  This paper designs methods for decentralized multiple hypothesis testing on
graphs that are equipped with provable guarantees on the false discovery rate
(FDR). We consider the setting where distinct agents reside on the nodes of an
undirected graph, and each agent possesses p-values corresponding to one or
more hypotheses local to its node. Each agent must individually decide whether
to reject one or more of its local hypotheses by only communicating with its
neighbors, with the joint aim that the global FDR over the entire graph must be
controlled at a predefined level. We propose a simple decentralized family of
Query-Test-Exchange (QuTE) algorithms and prove that they can control FDR under
independence or positive dependence of the p-values. Our algorithm reduces to
the Benjamini-Hochberg (BH) algorithm when after graph-diameter rounds of
communication, and to the Bonferroni procedure when no communication has
occurred or the graph is empty. To avoid communicating real-valued p-values, we
develop a quantized BH procedure, and extend it to a quantized QuTE procedure.
QuTE works seamlessly in streaming data settings, where anytime-valid p-values
may be continually updated at each node. Last, QuTE is robust to arbitrary
dropping of packets, or a graph that changes at every step, making it
particularly suitable to mobile sensor networks involving drones or other
multi-agent systems. We study the power of our procedure using a simulation
suite of different levels of connectivity and communication on a variety of
graph structures, and also provide an illustrative real-world example.
