Statistical mechanics of scale-free networks at a critical point: complexity without irreversibility?

Based on a rigorous extension of classical statistical mechanics to networks, we study a specific microscopic network Hamiltonian. The form of this Hamiltonian is derived from the assumption that individual nodes increase or decrease their utility by linking to nodes with a higher or lower degree than their own. We interpret utility as an equivalent to energy in physical systems and discuss the temperature dependence of the emerging networks. We observe the existence of a critical temperature Tc where total energy (utility) and network architecture undergo radical changes. Along this topological transition we obtain ensemble averages of scale-free networks with complex hierarchical topology. The scale-free nature emerges strictly within equilibrium, with a clearly defined microcanonical ensemble and the principle of detailed balance fulfilled. This provides evidence that "complex" networks may arise without irreversibility. The utility approach establishes a link between classical statistical physics and a wide variety of applications in socioeconomic statistical systems.