WWW and Internet models from 1955 till our days and the ``popularity is attractive'' principle

TL;DR

This paper compares different models of WWW and internet growth, emphasizing the role of the 'popularity is attractive' principle in network self-organization and providing exact solutions for node degree distributions.

Contribution

It demonstrates that the models of Bornholdt and Ebel and Barabási and Albert are similarly capable of modeling individual node growth and offers exact solutions for their degree distributions.

Findings

Models can estimate deviations of the scaling exponent from 2.

Both models provide similar possibilities for individual node growth.

The 'popularity is attractive' principle is fundamental in network self-organization.

Abstract

We note that the model discussed in the communication of S. Bornholdt and H. Ebel (World Wide Web scaling exponent from Simon's 1955 model, cond-mat/0008465) is the particular case of the model considered and solved exactly in our paper, cond-mat/0004434. These models may be used for estimation of the order of the deviation of the scaling exponent from 2 both for the distributions of incoming links and links coming out from nodes but not for the obtaining some specific values of the exponents from the WWW growth data. We emphasize that, unlike the statement of Bornholdt and Ebel, both the network under consideration and the model of Barab\'{a}si and Albert provide quite equal possibilities for individual growth. There is no great difference between them in this respect. The resulting distributions for individual nodes and arising scaling relations have been obtained in our paper, cond-mat/0004434. We discuss briefly the modern state of art in the physics of the evolving networks and the great role of the general principle -- {\em popularity is attractive} -- in the self-organization of complex communications networks, in physics of nonequilibrium phenomena, and in Nature.