We develop a general framework for design-based causal inference under interference in cluster experiments conducted via two-stage randomization on a network of interconnected units, without relying on exposure mapping assumptions, exclusion of cross-cluster interference, or Bernoulli treatment assignments. Within this framework, we establish a complete characterization of linear weighted estimators (LW) as defined by Godambe (1955) that achieve identification of various network causal effects under interference. This general class includes several new estimators with improved theoretical guarantees and superior finite-sample performance relative to existing approaches such as standard inverse-probability-of-treatment weighting. For most estimators in this class, we establish central limit theorems and conservative variance estimators, which allows us to describe the distinct asymptotic behavior exhibited by different weighting schemes potentially of interest. In particular, we study how randomization at the cluster-level affects the asymptotic behavior of various estimators, and we identify a subclass of cluster-agnostic LW estimators whose convergence rates are independent of the number of clusters and attain the optimal root-N rate, where N denotes the total number of units. Notably, for complete randomization we develop new techniques that may be of independent interest, both to establish a central limit theorem for sums of general dependent statistics and to construct conservative and bias-corrected variance estimators. We complement our theoretical results with extensive simulation studies that offer practical guidance on the choice of weighting method and experimental design under a wide range of interference structures.

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