Motifs for processes on networks

TL;DR

This paper introduces the concept of process motifs as structured sets of walks on networks, providing a new way to understand the mechanisms behind network functions, exemplified through a multivariate Ornstein-Uhlenbeck process.

Contribution

It distinguishes between structure motifs and process motifs, offering a quantitative framework to analyze mechanisms in network dynamics.

Findings

Process motifs help elucidate mechanisms in network functions.

Steady-state covariances reveal insights into process motifs.

Framework applicable to various dynamical systems on networks.

Abstract

The study of motifs in networks can help researchers uncover links between the structure and function of networks in biology, sociology, economics, and many other areas. Empirical studies of networks have identified feedback loops, feedforward loops, and several other small structures as "motifs" that occur frequently in real-world networks and may contribute by various mechanisms to important functions in these systems. However, these mechanisms are unknown for many of these motifs. We propose to distinguish between "structure motifs" (i.e., graphlets) in networks and "process motifs" (which we define as structured sets of walks) on networks and consider process motifs as building blocks of processes on networks. Using the steady-state covariances and steady-state correlations in a multivariate Ornstein--Uhlenbeck process on a network as examples, we demonstrate that the distinction between structure motifs and process motifs makes it possible to gain quantitative insights into mechanisms that contribute to important functions of dynamical systems on networks.