Randomized graph cluster randomization

The global average treatment effect (GATE) is a primary quantity of interest in the study of causal inference under network interference. With a correctly specified exposure model of the interference, the Horvitz–Thompson (HT) and Hájek estimators of the GATE are unbiased and consistent, respectively, yet known to exhibit extreme variance under many designs and in many settings of interest. With a fixed clustering of the interference graph, graph cluster randomization (GCR) designs have been shown to greatly reduce variance compared to node-level random assignment, but even so the variance is still often prohibitively large. In this work, we propose a randomized version of the GCR design, descriptively named randomized graph cluster randomization (RGCR), which uses a random clustering rather than a single fixed clustering. By considering an ensemble of many different clustering assignments, this design avoids a key problem with GCR where the network exposure probability of a given node can be exponentially small in a single clustering. We propose two inherently randomized graph decomposition algorithms for use with RGCR designs, randomized 3-net and 1-hop-max, adapted from the prior work on multiway graph cut problems and the probabilistic approximation of (graph) metrics. We also propose weighted extensions of these two algorithms with slight additional advantages. All these algorithms result in network exposure probabilities that can be estimated efficiently. We derive structure-dependent upper bounds on the variance of the HT estimator of the GATE, depending on the metric structure of the graph driving the interference. Where the best-known such upper bound for the HT estimator under a GCR design is exponential in the parameters of the metric structure, we give a comparable upper bound under RGCR that is instead polynomial in the same parameters. We provide extensive simulations comparing RGCR and GCR designs, observing substantial improvements in GATE estimation in a variety of settings.