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 all the degrees, to ensure its consistency with their joint distribution function. That this has not been done implies that there is a flaw in the existing degree-based models. Also, we demonstrate, in developing a degree-based model, that finding the domain of definition for the joint distribution of all the degrees of all the nodes in a network is as impossible as finding, in closed-form, the joint distribution itself, and highlights a possibly simpler way to analyse a network by making direct assumptions about the underlying processes of how it is formed rather than to develop a degree-based model for it. Finally and perhaps most importantly, we have argued that, for any directed or undirected network, one must deal with, either its un-ordered degrees, or those ordered in permissible ways, if one must deal with them at all. Therefore, the present approach to studying the degree distribution of a very general network, including scale-free networks, needs to be reconsidered.

Bernoulli random graph, high-order covariance, high-order correlation, scale-free networks, six degrees of separation, preferential attachment models.