  High-density (HD) percolation describes the percolation over specific
$\kappa$ -clusters, which are the compact sets of sites each connected to
$\kappa$ nearest filled sites at least. It takes place in the classical
patterns of independently distributed sites or bonds in which the ordinary
percolation transition also exsists. Hence, the study of series of $\kappa$
-type percolations amounts to the description of structure of classical
clusters for which $\kappa$ -clusters constitute $\kappa$ -cores nested one
into another. Such data are needed for description of a number of physical,
biological information and other properties of complex systems on random
lattices, graphs and networks. They range from magnetic properties of
semiconductor alloys to anomalies in supercooled water and clustering in
biological and social networks. Here we present the statistical mechanics
approach to study HD bond percolation on arbitrary graph. It is shown that
generating function for $\kappa$ -clusters' size distribution can be obtained
from partition function of specific $q$-state Potts-Ising model in $q \to 1$
limit. Using this approach we find exact $\kappa$ -clusters' size distribution
for Bethe lattice and Erdos Renyi graph. The application of the method to
Euclidean lattices is also discussed.
