sna (version 0.41)

efficiency: Compute Graph Efficiency Scores

Description

efficiency takes a graph stack (dat) and returns the Krackhardt efficiency scores for the graphs selected by g.

Usage

efficiency(dat, g=1:stackcount(dat), diag=FALSE)

Arguments

dat

A graph or graph stack

g

Index values for the graphs to be utilized; by default, all graphs are selected

diag

TRUE if the diagonal contains valid data; by default, diag==FALSE

Value

• A vector of efficiency scores

Details

Let $G={G_1,G_2,\dots,G_n}$ be a digraph with weak components $G_1,G_2,\dots,G_n$. For convenience, we denote the cardinalities of these graphs' vertex sets by $|V(G)|=N$ and $|V(G_i)|=N_i$, $\forall i \in 1,\dots,n$. Then the Krackhardt efficiency of $G$ is given by

$$1-\frac{|E(G)| - \sum_{i=1}^n \left(N_i-1\right)}{\frac{N^2-N}{2}\sum_{i=1}^n \left(N_i-1\right)}$$

which can be interpreted as 1 minus the proportion of possible ``extra'' edges (above those needed to connect the existing components) actually present in the graph. A graph which an efficiency of 1 has precisely as many edges as are needed to connect its components; as additional edges are added, efficiency gradually falls towards 0.

Efficiency is one of four measures (connectedness, efficiency, hierarchy, and lubness) suggested by Krackhardt for summarizing hierarchical structures. Each corresponds to one of four axioms which are necessary and sufficient for the structure in question to be an outtree; thus, the measures will be equal to 1 for a given graph iff that graph is an outtree. Deviations from unity can be interpreted in terms of failure to satisfy one or more of the outtree conditions, information which may be useful in classifying its structural properties.

References

Krackhardt, David. (1994). ``Graph Theoretical Dimensions of Informal Organizations.'' In K. M. Carley and M. J. Prietula (Eds.), Computational Organization Theory, 89-111. Hillsdale, NJ: Lawrence Erlbaum and Associates.

See Also

connectedness, efficiency, hierarchy, lubness, gden

Examples

#Get efficiency scores for graphs of varying densities
efficiency(rgraph(10,5,tprob=c(0.1,0.25,0.5,0.75,0.9)))