Long ties accelerate noisy threshold-based contagions

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 individual's 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, thereby harmonizing qualitative facts about how network structure affects contagion. To model the trade-off between long and short edges we analyze the rate of spread over networks that are the union of circular lattices and random graphs on $n$ nodes. Allowing for noise in adoption decisions (i.e., 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. This conclusion continues to hold true under partial but frequent enough rewiring and when adoption decisions are reversible but infrequently so. Simulations illustrate the robustness of these results to several variations on this 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.