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Propagation in Networks

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Day 1Day 2Day 3Day 4Day 5Day 6Day 7Day 8Day 1Day 2Day 3Day 4Day 5Day 6Day 7Day randomly chosen users500 most active users Propagation in Networks Network Science: Applications to Global Communications, Albert-Laszlo Barabasi

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Firefighter Problem A simple network - a grid where each intersection point is a node. 1. Fire starts at one point 2. 1 Firefighter can be deployed to protect a point at each time step 3. Fire spreads to all unprotected adjacent vertices in the next time step. 4. Repeat 3

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Firefighter Problem Strategies Repeat the example exercise with different firefighter placement How much of the network can you protect?

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Disease Models S – Susceptible I – Infectious R – Recovered / removed E – Exposed

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Disease Models SI Susceptible, and once you catch the disease, you remain infectious for the rest of your life. HIV, Herpes SIR Susceptible, and then you catch the disease. You are infectious for a while, but once recovered, you cannot catch the disease again. Mono, Chicken Pox

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Disease Models SIRS / SIS A susceptible person gets sick and is infectious. After recovering (and possibly enjoying a period of temporary immunity, indicated by R), the person is susceptible to the infection again. Strep throat SEIR After becoming infected, the person has a period where they are not contagious. This period of exposure is indicated with E Incorporates exposed but non infectious period

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How Diseases Track Information Same models that describe disease spread describe the spread of rumors, fads, links, etc. in social media.

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Discuss How do S/I/R models apply here. What does it mean to be susceptible? What does it mean to be infectious? What does it mean to be recovered? What does it mean if you have an SIRS model and go from recovered to susceptible again?

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k-threshold Models Disease is transmitted if k adjacent nodes are infected. 1-threshold C is infected if either A or B is infected A B C

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k-threshold Models 2-threshold C is infected only if 2 neighbors (both A and B) are infected A B C

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Application to Information - Discuss How do k-thresholds work for information spreading? What does it mean to have a 2-threshold? How can you use this to build strategies?

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Apply S/I/R Models and k-thresholds

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Exercise The disease will spread. Then, you can immunize uninfected nodes. Repeat. Assume a 1-threshold SI model How many nodes do you immunize and how many are saved? 1. You may immunize 1 node at each time period. Disease starts at YY. Bonus for protecting OO and DD. 2. You may immunize 1 node at each time period. Disease starts at both OO and NN. 3. You may immunize 2 nodes at each time period. Disease starts at B

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M M L L CC BB K K J J EE F F AA Z Z GG FF 6 6 DD I I Y Y B B HH 3 3 E E 2 2 II U U A A I I Q Q NN MM C C V V JJ KK LL RR VV YY ZZ OO QQ TT G G WW VV H H UU SS X X PP D D S S N N P P W W O O R R T T

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Exercise Now assume a 2-threshold model How many nodes do you immunize and how many are saved? 1. You may immunize 1 node at each time period. Disease starts a both OO and NN. 2. You may immunize 2 nodes at each time period. Disease starts at OO and NN.

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Exercise Assume someone can immunize 2 people in each round. Assume a 1-threshold model You can start the disease in 2 places. Choose them to cause the largest possible spread. Assume a 2-threshold model You can start the disease in 2 places. Choose them to cause the largest possible spread.

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Exercise Repeat all exercises for SIR model (once recovered, the node is immune) SIS model (node is infected for 1 step, then uninfected but susceptible again) SIRS model (node is infected for 1 step, then immune for 1 step, then susceptible again)

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