THE CORRELATION OF THE DEGREES OF BERNOULLI RANDOM GRAPHS (original) (raw)

Networks offer a way of describing and exploring certain systems in many branches of science. These include the ecological niches in food webs, the modules in biochemical networks, the communities in social networks, the associations between terrorists, and the online interactions of people on the Internet. Network models can be traced back to the undirected Bernoulli random graph without loops, often called the simple random graph. Such a Bernoulli random graph is the simplest model for a network, forms a basis for developing the often complex and analytically intractable network models, and provides them with the crucial ground truthing and a reality check. Of these, almost all degree-based models, to which belong scale-free models, assume that the degrees of a network are jointly independent. Here we demonstrate that this assumption is invalid for the degrees of the nodes of a Bernoulli graph. This is done, by computing, in closed-form, what turns out to be the elegant expressions of an arbitrarily high-order covariance of, and correlation among, the degrees of its nodes. We notice that those degrees are correlated non-negatively. Thus, if a degree-based model is to be developed, then it must include