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Thanks Mark for the clarification and note about what might eventually
be interesting about the scale-free research.  I should say that, when I
said that human interaction networks are obviously not scale-free, I
certainly did not mean to suggest that they are normally distributed.
As we well know from many network studies, indegree is usually skewed to
the right and sometimes very much so (I have even co-authored an obscure
paper that shows this to be the case in a college campus network and
that members of the network misperceive the implications for their
relative popularity).  And yet, as Martina pointed out, there are
obvious limits to how skewed they can be-- limits that do not apply to
such things as the network of hyperlinks that constitute the web.  And
Carter's observations reinforce this point.

A final thought per Mark's example using outdegree: I would hypothesize
that outdegree and indegree are actually pretty different in terms of
their right-skewness. Consider the "fame network" I alluded to [Name
generator asked of all people on the planet about one another: "Please
identify all other people that you recognize."].  I would submit that
indegree will be a lot more right-skewed than outdegree.

-----Original Message-----
From: Social Networks Discussion Forum [mailto:[log in to unmask]] On
Behalf Of Mark S. Handcock
Sent: Monday, January 27, 2003 3:26 PM
To: [log in to unmask]
Subject: Re: Erroneous facts / NyT article on social networks (scale
free)

*****  To join INSNA, visit http://www.sfu.ca/~insna/  *****

Martina and Ezra make some important points about the importance of
interdisciplinary approaches to network models and the sociology of
science. In
that context, here is a brief response to David's particular questions.

The notion of "scale free networks" is often used broadly and vaguely.
One very
common use focuses on the univariate distribution some characteristic of
a
network such as the aggregate distribution of out-degree of the nodes.
If the
distribution can be approximately summarized by a measure of level and
modest
symmetric variation about that level this literature refers to it as
having a
characteristic "scale". Often this is associated with a notion that the
distribution is Gaussian. The term "scale free" is used to describe
distributions that do not have this shape e.g., very non-Gaussian
distributions.
Hence networks were the distribution of out-degrees (say) is very right
skewed
are often called "scale free networks".

Much confusion occurs because of this. Many people with little
experience with
network processes find right-skewed distributions surprising. However, I
have
not spoken with a social network person who expected them all to be
approximately Gaussian in shape. In this sense it is not new nor
surprising. In
particular, statisticians are bemused by the entire enterprise and write
it of
as just more bad science. Statisticians made a cottage industry at the
beginning
of last century characterizing discrete distributions to represent such
behavior. I suggest looking at the presidential address of Maurice
Kendall in
his 1960 inaugural address to the Royal Statistical Society (on JSTOR
JRSS
1961). Kendall reflected that after many years of simple curve fitting
exercises
that for statistical modeling in the social sciences to mature as a
scientific
discipline,
it must move into tests of processual models.  Unfortunately, this
message has
been often been lost in the passage of time and the segmentation of
scientific
enterprise.  So here is what is new about "scale free" networks. Some of
the
more scientifically people in the "scale free" pool focus on the
stochastic
models that may underlay these phenomena, some are getting to the point
of
testing them empirically against alternative models. It is still at its
early
stages. I should also note that the focus has been on very simple models
for the
network (e.g., that it is randomly mixing with respect to the degree
distribution of the nodes). These are unlikely to be true in most real
networks
of interest where network structure and attribute based mixing are
important.

In musing on your second question: "what networks tend to be scale free,
and
what networks not?", the real question is what import is it if, say, the
degree
distribution is right-skewed. Or more generally, what can we say
usefully about
a network based on characterizing summary statistics like these. In this
sense
the social network community has been doing this for a long time. The
"scale
free" folks are just getting started.

Mark

-------------------------------------------------
Mark S. Handcock
Professor of Statistics and Sociology
Department of Statistics, C014-B Padelford Hall
University of Washington, Box 354322     Phone: (206) 221-6930
Seattle, WA 98195-4322.          FAX:   (206) 685-7419
Web: www.stat.washington.edu/handcock
internet: [log in to unmask]

----- Original Message -----
From: "David Lazer" <[log in to unmask]>
To: <[log in to unmask]>
Sent: Monday, January 27, 2003 8:58 AM
Subject: Re: Erroneous facts / NyT article on social networks


> *****  To join INSNA, visit http://www.sfu.ca/~insna/  *****
>
> A couple of questions:
>
> (1) what exactly is and is not new in the recent research in scale
free
> networks?  clearly, the importance of "hubs" has been known for a long
> time, although the power law research clearly accents those findings.
As
> the e-mails below discuss, there was some work on this going back on
"scale
> free" type ideas to at least the 1940s, and I've heard some allusions
to
> work going back to the 1920s.  exactly what was found, however, is
still a
> vague to me, and certainly had not been part of the core of social
network
> attention for a while.  Some other power law type research, e.g. city
size,
> earthquake distribution, war casualty distributions, etc, goes back
many
> decades, with spikes in attention with the work on "self-organized
> criticality" in the 90s, and, before that, on firm size in the econ
lit in
> the 1950s and 60s.  (I also bet that number of responses to socnet
e-mails
> is power law distributed, most queries generating few responses, and a
few,
> such as this generating a large number.)
>
> (2) what networks tend to be scale free, and what networks not?  The
> interpersonal data I tend to work with I'm pretty sure tend to be
normally
> distributed.  Many other kinds of networks, as Barabasi and others
have
> shown, are power law distributed in in-degree.  If one were to survey
> social network data sets, and categorize them by type of distribution
of
> in-degree, what would the categories be, and what would be the
variables
> underlying those categories?  Has this been done?
>
> David
>
>
>
>                       [log in to unmask]
>                       Sent by:                 To:
[log in to unmask]
>                       [log in to unmask]         cc:
>                       .EDU                     Subject: Re: Erroneous
facts /
NyT article on social networks
>
>
>                       01/26/2003 09:24
>                       PM
>                       Please respond
>                       to buttsc
>
>
>
>
>
>
> *****  To join INSNA, visit http://www.sfu.ca/~insna/  *****
>
> Mark Newman wrote:
> > And Rapoport touched on the same ideas even earlier in his work
> > on friendship networks, although he didn't specifically discuss
> > power-law degree sequences.
> >
>
> Indeed.  For that matter, there was a lot of very wonderful technical
> work by physicists, biologists, and others on both social and
biological
> networks back in the late 1940s/early 1950s in the _Bulletin of
> Mathematical Biophysics_ (of which Rapoport's work was part).  My
sense
> is that there is a fair amount of awareness of this literature within
> the modern network community, but I'm not sure to what extent the
> "scale-free" crowd is cognizant of it....
>
> > Still, as David Gibson points out, one shouldn't blame Duncan Watts
for
> > this.  In fact, Duncan gives ample credit to the pioneers of the
field
> > in his new book.
>
> I also noted that he was quoted as asking people to tone down the
> hype....not that I expect the message to sink in.  Looks like we're in
> for a bubble/crash cycle here -- I hope someone here is collecting
data
> on this!
>
> -Carter
>
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