Eur. Phys. J. B (2017) 90: 226
https://doi.org/10.1140/epjb/e2017-80185-5

Regular Article

Constraints and entropy in a model of network evolution

¹ Moogsoft Inc, 1265 Battery Street, San Francisco, California 94111, USA
² School of Engineering and Informatics, University of Sussex, Brighton BN1 9RH, UK
³ Centre for Networks and Collective Behaviour, and Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
⁴ Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Brighton BN1 9QH, UK

^a e-mail: p.tee@sussex.ac.uk

Received: 27 March 2017
Received in final form: 20 July 2017
Published online: 20 November 2017

Abstract

Barabási–Albert’s “Scale Free” model is the starting point for much of the accepted theory of the evolution of real world communication networks. Careful comparison of the theory with a wide range of real world networks, however, indicates that the model is in some cases, only a rough approximation to the dynamical evolution of real networks. In particular, the exponent γ of the power law distribution of degree is predicted by the model to be exactly 3, whereas in a number of real world networks it has values between 1.2 and 2.9. In addition, the degree distributions of real networks exhibit cut offs at high node degree, which indicates the existence of maximal node degrees for these networks. In this paper we propose a simple extension to the “Scale Free” model, which offers better agreement with the experimental data. This improvement is satisfying, but the model still does not explain why the attachment probabilities should favor high degree nodes, or indeed how constraints arrive in non-physical networks. Using recent advances in the analysis of the entropy of graphs at the node level we propose a first principles derivation for the “Scale Free” and “constraints” model from thermodynamic principles, and demonstrate that both preferential attachment and constraints could arise as a natural consequence of the second law of thermodynamics.