  We propose a new testing framework applicable to both the two-sample problem
on point processes and the community detection problem on rectangular arrays of
point processes, which we refer to as longitudinal networks; the latter problem
is useful in situations where we observe interactions among a group of
individuals over time. Our framework is based on a multiscale discretization
scheme that consider not just the global null but also a collection of nulls
local to small regions in the domain; in the two-sample problem, the local
rejections tell us where the intensity functions differ and in the longitudinal
network problem, the local rejections tell us when the community structure is
most salient. We provide theoretical analysis for the two-sample problem and
show that our method has minimax optimal power under a Holder continuity
condition. We provide extensive simulation and real data analysis demonstrating
the practicality of our proposed method.
