  Degree distribution of nodes, especially a power law degree distribution, has
been regarded as one of the most significant structural characteristics of
social and information networks. Node degree, however, only discloses the
first-order structure of a network. Higher-order structures such as the edge
embeddedness and the size of communities may play more important roles in many
online social networks. In this paper, we provide empirical evidence on the
existence of rich higherorder structural characteristics in online social
networks, develop mathematical models to interpret and model these
characteristics, and discuss their various applications in practice. In
particular, 1) We show that the embeddedness distribution of social links in
many social networks has interesting and rich behavior that cannot be captured
by well-known network models. We also provide empirical results showing a clear
correlation between the embeddedness distribution and the average number of
messages communicated between pairs of social network nodes. 2) We formally
prove that random k-tree, a recent model for complex networks, has a power law
embeddedness distribution, and show empirically that the random k-tree model
can be used to capture the rich behavior of higherorder structures we observed
in real-world social networks. 3) Going beyond the embeddedness, we show that a
variant of the random k-tree model can be used to capture the power law
distribution of the size of communities of overlapping cliques discovered
recently.
