1.  Propagation in Networks

2. Network Structure and Propagation  Diseases, fads, rumors, viral social media content all spread the same way in networks  Models for understanding disease let us understand how other things spread in networks  Questions  How does this relate to network structure?  How can we spread things better (information) or prevent the spread (viruses)?  Review some existing models and data

3. Day 1Day 2Day 3Day 4Day 5Day 6Day 7Day 8Day 1Day 2Day 3Day 4Day 5Day 6Day 7Day 8 500 randomly chosen users500 most active users Propagation in Networks “Network Science: Applications to Global Communications”, Albert-Laszlo Barabasi

4. 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 4

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

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

13. 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

14. 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

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

16. 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?

17. 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

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

19. 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?

20. Apply S/I/R Models and k-thresholds

21. 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

22. 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 5 5 4 4 7 7 8 8 1 1 S S N N P P W W O O R R T T

23. 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.

24. 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.

25. 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)