Summary: Comment on "Families and clustering in a natural numbers network"
Jeffrey D. Achter*
Colorado State University, Fort Collins, Colorado 80523, USA
(Received 23 April 2004; published 29 November 2004)
Corso [Phys. Rev. E 69 036106 (2004)] constructs a family of graphs from subsets of the natural numbers,
and numerically estimates diameter, degree and clustering. We give exact asymptotic formulas for these
quantities, and thereby argue that number theory is a more appropriate tool than simulation.
DOI: 10.1103/PhysRevE.70.058103 PACS number(s): 89.75.Fb, 02.10.De
In , the author examines an infinite class of finite
graphs constructed from the natural numbers. Small-world
networks--that is, those which are simultaneously of low
connectivity, small distance, and high clustering coefficient
--arise in stunningly diverse contexts. Given this, one
would like to estimate degree distributions and related data
for natural families of graphs.
For a given natural number X,  constructs a graph M
=M X . The vertices are the natural numbers 2, ... ,X, and
vertices (corresponding to) m and n are connected exactly if
they share a nontrivial divisor. Using a combination of nu-
merical experiments and heuristic arguments, Ref.  ad-