Hamiltonian Dynamics of Preferential Attachment

K. Zuev, F. Papadopoulos, and D. Krioukov
Journal of Physics A: Mathematical and Theoretical
v.49, n.10, p.105001, 2016
February 1, 2016

Abstract

Prediction and  control of network dynamics are grand-challenge problems in network science.  The lack of understanding of fundamental laws driving the dynamics of  networks is among the reasons why many practical problems of great  significance remain unsolved for decades. Here we study the dynamics of  networks evolving according to preferential attachment, known to approximate  well the large-scale growth dynamics of a variety of real networks. We show  that this dynamics is Hamiltonian, thus casting the study of complex networks  dynamics to the powerful canonical formalism, in which the time evolution of  a dynamical system is described by Hamilton's equations. We derive the  explicit form of the Hamiltonian that governs network growth in preferential  attachment. This Hamiltonian turns out to be nearly identical to graph energy  in the configuration model, which shows that the ensemble of random graphs  generated by preferential attachment is nearly identical to the ensemble of  random graphs with scale-free degree distributions. In other words,  preferential attachment generates nothing but random graphs with power-law  degree distribution. The extension of the developed canonical formalism for  network analysis to richer geometric network models with non-degenerate  groups of symmetries may eventually lead to a system of equations describing  network dynamics at small scales.

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