  The ability to detect weak distributed activation patterns in networks is
critical to several applications, such as identifying the onset of anomalous
activity or incipient congestion in the Internet, or faint traces of a
biochemical spread by a sensor network. This is a challenging problem since
weak distributed patterns can be invisible in per node statistics as well as a
global network-wide aggregate. Most prior work considers situations in which
the activation/non-activation of each node is statistically independent, but
this is unrealistic in many problems. In this paper, we consider structured
patterns arising from statistical dependencies in the activation process. Our
contributions are three-fold. First, we propose a sparsifying transform that
succinctly represents structured activation patterns that conform to a
hierarchical dependency graph. Second, we establish that the proposed transform
facilitates detection of very weak activation patterns that cannot be detected
with existing methods. Third, we show that the structure of the hierarchical
dependency graph governing the activation process, and hence the network
transform, can be learnt from very few (logarithmic in network size)
independent snapshots of network activity.
