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

Abstract

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/decrease their utility by linking to nodes with a higher/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 $T_c$ where total energy (utility) and network-architecture undergo radical changes. Along this topological transition we obtain scale-free networks with complex hierarchical topology. In contrast to models for scale-free networks introduced so far, the scale-free nature emerges within equilibrium, with a clearly defined microcanonical ensemble and the principle of detailed balance strictly fulfilled. This provides clear evidence that 'complex' networks may arise without irreversibility. The results presented here should find a wide variety of applications in socio-economic statistical systems.

4 pages, 5 figures

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Cited by in corpus (6)

• The statistical mechanics of networks
• Entropy of stochastic blockmodel ensembles
• Percolation transition in networks with degree-degree correlation
• Unified model for network dynamics exhibiting nonextensive statistics
• Topology of Cell-Aggregated Planar Graphs
• Tsallis mapping in growing complex networks with fitness