Abstract
Many social phenomena can be modeled as cascades in networks, where nodes adopt a behavior in response to peers adopting. When studying cascades, researchers typically assume that the number of active peers when a node adopts is equivalent to the node's threshold for adoption. This assumption is rarely justified due to the “opacity problem”: networked cascades reveal intervals which contain thresholds, rather than point estimates. Existing approaches take the maximum of each node's threshold interval, which biases models of social influence. Opacity is inevitable in many small graphs when using the threshold model, resulting from the networked process itself rather than data collection techniques. Using simulation, we extend this finding to the probabilistic SI (independent cascade) model. We confirm these theoretical results by studying 50 large hashtag cascades among 3.2 million Twitter users, finding that 20% of adoptions suffer from opacity. Different assumptions in response to opacity qualitatively change conclusions about peer influence. While opacity is a far-reaching problem, it can be addressed. Using information from nodes who have tightly bounded intervals allows building models to reduce error in estimating node thresholds.
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Published Version
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