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Propagation in Networks
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Propagation in Networks
500 randomly chosen users
500 most active users
Day 8
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“Network Science: Applications to Global Communications”, Albert-Laszlo Barabasi
Day 8
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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
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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?
Apply S/I/R Models and k-thresholds
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
DD
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EE
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CC
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GG
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K
A
B
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YY
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Q
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NN
VV
KK
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MM
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LL
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OO
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ZZ
JJ
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O
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HH
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D
AA
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BB
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PP
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RR
QQ
TT
SS
UU
H
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.
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)