Abstract
Networks are fundamental for our understanding of complex systems. The study of networks has uncovered common principles that underlie the behavior of vastly different fields of study, including physics, biology, sociology, and engineering. One of these common principles is the existence of network motifs-small recurrent patterns that can provide certain features that are important for the specific network. However, it remains unclear how network motifs are joined in real networks to make larger circuits and what properties emerge from interactions between network motifs. Here, we develop a framework to explore the mesoscale-level behavior of complex networks. Considering network motifs as hypernodes, we define the rules for their interaction at the network's next level of organization. We develop a method to infer the favorable arrangements of interactions between network motifs into hypermotifs from real evolved and designed network data. We mathematically explore the emergent properties of these higher-order circuits and their relations to the properties of the individual minimal circuit components they combine. We apply this framework to biological, neuronal, social, linguistic, and electronic networks and find that network motifs are not randomly distributed in real networks but are combined in a way that both maintains autonomy and generates emergent properties. This framework provides a basis for exploring the mesoscale structure and behavior of complex systems where it can be used to reveal intermediate patterns in complex networks and to identify specific nodes and links in the network that are the key drivers of the network's emergent properties.