  There is increasing evidence that dense networks occur in on-line social
networks, recommendation networks and in the brain. In addition to being dense,
these networks are often also scale-free, i.e. their degree distributions
follow $P(k)\propto k^{-\gamma}$ with $\gamma\in(1,2]$. Models of growing
networks have been successfully employed to produce scale-free networks using
preferential attachment, however these models can only produce sparse networks
as the numbers of links and nodes being added at each time-step is constant.
Here we present a modelling framework which produces networks that are both
dense and scale-free. The mechanism by which the networks grow in this model is
based on the Pitman-Yor process. Variations on the model are able to produce
undirected scale-free networks with exponent $\gamma=2$ or directed networks
with power-law out-degree distribution with tunable exponent $\gamma \in
(1,2)$. We also extend the model to that of directed $2$-dimensional simplicial
complexes. Simplicial complexes are generalization of networks that can encode
the many body interactions between the parts of a complex system and as such
are becoming increasingly popular to characterize different data sets ranging
from social interacting systems to the brain. Our model produces dense directed
simplicial complexes with power-law distribution of the generalized out-degrees
of the nodes.
