we find the degree distribution in the steady state. Unexpectedly, it is not a Gaussian (around k). We are able to find an approximate theory which explains these distributions quite well. Introducing a second network and coupling the two in various ways, we find both understandable and puzzling features. In the third part, we consider overlaying an SIS model of epidemics on a single adaptive network, allowing k to depend on the fraction of the infected population. The gross features of the resulting steady states can be well explained by a mean field like theory, balancing the rates of recovery and infection. Various avenues for further investigations are proposed.