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Hello,

I have a query regarding clustering coefficients. Watts and Strogatz[1]
define it as follows:

"Suppose that a vertex v has kv neighbours; then at most kv(kv-1)/2
edges can exist between them (this occurs when every neighbour of v is
connected to every other neighbour of v). Let Cv denote the fraction of
these allowable edges that actually exist. Define C as the average of
Cv over all v."

And Watts's book "Small Worlds" has a very similar definition. Consider
a node with 1 neighbour; Cv for that node is clearly ill-defined (as it
involves a divide by 0). There are two approaches to take when
calculating C in a network with n nodes:

i)add up Cv for all nodes where Cv is defined, and divide by n
ii)add up Cv for all nodes where Cv is defined, and divide by the
number of nodes for which Cv was defined

i) seems to me to be the "obvious" interpretation of Watt's definition,
but has the property that a large number of nodes of degree 1 will
cause C to be small.

I'd be interested to know what others think about this (or indeed to be
pointed at a publication that discusses the two approaches).

Thanks,

Matthew

[1] nature 393:440-442,1998
--
Matthew Vernon MA VetMB LGSM MRCVS
Farm Animal Epidemiology and Informatics Unit
Department of Veterinary Medicine, University of Cambridge

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