Feedback through graph motifs relates structure and function in complex networks

TL;DR

This paper develops a theoretical framework linking local connectivity motifs to the overall function of complex networks with linear dynamics, enabling simplified analysis and understanding of network behavior.

Contribution

It introduces a motif-based reduced order model that relates network structure to function, applicable to both homogeneous and heterogeneous networks.

Findings

Motifs determine network feedback and response.

Simplifies analysis of Erdős-Rényi graphs.

Extends to heterogeneous networks and real-world data.

Abstract

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.