In physics, biology and engineering, network systems abound. How does the
connectivity of a network system combine with the behavior of its individual
components to determine its collective function? We approach this question for
networks with linear time-invariant dynamics by relating internal network
feedbacks to the statistical prevalence of connectivity motifs, a set of
surprisingly simple and local statistics of connectivity. This results in a
reduced order model of the network input-output dynamics in terms of motifs
structures. As an example, the new formulation dramatically simplifies the
classic Erdos-Renyi graph, reducing the overall network behavior to one
proportional feedback wrapped around the dynamics of a single node. For general
networks, higher-order motifs systematically provide further layers and types
of feedback to regulate the network response. Thus, the local connectivity
shapes temporal and spectral processing by the network as a whole, and we show
how this enables robust, yet tunable, functionality such as extending the time
constant with which networks remember past signals. The theory also extends to
networks composed from heterogeneous nodes with distinct dynamics and
connectivity, and patterned input to (and readout from) subsets of nodes. These
statistical descriptions provide a powerful theoretical framework to understand
the functionality of real-world network systems, as we illustrate with examples
including the mouse brain connectome.Comment: 31 pages, 20 figure
