Organization of complex networks without multiple connections

We find a new structural feature of equilibrium complex random networks without multiple and self-connections. We show that if the number of connections is sufficiently high, these networks contain a core of highly interconnected vertices. The number of vertices in this core varies in the range between constxN(1/2) and constxN(2/3), where N is the number of vertices in a network. At the birth point of the core, we obtain the size-dependent cutoff of the distribution of the number of connections and find that its position differs from earlier estimates.