  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.
