qp.clique {qp}R Documentation

Relationship between non-rejection rate and maximum clique size

Description

Using the output of qp.search this function calculates the maximum clique size as a function of the minimum threshold on the non-rejection rate for removing an edge

Usage

qp.clique(qp.output, N, threshold.lim=c(0,1), breaks=5,
          plot.image=TRUE, exact.calculation=FALSE,
          approximation.iterations=100)

Arguments

qp.output output of qp.search
N sample size
threshold.lim range of the non-rejection rate threshold on which calculate the funcion
breaks one of:
  • a vector giving the breakpoints along the range defined by threshold.lim,
  • a single number giving the number of equidistant breakpoints that divide the range defined by threshold.lim.
plot.image when this flag is set to TRUE, the qp.clique plot is produced
exact.calculation when this flag is set to TRUE, the exact maximum clique size is calculated and when set to FALSE a lower bound is calculated instead
approximation.iterations number of iterations performed to calculate the lower bound on the clique number of each graph. It applies only when exact.calculation=FALSE

Details

The qp.clique plot provides information on the graphs potentially selected by specifying different values of the threshold. Every circle in the plot corresponds to a graph and has three values associated with it: the threshold value used to construct the graph (horizontal axis); the number of vertices of the largest clique of the graph (vertical axis); the percentage of present edges in the graph (number inside the plot, beside the circle). Furthermore, adjacent circles are joined by a line and the dotted horizontal line corresponds to the sample size N.

Beware that setting exact.calculation=TRUE and giving breakpoints between 0.95 and 1.0, may result into very dense graphs which can lead to a very long time of computation due to the NP-completeness of the problem of calculating the size of the largest clique which is therefore bounded by an exponential growth of the running time as function of the graph density (cf.~Pardalos and Xue, 1994).

The lower bound on the maximum clique size is calculated by ranking the vertices by their connectivity degree, put the first vertex in a set and go through the rest of the ranking adding those vertices to the set that form a clique with the vertices currently within the set. Once the entire ranking has been examined a large clique should have been built and hopefully the largest one. This process is repeated a number of times (approximation.iterations) each of which the ranking is altered with increasing levels of randomness acyclically (altering 1 to $p$ vertices and again). Larger values of approximation.iterations should provide tighter lower bounds and eventually the exact maximum clique size (the clique number).

Value

threshold threshold on the non-rejection rate that provides the maximum clique size that is strictly smaller than the sample size N
size maximum clique size strictly smaller than the sample size N

Author(s)

Robert Castelo and Alberto Roverato

References

Castelo, R. and Roverato, A. (2006). A robust procedure for Gaussian graphical model search from microarray data with p larger than n, J. Mach. Learn. Res., 7:2621-2650

Pardalos, P.M. and Xue, J. (1994). The maximum clique problem, J. Global Optim., 4:301-328

See Also

qp.search

Examples

data(jmlr06data)
qp.clique(qp.out.bd5.N20.q10,20)

[Package qp version 0.2-1 Index]