Graph cluster randomization: network exposure to multiple universes

A/B testing is a standard approach for evaluating the effect of online experiments; the goal is to estimate the `average treatment effect' of a new feature or condition by exposing a sample of the overall population to it. A drawback with A/B testing is that it is poorly suited for experiments involving social interference, when the treatment of individuals spills over to neighboring individuals along an underlying social network. In this work, we propose a novel methodology using graph clustering to analyze average treatment effects under social interference. To begin, we characterize graph-theoretic conditions under which individuals can be considered to be `network exposed' to an experiment. We then show how graph cluster randomization admits an efficient exact algorithm to compute the probabilities for each vertex being network exposed under several of these exposure conditions. Using these probabilities as inverse weights, a Horvitz-Thompson estimator can then provide an effect estimate that is unbiased, provided that the exposure model has been properly specified. Given an estimator that is unbiased, we focus on minimizing the variance. First, we develop simple sufficient conditions for the variance of the estimator to be asymptotically small in n, the size of the graph. However, for general randomization schemes, this variance can be lower bounded by an exponential function of the degrees of a graph. In contrast, we show that if a graph satisfies a restricted-growth condition on the growth rate of neighborhoods, then there exists a natural clustering algorithm, based on vertex neighborhoods, for which the variance of the estimator can be upper bounded by a linear function of the degrees. Thus we show that proper cluster randomization can lead to exponentially lower estimator variance when experimentally measuring average treatment effects under interference.Comment: 9 pages, 2 figure

Competing epidemics on complex networks

Human diseases spread over networks of contacts between individuals and a substantial body of recent research has focused on the dynamics of the spreading process. Here we examine a model of two competing diseases spreading over the same network at the same time, where infection with either disease gives an individual subsequent immunity to both. Using a combination of analytic and numerical methods, we derive the phase diagram of the system and estimates of the expected final numbers of individuals infected with each disease. The system shows an unusual dynamical transition between dominance of one disease and dominance of the other as a function of their relative rates of growth. Close to this transition the final outcomes show strong dependence on stochastic fluctuations in the early stages of growth, dependence that decreases with increasing network size, but does so sufficiently slowly as still to be easily visible in systems with millions or billions of individuals. In most regions of the phase diagram we find that one disease eventually dominates while the other reaches only a vanishing fraction of the network, but the system also displays a significant coexistence regime in which both diseases reach epidemic proportions and infect an extensive fraction of the network.Comment: 14 pages, 5 figure

Design and Analysis of Experiments in Networks: Reducing Bias from Interference

Estimating the effects of interventions in networks is complicated due to interference, such that the outcomes for one experimental unit may depend on the treatment assignments of other units. Familiar statistical formalism, experimental designs, and analysis methods assume the absence of this interference, and result in biased estimates of causal effects when it exists. While some assumptions can lead to unbiased estimates, these assumptions are generally unrealistic in the context of a network and often amount to assuming away the interference. In this work, we evaluate methods for designing and analyzing randomized experiments under minimal, realistic assumptions compatible with broad interference, where the aim is to reduce bias and possibly overall error in estimates of average effects of a global treatment. In design, we consider the ability to perform random assignment to treatments that is correlated in the network, such as through graph cluster randomization. In analysis, we consider incorporating information about the treatment assignment of network neighbors. We prove sufficient conditions for bias reduction through both design and analysis in the presence of potentially global interference; these conditions also give lower bounds on treatment effects. Through simulations of the entire process of experimentation in networks, we measure the performance of these methods under varied network structure and varied social behaviors, finding substantial bias reductions and, despite a bias–variance tradeoff, error reductions. These improvements are largest for networks with more clustering and data generating processes with both stronger direct effects of the treatment and stronger interactions between units. Keywords: causal inference; field experiments; peer effects; spillovers; social contagion; social network analysis; graph partitionin

Random graphs containing arbitrary distributions of subgraphs

Traditional random graph models of networks generate networks that are locally tree-like, meaning that all local neighborhoods take the form of trees. In this respect such models are highly unrealistic, most real networks having strongly non-tree-like neighborhoods that contain short loops, cliques, or other biconnected subgraphs. In this paper we propose and analyze a new class of random graph models that incorporates general subgraphs, allowing for non-tree-like neighborhoods while still remaining solvable for many fundamental network properties. Among other things we give solutions for the size of the giant component, the position of the phase transition at which the giant component appears, and percolation properties for both site and bond percolation on networks generated by the model.Comment: 12 pages, 6 figures, 1 tabl