  A fundamental property of complex networks is the tendency for edges to
cluster. The extent of the clustering is typically quantified by the clustering
coefficient, which is the probability that a length-2 path is closed, i.e.,
induces a triangle in the network. However, higher-order cliques beyond
triangles are crucial to understanding complex networks, and the clustering
behavior with respect to such higher-order network structures is not well
understood. Here we introduce higher-order clustering coefficients that measure
the closure probability of higher-order network cliques and provide a more
comprehensive view of how the edges of complex networks cluster. Our
higher-order clustering coefficients are a natural generalization of the
traditional clustering coefficient. We derive several properties about
higher-order clustering coefficients and analyze them under common random graph
models. Finally, we use higher-order clustering coefficients to gain new
insights into the structure of real-world networks from several domains.
