willinger - Internet Mathematics Vol. 2, No. 4: 431-523...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Internet Mathematics Vol. 2, No. 4: 431-523 Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications Lun Li, David Alderson, John C. Doyle, and Walter Willinger Abstract. There is a large, popular, and growing literature on “scale-free” networks with the Internet along with metabolic networks representing perhaps the canonical examples. While this has in many ways reinvigorated graph theory, there is unfortu- nately no consistent, precise definition of scale-free graphs and few rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and that the most celebrated claims regarding the Internet and biology are verifiably false. In this paper, we introduce a structural metric that allows us to differentiate between all simple, connected graphs having an identical degree sequence, which is of particular interest when that sequence satisfies a power law relationship. We demonstrate that the proposed structural metric yields considerable insight into the claimed properties of SF graphs and provides one possible measure of the extent to which a graph is scale-free . This structural view can be related to previ- ously studied graph properties such as the various notions of self-similarity, likelihood, betweenness and assortativity. Our approach clarifies much of the confusion surround- ing the sensational qualitative claims in the current literature, and offers a rigorous and quantitative alternative, while suggesting the potential for a rich and interesting theory. This paper is aimed at readers familiar with the basics of Internet technology and comfortable with a theorem-proof style of exposition, but who may be unfamiliar with the existing literature on scale-free networks. 1. Introduction One of the most popular topics recently within the interdisciplinary study of complex networks has been the investigation of so-called “scale-free” graphs. © A K Peters, Ltd. 1542-7951/05 $ 0.50 per page 431
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
432 Internet Mathematics Originally introduced by Barab´ asi and Albert [Barab´ asi and Albert 99], scale- free (SF) graphs have been proposed as generic yet universal models of network topologies that exhibit power law distributions in the connectivity of network nodes. As a result of the apparent ubiquity of such distributions across many naturally occurring and man-made systems, SF graphs have been suggested as representative models of complex systems in areas ranging from the social sci- ences (collaboration graphs of movie actors or scientific coauthors) to molecular biology (cellular metabolism and genetic regulatory networks) to the Internet (web graphs, router-level graphs, and AS-level graphs). Because these models exhibit features not easily captured by traditional Erd¨ os-Reny´ ı random graphs [Erd¨ os and Renyi 59], it has been suggested that the discovery, analysis, and ap- plication of SF graphs may even represent a “new science of networks” [Barab´ asi 02, Dorogovtsev and Mendes 03].
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 93

willinger - Internet Mathematics Vol. 2, No. 4: 431-523...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online