  Networks are a fundamental model of complex systems throughout the sciences,
and network datasets are typically analyzed through lower-order connectivity
patterns described at the level of individual nodes and edges. However,
higher-order connectivity patterns captured by small subgraphs, also called
network motifs, describe the fundamental structures that control and mediate
the behavior of many complex systems. We develop three tools for network
analysis that use higher-order connectivity patterns to gain new insights into
network datasets: (1) a framework to cluster nodes into modules based on joint
participation in network motifs; (2) a generalization of the clustering
coefficient measurement to investigate higher-order closure patterns; and (3) a
definition of network motifs for temporal networks and fast algorithms for
counting them. Using these tools, we analyze data from biology, ecology,
economics, neuroscience, online social networks, scientific collaborations,
telecommunications, transportation, and the World Wide Web.
