qp.clique {qp} | R Documentation |
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
qp.clique(qp.output, N, threshold.lim=c(0,1), breaks=5, plot.image=TRUE, exact.calculation=FALSE, approximation.iterations=100)
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:
|
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 |
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).
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 |
Robert Castelo and Alberto Roverato
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
data(jmlr06data) qp.clique(qp.out.bd5.N20.q10,20)