  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 (on average 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
${n}^{-1/\theta}$ probability along at least some "short" cycle edges is enough
to ensure that random rewiring accelerates the spread of a noisy
threshold-$\theta$ contagion. This conclusion also holds under partial but
frequent enough rewiring and when adoption decisions are reversible but
infrequently so, as well as in high-dimensional lattice structures that
facilitate faster-expanding contagions. 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.
