  The micro-randomized trial (MRT) is an experimental design that can be used
to develop optimal mobile health interventions. In MRTs, interventions in the
form of notifications or messages are sent through smart phones to individuals,
targeting a health-related outcome such as physical activity or weight
management. Often, mobile health interventions have a social media component;
an individual's outcome could thus depend on other individuals' treatments and
outcomes. In this paper, we study the micro-randomized trial in the presence of
such cross-unit interference. We model the cross-unit interference with a
network interference model; the outcome of one individual may affect the
outcome of another individual if and only if they are connected by an edge in
the network. Assuming the dynamics can be represented as a Markov decision
process, we analyze the behavior of the outcomes in large sample asymptotics
and show that they converge to a mean-field limit when the sample size goes to
infinity. Based on the mean-field result, we give characterization results and
estimation strategies for various causal estimands including the short-term
direct effect of a binary intervention, its long-term direct effect and its
long-term total effect.
