  Real-world networks are rarely static. Recently, there has been increasing
interest in both network growth and network densification, in which the number
of edges scales superlinearly with the number of nodes. Less studied but
equally important, however, are scaling laws of higher-order cliques, which can
drive clustering and network redundancy. In this paper, we study how cliques
grow with network size, by analyzing several empirical networks from emails to
Wikipedia interactions. Our results show superlinear scaling laws whose
exponents increase with clique size, in contrast to predictions from a previous
model. We then show that these results are in qualitative agreement with a new
model that we propose, the Local Preferential Attachment Model, where an
incoming node links not only to a target node but also to its higher-degree
neighbors. Our results provide new insights into how networks grow and where
network redundancy occurs.
