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.