Internet Mathematics Vol. 2, No. 4: 431523
Towards a Theory of
ScaleFree 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 “scalefree” 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 scalefree 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 scalefree
. This structural view can be related to previ
ously studied graph properties such as the various notions of selfsimilarity, 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 theoremproof style of exposition, but who may be unfamiliar
with the existing literature on scalefree networks.
1.
Introduction
One of the most popular topics recently within the interdisciplinary study of
complex networks has been the investigation of socalled “scalefree” graphs.
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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 manmade 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, routerlevel graphs, and ASlevel graphs). Because these models
exhibit features not easily captured by traditional Erd¨
osReny´
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].
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 Spring '07
 DavidAldous
 Graph Theory, Normal Distribution, The Land, Probability theory, Cumulative distribution function, Lun Li

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