concluding remarks. 2 Methods 2.1 Networks This section provides some necessary background on network concepts and the notation used throughout this paper. We assume that the true data generating mechanism is a contagion process spreading through a network. We identify a network as an independent cluster, such that outcomes within a network are correlated whereas those across networks (across clusters) are independent. A simple network G consists of set of nodes N = { 1 , ..., n } and edges E ⊆ N × N . The placement of edges may be described by an n × n adjacency matrix e , where element e$_{jj}$ ′ = e$_{j}$ $^{′}$j has value 1 if an edge exists between nodes j and j ′ and is 0 otherwise. Complete network and contagion processes are not generally observable; but even when they are not, it may nevertheless be possible to characterize certain features of networks. Let e$_{i}$ represent the n$_{i}$ × n$_{i}$ adjacency matrix for the network in cluster i consisting of n$_{i}$ nodes. The degree of a node is the number of edges that are adjacent to it: k$_{ij}$ = ∑ j ′ e$_{ijj}$ ′ . Mean neighbor degree ∑ j ′ e$_{ijj}$ ′ k$_{ij}$ ′ k$_{ij}$ is the unweighted average of a node’s neighbors’ degrees. Degree assortativity is a composite measure of mean neighbor degree across the entire network, defined as the Pearson correlation coefficient of degrees of adjacent nodes taken over all network edges$^{10}$, which can be calculated for a given network using a weighted and normalized sum across all degree pairs 11 as ∑ jk jk ( e$_{jk}$ − q$_{j}$q$_{k}$ ) σ 2 q , where j and k are given node degrees, e$_{jk}$ is the probability of a randomly-selected edge connecting a node with degree j + 1 and a node with degree k + 1 (known as excess degree ), q$_{j}$ (and q$_{k}$ ) is the probability mass of the excess degree distribution, and σ 2 q is the variance of the excess degree distribution. A connected component is a maximal subset of nodes for which a path exists between each pair of nodes. A path exists between two node j and j ′ if and only if there exists a subset of edges E$_{ijj}$ ′ ⊆ E$_{i}$ in the network that connect nodes j and j $^{′}$. The components of the network c ∈ 1 , ..., C$_{i}$ are assumed to be ordered from largest to smallest, and node j in network i is belongs to connected component c$_{ij}$ . The largest component in the network contains ∑ j $_{I}$( c$_{ij}$ = 1) nodes. The mean component size is n$_{i}$/C$_{i}$ , and the size of the component node j in network i belongs to is ∑ j ′ $_{I}$( c$_{ij}$ = c$_{ij}$ ′ ) . The contagion status of each node or that of the node’s network neighbors might also be observed at baseline. We describe a person who has been impacted by the contagion process as affected ( e.g. , infected if the process is infectious or impacted if the process alters behavior), and we use I$_{ij}$ ( t ) to denote the binary process outcome for node j in network i at time t . One simple metric of contagion status is the number of affected neighbors at baseline for each node ∑ j ′ e$_{ijj}$ ′ I$_{ij}$ ′ (0) , or the number of affected individuals at baseline belonging to the same component as a given node ∑ j $^{′}$S $_{I}$( c$_{ij}$ = c$_{ij}$ ′ ) I$_{ij}$ ′ (0) . Another metric is the length of the shortest path between each node j and each infected individual j ′ in the network at baseline. The shortest 3