Although Pat has pretty well summarized my basic position (the network
boundaries of interest to you should be based on your prior knowledge of
the processes under study), I wanted to use a few brave statements of
Bill's to add to the profusion of soapbox commentary:
Bill Richards wrote:
> Stacy --
> You are right about 10 being too small. There is simply not enough
> room in a network of 10 people for anything particularly
> interesting or complex to happen.
Irrespective of the truth or falsity of this (what does one mean by
"interesting or complex"?), I think it is important to point out that
this is irrelevant for many research questions (the present one possibly
excepted). If one is studying a phenomenon which acts on networks of
size 10 or less, I would think that one would then need to study
networks of size 10 or less; if this means that certain "interesting" or
"complex" things are eliminated, then so what? Many "interesting"
things can't happen within certain types of systems, but this doesn't
give one license to arbitrarily decide that one's system is other than
the data says it is. (You can't have attractors within Hamiltonian
systems, for example, but this doesn't mean that we can declare the
undamped pendulum to be non-Hamiltonian because we're bored with it.
Undamped pendula _are_ boring! :-)) Note that I would say that you
certainly _should_ switch models if the _data_ suggests that your
initial model is too restrictive....but the decision should not be based
on how "interesting" or otherwise aesthetically appealing the new model
Incidentally, I don't think that Bill intended his comments to apply
beyond the present context (in which the poster indicated an explicit
desire to maximize differences in scores), but I thought it important to
pontificate about this just in case. :-)
> A network with 120 will be large enough for a range of interesting
> phenomena to be seen. Do you want to just demonstrate an analytic
> technique or are you interested in studying real-world phenomena?
> For the former, use a network that is big enough to let you
> demonstrate the abilities of your technique. For the latter, use a
> network that is more than "just large enough" to show the kind of
> things you are interested in. If you don't, you take the risk of
> trivializing your problem or stacking the dice by limiting the
> range of what is possible.
> Personally, I like bigger better than smaller.
I wonder about this. While there are certainly more large networks than
small ones (and more ways to get unusual behavior), the distributions of
many network features tend to become much more concentrated in large
graphs; hence, a priori you may find that there is _less_ to see in a
large structure than in a small one. (Again, whether or not that is a
legitimate concern is another matter.) Certainly, this seems to be the
for a number of graph-level indices (degree centralization, Krackhardt
hierarchy, Krackhardt connectedness, betweenness centralization) over
the set of all graphs conditional on order and density. For degree (and
hence, with some tweaks, degree centralization) the convergence follows
pretty obviously from the law of large numbers, but simulation results
suggest increasing concentration for the others as well. (Everett,
Borgatti, and whomever also have some interesting findings about the
very high correlations between (node-level) degree and betweenness
scores in large graphs, which I wish they'd publish. Hint hint.)
Of course, social networks could be exceptions to the rule, but most of
the ones I've looked at don't seem to be. <shrug> Given the
combinatorial constraints, large interpersonal networks would almost
_have_ to exhibit increasing concentration on most of the standard
measures. Someone with a counterexample is probably out there somewhere
waiting to whack me over the head, but I'd still tend to bet with the
house on this one.
OK, I'll get off the soapbox now. Prepare the 2x4....