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In physics, biology and engineering, network systems abound. How does theconnectivity of a network system combine with the behavior of its individualcomponents to determine its collective function? We approach this question fornetworks with linear time-invariant dynamics by relating internal networkfeedbacks to the statistical prevalence of connectivity motifs, a set ofsurprisingly simple and local statistics of connectivity. This results in areduced order model of the network input-output dynamics in terms of motifsstructures. As an example, the new formulation dramatically simplifies theclassic Erdos-Renyi graph, reducing the overall network behavior to oneproportional feedback wrapped around the dynamics of a single node. For generalnetworks, higher-order motifs systematically provide further layers and typesof feedback to regulate the network response. Thus, the local connectivityshapes temporal and spectral processing by the network as a whole, and we showhow this enables robust, yet tunable, functionality such as extending the timeconstant with which networks remember past signals. The theory also extends tonetworks composed from heterogeneous nodes with distinct dynamics andconnectivity, and patterned input to (and readout from) subsets of nodes. Thesestatistical descriptions provide a powerful theoretical framework to understandthe functionality of real-world network systems, as we illustrate with examplesincluding the mouse brain connectome.

Feedback through graph motifs relates structure and function in complex networks