its cluster; we describe such a modification in Appendix \[appendix:hole\_punching\].

Graph cluster randomization could be applied to any mapping $C(\cdot)$ of vertices to clusters. One such mapping, which we use for the simulations reported in this paper, is formed by $\epsilon$-net clustering as previously considered by @ugander_graph_2013. An $\epsilon$-net in the graph distance metric is a set of vertices such that no two vertices in the set are less than $\epsilon$ hops of each other, and every vertex outside the set is within $\epsilon$ hops (in fact, $\epsilon-1$ hops) of a vertex in the set. An $\epsilon$-net can be formed by repeatedly selecting a vertex and removing it and every vertex within distance $\epsilon-1$ from the network, until all vertices have been removed. Having completed this step, the population of selected vertices forms an $\epsilon$-net. An $\epsilon$-net clustering can be formed by assigning each vertex to the closest vertex in the $\epsilon$-net, and breaking the possible ties through some arbitrary rule. Different selection and assignment rules and different values of $\epsilon$ correspond to different experimental designs. We compare clustered random assignment using $\epsilon$-nets to independent random assignment, where vertices are independently assigned to treatment and control.

Other mappings of vertices 