The first-order properties like degree distribution of nodes and the clustering co-efficient have been the prime focus of research in the study of structural properties of networks. The presence of a power law in the degree distribution of nodes has been considered as an important structural characteristic of social and information networks. Higher-order structural properties such as edge embeddedness may also play a more important role in many on-line social networks but have not been studied before. In this research, we study the distribution of higher-order structural properties of a network, such as edge embeddedness, in complex network models and on-line social networks. We empirically study the embeddedness distribution of a variety of network models and theoretically prove that a recently-proposed network model, the random $k$-tree, has a power-law embedded distribution. We conduct extensive experiments on the embeddedness distribution in real-world networks and provide evidence on the correlation between embeddedeness and communication patterns among the members in an on-line social network. / Graduate