Hamiltonian Dynamics of Preferential Attachment

TL;DR

This paper reveals that the dynamics of networks evolving through preferential attachment are Hamiltonian, linking network growth to classical mechanics and showing their equivalence to certain random graph models.

Contribution

It introduces a Hamiltonian formalism for preferential attachment networks and derives the explicit Hamiltonian governing their growth, connecting it to graph energy and scale-free properties.

Findings

Preferential attachment network dynamics are Hamiltonian.

The derived Hamiltonian closely resembles graph energy in the configuration model.

Preferential attachment generates random graphs with power-law degree distributions.

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.