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Multi-stage complex contagions
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58.For clarity, in all 100 realizations, we initially S1-activate the same set of 348 active nodes chosen at random (which corresponds to ). We average over multiple realizations because nodes are updated in a random order and thus the time evolution of the system is different in each realization. However, the final state of the system is the same for all realizations because it is determined by the set of initially active nodes. For the particular set of 348 active nodes that we use, the cascade occurs in the multi-stage case but not in the single-stage case. Generally, for the parameters used in Fig. 2, about 5% of various random sets of 348 nodes induce cascades in the single-stage case and about 46% of them do so in the multi-stage case.
59.Similar to Fig. 2, in all realizations, we use a single set of 348 initially active nodes that produces cascades in the multi-stage case but not in the single-stage case. For the parameters used in Fig. 3, about 0.1% of random sets of 348 active nodes induce cascades in the single-stage case and about 52% of them do so in the multi-stage case.
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62.However, assumption (i) should probably be relaxed for a model that includes “hipsters.”
63.This conditioning is important, as it helps one to account for correlations between the state of node A and those of its neighbors. This significantly improves the accuracy of the result in comparison to mean-field approximations in which such dynamical correlations are neglected.
65.Assuming that a random node A is not Si-active, denotes the probability that a random neighbor of A is Si-active and depends on the degree k of node A. By contrast, gives the probability that a degree- neighbor of A is Si-active and does not depend on the degree of node A.
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67.One can also apply a similar analysis to the joint degree-degree distribution .
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The spread of ideas across a social network can be studied using complex contagion models, in which agents are activated by contact with multiple activated neighbors. The investigation of complex contagions can provide crucial insights into social influence and behavior-adoption cascades on networks. In this paper, we introduce a model of a multi-stage complex contagion on networks. Agents at different stages—which could, for example, represent differing levels of support for a social movement or differing levels of commitment to a certain product or idea—exert different amounts of influence on their neighbors. We demonstrate that the presence of even one additional stage introduces novel dynamical behavior, including interplay between multiple cascades, which cannot occur in single-stage contagion models. We find that cascades—and hence collective action—can be driven not only by high-stage influencers but also by low-stage influencers.
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