As shown on Table 7, each social network structural construct defined in Chapter 3 is operationalized with a particular social network variable. The table indicates the construct name, variable name, data source, and reference in the social network analysis
literature. Using the notational system defined in the previous subsection, definitions are first presented for the social network measures of “density,” “nodal degree,” “mean nodal degree,” and “standardized actor degree centrality.” These definitions are then used in defining the formal specification for each of the six social network variables below.
Table 7
Social Network Variables
Construct Name
Variable Name Data Source Social Network Analysis Reference
Group Closure
Group density Public forums Wasserman and Faust 1994 Core
Closure Core density Public forums, Project membership records
Wasserman and Faust 1994 Peripheral Two-Mode Closure Peripheral two- mode density Public forums, Project membership records
Borgatti, et. al. 1998 Wasserman and Faust 1994 Core
Bridging Core membership degree
SourceForge membership records, Project membership records
Wasserman and Faust 1994
Administrator
Bridging Administrator membership degree
SourceForge membership records, Project membership records
Wasserman and Faust 1994
Administrator
Centrality Administrator class centrality Public forums, Project membership records
Everett and Borgatti 1999
Density. The “density” of a graph is the actual number of lines in a graph as a
proportion of the total possible number of lines in the graph. Denoting density as ∆, the calculation for density is specified by the formula:
Nodal degree. The “nodal degree” of a node ni, denoted by d(ni), is the number of
lines that are incident with the node ni (Wasserman and Faust 1994). A node is incident with a line if that node is one of the unordered pair of nodes which defines the line (Wasserman and Faust 1994). Using sociometric notation, nodal degree is defined for a one-mode network as:
d(ni) = ∑all j xij = ∑all i xij
The nodal degree for the mode-1 actors in an affiliation network is defined as: d(ni) = ∑all j xij
Mean nodal degree. The “mean nodal degree” of a graph, denoted by d^, is the
average nodal degree for all nodes in the network. Applied to the actors in an affiliation network, mean nodal degree is:
d^ = ∑from i=1 to g d(ni) / g = 2L / g.
Standardized actor degree centrality. The “standardized actor degree centrality”
of a node ni, denoted by C'D (ni), is defined as:
C'D (ni) = d(ni) / (g-1). (Wasserman and Faust 1994)
The general social network measures defined above are now used in defining the specific social network variables to be used in this research.
Group density. The “group density” (GD) is the density of the “total conversation
network,” which is a one-mode network where actors are members of the focal project community and the relation is forum conversation.
Core density. The “core density” (CD) is the density of the “core conversation
network,” which is a one-mode network where the actors are members of the core subgroup of the focal project community and the relation is forum conversation16.
Peripheral two-mode density. The “peripheral two-mode density” (PTD) is the
density of the “periphery-core conversation network,” which is a two-mode network where the mode-1 actors are members of the peripheral subgroup, the mode-2 actors are members of the core subgroup, and the relation is forum conversation which is only defined for actor pairs containing one core actor and one peripheral actor. Centralization of the total conversation network was considered as a candidate for operationalizing the Peripheral Two-Mode Closure construct. However, peripheral two-mode density was chosen instead because it takes advantage of the explicit definition of the core and peripheral subgroups, while centralization implicitly defines a core-periphery structure using network properties.
Core membership degree. The “core membership degree” (CMD) is the mean
nodal degree (defined for an affiliation network) for all actors in the “core project membership network,” which is an affiliation network where the actors are core subgroup
members of the focal project community, the events are SourceForge projects, and the relation is project membership. Class centrality measures (Everett and Borgatti, 1999) could also have been used to operationalize the bridging constructs. However, the decision was made not to process the entire SourceForge membership network and therefore the average degree measure was selected because it only requires the collection of project membership data for the focal project actors.
Administrator membership degree. The “administrator membership degree”
(AMD) is the mean nodal degree (defined for an affiliation network) for all actors in the “administrator project membership network”, which is an affiliation network where the actors are administrator subgroup members of the focal project community, the events are SourceForge projects, and the relation is project membership.
Administrator class centrality. The “administrator class centrality” (ACC) is the
standardized actor degree centrality of the super-node in the “administrator-other conversation network17,” which is a special type of two-mode network (Everett and Borgatti 1999) where the administrator subgroup members are represented as a single mode-1 “super-actor,” the mode-2 actors are the other members of the focal project community, and the relation is forum conversation which is only defined for actor pairs containing the single super-actor and a mode-2 actor18. Degree centrality was chosen
17 If the super-node contains only one actor, then administrator class centrality is equivalent to standardized
nodal degree centrality for the one actor.
18 In this definition, the effect of the mode-1 “super-actor” is that ties from a single mode-2 actor to
over other possible centrality measures such as closeness or betweenness because it is a well-tested measure and there is no compelling reason to make other choices.