A Simple Model of Cascades in Networks∗

We consider a linear threshold model of cascades in networks. An agent switches (e.g. adopts an innovation) if the proportion of his neighbors who have already switched exceeds his threshold. Agents’ thresholds are drawn randomly at the start of the cascade process. We present a result for the expected number of switches in arbitrary finite networks with any initial seeds. We define a new measure of an agent’s ability to influence a cascade in a given network, called cascade centrality, which is the expected size of the cascade when the agent is the only seed in the network. We then define contagion centrality, which is the probability that all agents switch when the node is a seed. For certain network topologies, we find analytic expressions for cascade centrality and contagion centrality and show that there may be tension between them. Yet nodes with high cascade and contagion centrality share an interesting property: they have many neighbors, but their neighbors have few neighbors. As an illustration of cascade centrality, we look at how the network structure affects optimal prices when a profitmaximizing firm tries to spread an innovation. Optimal price and seeding behavior can be counterintuitive when firms price irreversible cascades. Our tractable model can be extended in various ways. ∗First draft: 20 December 2014. We would like to thank Rediet Abebe, Daron Acemoglu, Elie Adam, Yann Bramoullé, Kimon Drakopoulos, Mikhail Drugov, Yuichiro Kamada, Michael Kearns, Jon Kleinberg, Daniel Margo, Stephen Morris, and Bassel Tarbush, as well as seminar and conference participants at Boston College, Harvard, Québec, Queen Mary, LMU, Oxford, Cambridge, GREQAM, HSE, 20th CTN Workshop, Conference on Information Transmission in Networks, 2014 INFORMS, 2015 SAET Conference, Monash, University of Sydney, and the 11th World Congress of the Econometric Society for their comments. †Operations Research Center and Laboratory for Information and Decision Systems, MIT. Email: [email protected]. ‡Department of Electrical Engineering and Computer Science and Laboratory for Information and Decision Systems, MIT. Email: [email protected]. §Institute for New Economic Thinking at the Oxford Martin School, University of Oxford. Email: [email protected].