For those who do or don't want to separate from the small world, here's my
few thoughts:

On Sun, 27 Jan, Ed Vielmetti wrote:
>Ah, but real world graphs are not random.


Precisely. Human social connections are most unlikely to be formed on a
random basis. There are plenty of other more plausible (but still
stochastically-based) mechanisms for locally generated social processes
that could result in tie formation. For instance, a tendency towards
transitivity, or clustering as it is usually referred to in small world
studies. Once we allow transitivity, then the resulting distribution of
graphs can have noticeably different properties from a distribution of
Bernoulli graphs (or U given L if we want to control for the number of
edges.) And it’s a pretty simple matter to produce graph distributions that
are not small world. In a p* framework, with Markov random graph
parameters, for instance, it is easy enough to simulate Duncan Watts’s
“caveman” graphs by using a heavy transitivity parameter. But in some
current work, we've also found some more interesting (and non-degenerate)
non-small world distributions. So the relevance of "the small world", or
any of these other distributions, remains an empirical question.

In this recent discussion, Blyden Potts suggested that the quality of the
tie, beyond simply “knowing the person”, may be crucial. It’s a good
comment. Which type of social network are we dealing with? For most of us,
a network of “knowing the person” is just not where the action is. Many of
us deal with networks such as trust or sexual partnerships or working
relationships. Different types of network surely implicate different types
of local social processes, and the global outcomes are not at all clear.
And “global” here need not be global in the sense of  the whole human
species -  if we're talking about an organisation or a political system,
then that's what counts as global.

For me, the small world impetus has been good in helping (indeed, making
me) think about this local/global nexus in new ways. Clearly the global
structure is in some sense important (else why do we network analysts
bother?), but that sense will again depend on the type of network and what
it transmits or constrains. I doubt that we have exhausted the
possibilities of thinking about the global, but connectivity and the
distribution of geodesics is clearly one way, hence the relevance of small
world. But I also have some sympathy for those who get irritated with
continuing focus on small world per se. Because it's not the only type of
global pattern, nor is it necessarily the most important (nor is it
necessarily even categorical: can’t graphs display varying degrees of
“small-worldedness”?). There are many other types of “worlds” that are to
be seen in our graphs. We need to start with some reasonable hypotheses
about local social processes, and see how these play out in terms of global
structures - and if those structures are not small worlds, so much the
worse for the small world. But the further step, and perhaps the hardest,
is to think about the ramifications of the global pattern given the
substantive content of the network under consideration. I guess that’s
where the payoff is.

Thanks,

Garry

____________________
Dr Garry Robins,
Department of Psychology,
School of Behavioural Science,
The University of Melbourne,
Victoria 3010,
Australia.

Tel: 61 3 8344 6354
Fax: 61 3 9347 6618
Web: http://www.psych.unimelb.edu.au/staff/robins.html