Distances in random graphs with finite mean and infinite variance degrees
Remco W. van der Hofstad, Eindhoven University of Technology
Gerard Hooghiemstra, Delft University of Technology
Dmitri Znamenski, EURANDOM
Abstract
In this paper we study typical distances in
random graphs with i.i.d. degrees of which the tail of the
common distribution function is regularly varying with
exponent 1-τ.
Depending on the value of the parameter τ
we can distinct three cases: (i) τ>3, where
the degrees have finite variance, (ii) τ in
(2,3), where the degrees have infinite variance,
but finite mean, and (iii) τ in (1,2), where
the degrees have infinite mean.
The distances between two randomly chosen nodes
belonging to the same connected component,
for τ>3 and τ in (1,2), have been studied in
previous publications, and we survey these results here.
When τ in (2,3), the graph distance centers
around 2 log log{N}/| log(τ-2)|. We present
a full proof of this result, and study the fluctuations
around this asymptotic means, by describing the asymptotic
distribution. The results presented here improve upon
results of Reittu and Norros, who prove an upper bound only.
The random graphs studied here can serve as models for
complex networks where degree power laws are observed;
this is illustrated by comparing the typical distance in
this model to Internet data, where a degree power law
with exponent τ approximately 2.2 is observed
for the so-called Autonomous Systems (AS) graph.
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