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Title: Hamiltonian dynamics of preferential attachment
Authors: Zuev, Konstantin 
Papadopoulos, Fragkiskos 
Krioukov, Dmitri V 
Keywords: Complex networks;Exponential random graphs;Hamiltonian dynamics;Preferential attachment
Category: Computer and Information Sciences;Electrical Engineering - Electronic Engineering - Information Engineering
Field: Engineering and Technology
Issue Date: 27-Jan-2016
Publisher: Institute of Physics Publishing
Source: Journal of Physics A: Mathematical and Theoretical, 2016, vol. 49, no. 10,
Journal: Journal of Physics A: Mathematical and Theoretical 
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 (PA), 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 PA. 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 PA is nearly identical to the ensemble of random graphs with scale-free degree distributions. In other words, PA 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.
ISSN: 1751-8113
DOI: 10.1088/1751-8113/49/10/105001
Rights: © 2016 IOP Publishing Ltd
Type: Article
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