  We consider the problem of detecting whether or not, in a given sensor
network, there is a cluster of sensors which exhibit an "unusual behavior."
Formally, suppose we are given a set of nodes and attach a random variable to
each node. We observe a realization of this process and want to decide between
the following two hypotheses: under the null, the variables are i.i.d. standard
normal; under the alternative, there is a cluster of variables that are i.i.d.
normal with positive mean and unit variance, while the rest are i.i.d. standard
normal. We also address surveillance settings where each sensor in the network
collects information over time. The resulting model is similar, now with a time
series attached to each node. We again observe the process over time and want
to decide between the null, where all the variables are i.i.d. standard normal,
and the alternative, where there is an emerging cluster of i.i.d. normal
variables with positive mean and unit variance. The growth models used to
represent the emerging cluster are quite general and, in particular, include
cellular automata used in modeling epidemics. In both settings, we consider
classes of clusters that are quite general, for which we obtain a lower bound
on their respective minimax detection rate and show that some form of scan
statistic, by far the most popular method in practice, achieves that same rate
to within a logarithmic factor. Our results are not limited to the normal
location model, but generalize to any one-parameter exponential family when the
anomalous clusters are large enough.
