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Regulatory networks in cells may comprise a variety of types of molecular interactions. The most basic are pairwise interactions, in which one species controls the behaviour of another (e.g. a transcription factor activates or represses a gene). Higher-order interactions, while more subtle, may be important for determining the function of networks. Here, the authors systematically expand a simple master equation model for a gene to derive an approach for robustly assessing the cooperativity (effective copy number) with which a transcription factor acts. The essential idea is that moments of a joint distribution of protein copy numbers determine the Hill coefficient of a cis-regulatory input function without non-linear fitting. The authors show that this method prescribes a definition of cooperativity that is meaningful even in highly complex situations in which the regulation does not conform to a simple Hill function. To illustrate the utility of the method, the authors measure the cooperativity of the transcription factor CI in simulations of phage-λ and show how the cooperativity accurately reflects the behaviour of the system. The authors numerically assess the effects of deviations from ideality, as well as possible sources of error. The relationship to other definitions of cooperativity and issues for experimentally realising the procedure are discussed.