Propagation in Networks 500 randomly chosen users 500 most active users Day 8 Day 6 Day 5 Day 7 Day 3 Day 2 Day 1 Day 4 Day 6 Day 8 Day 7 Day 2 Day 5 Day 4 Day 3 Day 1 “Network Science: Applications to Global Communications”, Albert-Laszlo Barabasi
Firefighter Problem A simple network - a grid where each intersection point is a node. • Fire starts at one point • 1 Firefighter can be deployed to protect a point at each time step • Fire spreads to all unprotected adjacent vertices in the next time step. • Repeat
Firefighter Problem Strategies • Repeat the example exercise with different firefighter placement • How much of the network can you protect?
Disease Models • S – Susceptible • I – Infectious • R – Recovered / removed • E – Exposed
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
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
How Diseases Track Information • Same models that describe disease spread describe the spread of rumors, fads, links, etc. in social media.
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?
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 C B
k-threshold Models • 2-threshold • C is infected only if 2 neighbors (both A and B) are infected A C B
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?
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? • You may immunize 1 node at each time period. Disease starts at YY. • Bonus for protecting OO and DD. • You may immunize 1 node at each time period. Disease starts at both OO and NN. • You may immunize 2 nodes at each time period. Disease starts at B
6 DD EE 5 7 FF E M CC 8 F 4 GG 3 L BB 1 D AA 2 J K HH ZZ Z B JJ I YY VV A Y II KK U I VV WW Q NN MM LL N S T H V C G P RR TT W OO X QQ UU O SS R PP
Exercise • Now assume a 2-threshold model • How many nodes do you immunize and how many are saved? • You may immunize 1 node at each time period. Disease starts a both OO and NN. • You may immunize 2 nodes at each time period. Disease starts at OO and NN.
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.
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)