  Percolation analysis has long been used to quantify the connectivity of the
cosmic web. Most of the previous work is based on density fields on grids. By
smoothing into fields, we lose information about galaxy properties like shape
or luminosity. Lack of mathematical model also limits our understanding of
percolation analysis. In order to overcome these difficulties, we have studied
percolation analysis based on discrete points. Using a Friends-of-Friends (FoF)
algorithm, we generate the S-bb relation, between the fractional mass of the
largest connected group (S) and the FoF linking length (bb). We propose a new
model, the Probability Cloud Cluster Expansion Theory (PCCET) to relate the
S-bb relation with correlation functions. We show that the S-bb relation
reflects a combination of all orders of correlation functions. Using N-body
simulation, we find that the S-bb relation is robust against redshift
distortion and incompleteness in observation. From the Bolshoi simulation, with
Halo Abundance Matching (HAM), we have generated a mock galaxy catalogue. Good
matching of the projected two-point correlation function with observation is
confirmed. However, comparing the mock catalogue with the latest galaxy
catalogue from SDSS DR12, we have found significant differences in their S-bb
relations. This indicates that the mock galaxy catalogue cannot accurately
retain higher order correlation functions than the two-point correlation
function, which reveals the limit of HAM method. As a new measurement, S-bb
relation is applicable to a wide range of data types, fast to compute, robust
against redshift distortion and incompleteness, and it contains information of
all orders of correlation function.
