Journal
DISCRETE APPLIED MATHEMATICS
Volume 157, Issue 7, Pages 1615-1627Publisher
ELSEVIER SCIENCE BV
DOI: 10.1016/j.dam.2008.09.012
Keywords
Threshold models; Spread of disease; Spread of opinion; Social network; Firefighter problem
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Funding
- National Science Foundation [CCR 91-19999, BIR 94-12594, DBI 99-82983, SBR 97-09134, SES-92-11492, EIA-02-05116]
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We will consider models of the spread of disease or opinion through social networks, represented as graphs. In our models, vertices will be in one of two states, I (infected) or 0 (uninfected) and change of state will take place at discrete times. After describing several such models, we will concentrate on the model, called an irreversible k-threshold process, where a vertex enters state 1 if at least k of its neighbors are in state 1, and where a vertex never leaves state 1 once it is in it. We will seek sets of vertices with the property that, if they are in state 1 at the beginning, then eventually all vertices end up in state 1. Such vertex sets correspond to vertices that can be infected with a disease or opinion so as to guarantee saturation of the population with the disease or opinion. We will also discuss ways to defend against such saturating sets, for example by vaccination or designing network topologies. (C) 2008 Elsevier B.V. All rights reserved.
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