PURPOSE Calculate Krackhardt's graph theoretic dimension of organizations
DESCRIPTION Krackhardt has proposed four graph theoretic measures which capture the extent to which a network forms a hierarchical structure. These are based on the extent to which a network is the same as a connected out-tree. These are connectivity the extent to which the network is just one weak component. Hierarchy the extent to which paths are not reciprocated. Graph efficiency is the extent to which the underlying network has redundant edges that is edges whose deletion does not disconnect the component and Least upper boundedness (LUB) the extent to which pairs of actors have a unique common superior. The measures all range from zero to one and an out-tree will give a value of one on each measure.
Technically these are defined as follows:
The weak reachability matrix has an i,j th entry of one if there is a semipath connecting i with j and zero otherwise.
Connectivity is the density of the weak reachability matrix.
Hierarchy is one minus the dyad based reciprocity of the reachability matrix (not the weak matrix but the normal reachability matrix)
The inefficiency of a component of size k is the number of edges the underlying graph contains above (k-1) divided by (k-1)^2. Since a tree has k-1 edges the numerator is the number of redundant edges and the denominator is the maximum possible number of redundant edges. The efficiency is one minus the weighted average component inefficiency weighted by component size.
Two actors in the same component are said to have an upper bound if there is a path from one vertex to them both (this vertex may be one of the actors). If one of the upper bounds is always on a path from all the other upper bounds it is called a least upperbound. The least upper bound fraction is the number of pairs of actors in each component that do not have a least upper bound divided by the maximum number of pairs of vertices that could not have a least upper bound, this is (k-1)(k-2)/2 for a component of size k. The LUB score is one minus the weighted average least upper bound component fraction weighted by component size.
PARAMETERS Input dataset:
Name of file containing dataset to be analyzed. Data type: Digraph.
Output dataset:(Default =GTD)
Name of UCINET datafile containing all the GTD measures.
Different measures can be selected by ticking the required box, the default is that all will be calculated.
LOG FILE The connectedness, hierarchy, efficiency and LUBedness for the network
REFERENCES Krackhardt D (1994) Graph Theoretic Dimensions of Informal Organizations in Computational Organizational Theory (K Carley and M Prietula eds) Hillside NJ: Lawrence Erlbaum Assoc. pp 89-111.