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Subject: Re: Which measures completely describe a network?
From: "Carter T. Butts" <[log in to unmask]>
Reply-To:[log in to unmask]
Date:Thu, 8 Sep 2005 03:27:50 -0700
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*****  To join INSNA, visit http://www.insna.org  *****

Jorge Colazo wrote:
> *****  To join INSNA, visit http://www.insna.org  *****
>
> Hello,
>
> What is the set of measures that can be used in an empirical study to most
> accurately and completely describe a given network?
>
...
 > Is there a conceptual (other than Friedkin 1991 AJS and related work)
 > framework that would allow one to choose a set of measures with
 > minimum overlap and maximum comprehensiveness to describe a given
 > network?


Yes -- the eigenstructure of a network's adjacency matrix provides a
complete characterization of the network in question.  Andrew Seary and
Bill Richards have written a number of accessible pieces on the
eigendecomposition of graphs; see, for instance,

Seary, A.J. and Richards, W.D.  2003.  "Spectral Methods for Analyzing
and Visualizing Networks: an Introduction."  In _Dynamic Social Network
Modeling and Analysis: Workshop Summary and Papers_.  Washington, DC:
National Academies Press.

A somewhat less ambitious approach is to identify a class of properties
or indices whose limiting values identify certain interesting classes of
structures.  For instance, early work by Harary, Davis, Holland,
Leinhardt, and others characterized various families of graphs resulting
from extreme dyad, triad, and signed cycle properties.  (Several papers
in this vein are in Leinhardt's classic 1977 compilation.)  Another
effort of this kind (mentioned in an earlier SOCNET exchange) is
Krackhardt's family of outtree indices.  This family is described in

Krackhardt, David.  1994.  "Graph Theoretical Dimensions of Informal
Organizations." In _Computational Organization Theory_.  Hillsdale, NJ:
Lawrence Erlbaum and Associates.

Less ambitious still is to give up on characterizing graphs per se, and
to focus on characterizing probability distributions on graph sets.
Surprisingly, there are many cases in which empirically interesting
families of graph distributions can be fully characterized by relatively
small numbers of structural indices.  A classic example is the family of
fixed-order Markov graphs, which can be characterized via counts of star
and triangle configurations.  A good introduction to some of these ideas
can be found in the recent volume edited by Carrington et al.  (See
especially chapter 10.)

I am not sure if this is the answer for which you were looking, but
perhaps it will be of use.  The question you raise is actually a fairly
subtle one, and is connected with a number of problems of both practical
and theoretical interest.  So, if the answer is less than simple, at
least it stems from a fruitful question!

Hope that helps,

-Carter

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