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The thresholds j are drawn at random from a

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Unformatted text preview: scades on Random Networks Watts PNAS 2002 • Each node can be in one of two states, say {+1, −1}. • Start with almost all nodes in {−1}, but just one node (or a small fraction of nodes) in {+1}. • Nodes update state asynchronously. For node j if the fraction of its neighbors in state +1 is greater than a threshold function Φj , j switches to +1 and stays in that state forever. • The thresholds Φj are drawn at random from a distribution f (Φ) 1 which is normalized in the usual way: 0 f (Φ)dΦ = 1. • Local dependence, fractional threshold Φj , heterogeneous degree make this model differ from contact processes. Global cascades? • Question: for what kinds of networks and thresholds will a small perturbation (of even one node) cause a fraction of all nodes to flip? (i.e. a global cascade). • Some terms: Innovator – The first node(s) flipped to +1. Early adopter / vulnerable – A neighbor of innovator who flips right away. • Early adopter much have threshold Φj ...
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