Post on 03-Jan-2016
Propagation in Networks
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
Day 1Day 2Day 3Day 4Day 5Day 6Day 7Day 8 Day 1Day 2Day 3Day 4Day 5Day 6Day 7Day 8
500 randomly chosen users 500 most active users
Propagation in Networks
“Network Science: Applications to Global Communications”, Albert-Laszlo Barabasi
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
5
6
7
8
9
10
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
B
C
k-threshold Models
2-threshold C is infected only if 2 neighbors (both A and B) are
infected
A
B
C
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
M
L
CC
BB
KJ
EE
F
AA
Z
GG
FF
6DD
IY
B
HH
3
E
2
IIU
A
IQ NN MM
CV
JJ
KK
LL
RR
VV
YY
ZZ
OOQQ
TT
G
WW
VV
H
UUSS
X
PP
D
5
4
7
8
1
SN
PW
OR
T
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)