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Prediction and control of network dynamics are grand-challenge problems innetwork science. The lack of understanding of fundamental laws driving thedynamics of networks is among the reasons why many practical problems of greatsignificance remain unsolved for decades. Here we study the dynamics ofnetworks evolving according to preferential attachment, known to approximatewell the large-scale growth dynamics of a variety of real networks. We showthat this dynamics is Hamiltonian, thus casting the study of complex networksdynamics to the powerful canonical formalism, in which the time evolution of adynamical system is described by Hamilton's equations. We derive the explicitform of the Hamiltonian that governs network growth in preferential attachment.This Hamiltonian turns out to be nearly identical to graph energy in theconfiguration model, which shows that the ensemble of random graphs generatedby preferential attachment is nearly identical to the ensemble of random graphswith scale-free degree distributions. In other words, preferential attachmentgenerates nothing but random graphs with power-law degree distribution. Theextension of the developed canonical formalism for network analysis to richergeometric network models with non-degenerate groups of symmetries mayeventually lead to a system of equations describing network dynamics at smallscales.

Hamiltonian Dynamics of Preferential Attachment