Network structure can affect when and how widely new ideas, products, and behaviors are adopted. In widely-used models of biological contagion, interventions that randomly rewire edges (generally making them longer) accelerate spread. However, there are other models relevant to social contagion, such as those motivated by myopic best-response in games with strategic complements, in which an individuals behavior is described by a threshold number of adopting neighbors above which adoption occurs (i.e., complex contagions). Recent work has argued that highly clustered, rather than random, networks facilitate spread of these complex contagions. Here we show that minor modifications to this model, which make it more realistic, reverse this result: we allow very rare below-threshold adoption, i.e., rarely adoption occurs when there is only one adopting neighbor. To model the trade-off between long and short edges we consider networks that are the union of cycle-power-$k$ graphs and random graphs on $n$ nodes. Allowing adoptions below threshold to occur with order $1/sqrt{n}$ probability along some short cycle edges is enough to ensure that random rewiring accelerates spread. Simulations illustrate the robustness of these results to other commonly-posited models for noisy best-response behavior. Hypothetical interventions that randomly rewire existing edges or add random edges (versus adding short, triad-closing edges) in hundreds of empirical social networks reduce time to spread. This revised conclusion suggests that those wanting to increase spread should induce formation of long ties, rather than triad-closing ties. More generally, this highlights the importance of noise in game-theoretic analyses of behavior.
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