  We propose the n-clique network as a powerful tool for understanding global
structures of combined highly-interconnected subgraphs, and provide theoretical
predictions for statistical properties of the n-clique networks embedded in a
complex network using the degree distribution and the clustering spectrum.
Furthermore, using our theoretical predictions, we find that the statistical
properties are invariant between 3-clique networks and original networks for
several observable real-world networks with the scale-free connectivity and the
hierarchical modularity. The result implies that structural properties are
identical between the 3-clique networks and the original networks.
