Mathematics and the internet: A source of enormous confusion and great potential

There has been a lot of activity in recent years in attempting to model complex systems like the internet, or social networks.  In 1999, Barabasi and Albert proposed a model (BA model) in which the internet is described as a growing random graph, where new nodes are added, and joined to pre-existing nodes with a finite number of links, with higher probability of attaching to links with higher degree.  This model, called preferential attachment model has generated a lot of activity (over 15000 citations).  In particular, in the BA model, the nodes have a power-law in the degree-distribution of nodes, and the nodes with high degree (called hubs) are critical for connectivity of the network.

 In this paper, the authors have argued that the  BA model is based on bad analysis of data, and  gives a description of internet that is  qualitatively incorrect about their predicted susceptibility to sabotage. They also propose an alternative approach to modelling networks like the internet as a growing network, engineered using heuristic optimization.

References: W. Willinger, D. Alderson, and J. C. Doyle, Notices of the American Mathematical Society, vol. 56, no. 5, pp. 586-599, 2009