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

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5

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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

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

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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)