  In this article, we investigate several properties of high-dimensional random
Apollonian networks (HDRANs), including two types of degree profiles, the
small-world effect (clustering property), sparsity, and three distance-based
metrics. The characterizations of degree profiles are based on several rigorous
mathematical and probabilistic methods, such as a two-dimensional mathematical
induction, analytic combinatorics, and P\'{o}lya urns, etc. The small-world
property is uncovered by a well-developed measure---local clustering
coefficient, and the sparsity is assessed by a proposed Gini index. Finally, we
look into three distance-based properties; they are total depth, diameter and
Wiener index.
