Propagation in Networks

Propagation in Networks 500 randomly chosen users 500 most active users Day 8 Day 1 Day 7 Day 2 Day 3 Day 4 Day 6 Day 5 Day 8 Day 7 Day 2 Day 1 Day 3 Day 4 Day 6 Day 5 “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 SEIR 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? Notes: a 2-threshold could be a good model for a meme. The first time you see it, you may not recognize it as a meme. The second time you see it, you may notice that it is a repeated pattern and see the meme in it. For something not typically meme like, one view cannot indicate to someone that it’s a meme. Thus, you have a least a 2 threshold.

Apply S/I/R Models and k-thresholds Use an SI model How quickly will the disease spread (how many time steps?) If you can immunize a node between each step it spreads, what is the best strategy and why? Repeat with an SIS model. Then what happens? For a 1-threshold? For a 2-threshold? Note: If we save P, it can save most of the left cluster. Thus, hubs are important. Note: F and P are critical because they are bridges. Spreading there will reach across to the other cluster. Thus, bridges are important. In bigger networks, walling off a section of the network with immunized nodes can be effective.

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

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)