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There is a hierarchy of minimal constructions of complex networks: (i) the maximally random network with a given degree distribution P(k); (ii) the maximally random network with a given distribution P(k,k′) of the degrees of the nearest-neighbor vertices; (iii) the maximally random network with a given distribution P(k,k′,k″) of the degrees of a triple of connected vertices; and so on. All these are equilibrium networks with a treelike local structure. In this report we discuss only random graphs with a given degree-degree distribution P(k,k′), since any empirical data on P(k,k′,k″) and higher-order multivertex correlations is absent.
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