rolecensus.htm

2014-10-15

ROLECENSUS
– A Tool for describing and identifying patterns of embedding vertices in their
„triadic environment“.

In their
1987 paper Strukturbeschreibung von Positionen
in sozialen Beziehungsnetzen Hans J. Hummell and Wolfgang Sodeur proposed a conceptual tool to
describe patterns of embedding vertices
in their „triadic environment“, which they called *Positionenzensus*.

In
addition, Wolfgang Sodeur presented a computer routine that could enumerate all
possible patterns in simple binary directed network (with some 300 vertices in
total). This routine was included in a bundle of network algorithms by Wolfgang
Sodeur, called *NETZDIAL*, written in FORTRAN.

The general
idea of the *Positionenzensus* is to
characterize every vertex of a network („Ego“) by the configuration of
outgoing, incoming, reciprocated as well
as absent links to every pair of other
vertices („Alter 1“ and „Alter 2“) together with the configuration of links (incoming,
outgoing, reciprocated, absent) among these two Alteri.

This idea
is an extension of the well known
concept of *triadcensus* developed by
James A. Davis, Paul W. Holland and Samuel Leinhardt, software for which is
available in many packages for network analysis (among others in *UCINET*, *PAJEK* and the packages <*sna>/<statnet>*
or <*igraph>* for R.) It is a
census of triadic configurations, seen from the perspetive of the vertices
(nodes) of a triad. S. Messing et. al. later called it *node-level triadcensus* (see below).

In
comparison to the triadcensus, which classifies all triads into
16 classes of „structurally equivalent“ triads („types“ of triads), in *Positionenzensus* there are 36 structurally different „types“ of Ego’s embeddedness in his „triadic
environment“. Again, at one extreme there is a situation, where Ego is
unrelated to two Alteri, who are also unrelated; at the other extreme the
situation where Ego is „symmetrically“ related to both Alteri, who are also
„symmetrically“ related among themselves.

A table,
which maps all 64 labled triads, which are possible among three specific
vertices(nodes) A,B,C to the 36 different types of embeddedness of the nodes A, B and C into
their “triadic environment” can be found here. (N.B. This table uses the original numbering of “triadic positions” of
Hummell/Sodeur (1987))

In a
network with g vertices, every vertex can be combined with (g-1)*(g-2) pairs of
remaining vertices. Therefeore, every vertex can be characterized by the
frequency distribution of his (g-1)*(g-2) configurations along the above
mentioned 36 types of embeddedness. Thus, the *Positionenzensus* of a network with g vertices results in an
g*36-matrix which can be further analyzed, as e.g. in positional/role analyses.

In 1990
Ronald Burt introduced this concept to the English-speaking community in an
article „Detecting Role Equivalence“ published in Social Networks, 12,
83-97. His visual presentation and numbering of
the 36 types
rendered the concept, which more apropiately he called *rolecensus*, intuitively much more accessible. At the same time he
included the respective algorithm in his package of network routines, known as *STRUCTURE*.

Meanwhile
(2010) Solomon Messing, Sean J. Westwood, Mike Nowak, and Dan McFarland wrote a
routine *triadcensus* (N.B. without
dot) as part of their package *triads* which can be used under R. They
call this routine *node-level triad census
for networks* which is even more approriate.

Their
package <triads> which however depends on version 0.5 of the package
<igraph>, can still be found on *CRAN.*

Empirical
applications of the concept can be found in: Mark Trappmann, Hans J. Hummell and
Wolfgang Sodeur, 2011: Strukturanalyse sozialer Netzwerke: Konzepte, Modelle,
Methoden; Wiesbaden: Verlag für Sozialwissenschaften, where some comments are given
concerning the relations of the *rolecensus* to concepts of positional and
role analysis (especially to regular equivalence).

The
original intention of H.J. Hummell and W. Sodeur was not restricted to
analyzing the *rolecensus* in it’s
„raw“ form, giving all 36 columns the same weight, but to collapse/combine
columns according to theoretical considerations. Thus, they tentatively
classified the rolecensus with respect to concepts of domination/subordination,
clique-membership/isolation or mediation/brokerage etc.. Concerning the last
concept, there are obvious similarities with R.V. Gould and R.M. Fernandez’
(1989) concept of brokerage roles. The essential difference is that their
brokerage roles suppose in addition to the configuration of relations a
partition of the nework in two or more groups, whereas Hummell and Sodeur’s
concept focuses on the relational pattern per se.

If you use
either Wolfgang Sodeur’s *NETZDIAL* or Ronald Burts *STRUCTURE * or Solomon Messing
et.al. *triadcensus, * you should be aware that the sequence and numbering
of the 36 types differ. Sodeur’s *NETZDIAL* outputs both: the original sequence and numbering of Hummell/Sodeur
(1987) and the later one of R.Burt (1990) which
intuitively is the most plausible; Solomon Messing et.al. sequence and
labelling is also different.

Those using
R as a framework for statistical computing will find a script here
which runs independently of other R packages for network analysis but is very
slow.