  How does connectivity impact network dynamics? We address this question by
linking network characteristics on two scales. On the global scale we consider
the coherence of overall network dynamics. We show that such \emph{global
coherence} in activity can often be predicted from the \emph{local structure}
of the network. To characterize local network structure we use "motif
cumulants," a measure of the deviation of pathway counts from those expected in
a minimal probabilistic network model.
  We extend previous results in three ways. First, we give a new combinatorial
formulation of motif cumulants that relates to the allied concept in
probability theory. Second, we show that the link between global network
dynamics and local network architecture is strongly affected by heterogeneity
in network connectivity. However, we introduce a network-partitioning method
that recovers a tight relationship between architecture and dynamics. Third,
for a particular set of models we generalize the underlying theory to treat
dynamical coherence at arbitrary orders (i.e. triplet correlations, and
beyond). We show that at any order only a highly restricted set of motifs
impact dynamical correlations.
