  Synchronization phenomena on networks have attracted much attention in
studies of neural, social, economic, and biological systems, yet we still lack
a systematic understanding of how relative synchronizability relates to
underlying network structure. Indeed, this question is of central importance to
the key theme of how dynamics on networks relate to their structure more
generally. We present an analytic technique to directly measure the relative
synchronizability of noise-driven time-series processes on networks, in terms
of the directed network structure. We consider both discrete-time
auto-regressive processes and continuous-time Ornstein-Uhlenbeck dynamics on
networks. Our technique builds on computation of the network covariance matrix
in the space orthogonal to the synchronized state, enabling it to be more
general than previous work in not requiring either symmetric (undirected) or
diagonalizable connectivity matrices, and allowing arbitrary self-link weights.
More importantly, our approach quantifies the relative synchronisation
specifically in terms of the contribution of process motif (walk) structures.
We demonstrate that in general the relative abundance of process motifs with
convergent directed walks (including feedback and feedforward loops) hinders
synchronizability. We also reveal subtle differences between the motifs
involved for discrete or continuous-time dynamics. Our insights analytically
explain several known general results regarding synchronizability of networks,
including that small-world and regular networks are less synchronizable than
random networks.
