Topics In Social Computing (67810) Module 2 (Dynamics) Cascades, Memes, and Epidemics (Networks Crowds & Markets Ch. 21)

EPIDEMIC MODELS 2

Reminder Last time we spoke about the adoption of technologies in networks Was largely driven by the utility gained from a new technology. Each social contact that adopted made adoption more worthwhile. – This is called The Network Effect – Adoption by an agent affects the utility of others: Economists refer to this as an Externality 3

Probabilistic Contagion Some things (like viruses) can be contagious even from a single interaction with someone infected, and do not require a threshold to be crossed. The most basic model: Given an infected node, each of its neighbors is infected with probability p. For simplicity, assume that each edge is only used once and that we randomize i.i.d for different edges. 4

The Basic Reproductive Number 5

The Infection Process 6

An alternative view 7 Remove each edge with probability p (i.i.d). What is the size of the connected component of the root?

The Basic Reproductive Number 8

The Expected Number of Infected Individuals 9

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12 Proving part 2 of the Theorem: When does the epidemic survive to level n+1? – The root infect a child with prob. p – This child (which is the root of an identical subtree) has an epidemic that survives n levels or more. p

13 Specific child is infected & epidemic spreads n levels more All d children fail (independently) Not all d children fail

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Discussion In graphs that aren’t regular trees, the analysis we did is not correct. The main change: collisions (i.e., several paths that reach some nodes). Still, the basic reproductive number is useful in models such as G(n,p), small world graphs, and some random scale free network models – Similar results also apply there due to lack of collisions in early stages of the process 17

Source: Wikipedia. 18

SI model Nodes can be Suceptible or Infected. Infected nodes can continue to infect for their entire lifetime and never recover. Everyone is infected eventually. But how fast? Can you think of an example that would fit this model? (Computer malware, Rumors, “Friendly” Bacteria) 19

The Continuous Case 20

SIR Model A general model for the spread of an epidemic on a graph: Nodes can be in one of 3 states Susceptible Infected Removed ( /Recovered ) Once nodes are removed they never get infected again. Best suited for a disease contracted only once in a lifetime. 21

SIR Model – Continuous ver. 22 *Courtesy of Lev Muchnik

Example 23 *Courtesy of Lev Muchnik

SIR – A Discrete-Time Network Model 24

SIR - Network Model 25

SIR - Network Model 26

SIR - Network Model 27

SIR - Network Model 28

SIR - Network Model 29

SIR - Network Model 30 Some nodes never got infected.

SIS model 31

Example Ongoing study from 1948 examines individuals in 3 year intervals. Each person is classified as content/not content/ neutral (using questionaire) 32 Hill, A. L., Rand, D. G., Nowak, M. A., & Christakis, N. A. (2010). Emotions as infectious diseases in a large social network: the SISa model. Proceedings of the Royal Society B: Biological Sciences, 277(1701),

33 Hill, A. L., Rand, D. G., Nowak, M. A., & Christakis, N. A. (2010). Emotions as infectious diseases in a large social network: the SISa model. Proceedings of the Royal Society B: Biological Sciences, 277(1701),

Model: A variant of the the SIS model: A neutral person is susceptible. Becomes infected either spontaneously (Automatically) with no contacts or through infection by a social contact Two forms of infection: Becoming content or dis-content. 34 Hill, A. L., Rand, D. G., Nowak, M. A., & Christakis, N. A. (2010). Emotions as infectious diseases in a large social network: the SISa model. Proceedings of the Royal Society B: Biological Sciences, 277(1701),

35 Hill, A. L., Rand, D. G., Nowak, M. A., & Christakis, N. A. (2010). Emotions as infectious diseases in a large social network: the SISa model. Proceedings of the Royal Society B: Biological Sciences, 277(1701),

To establish the social network structure: Social ties on each individual included: – Family – (Self nominated) friends – Neighbors – Co-Workers 36

37 Hill, A. L., Rand, D. G., Nowak, M. A., & Christakis, N. A. (2010). Emotions as infectious diseases in a large social network: the SISa model. Proceedings of the Royal Society B: Biological Sciences, 277(1701),

Fitting parameters to the data: 38 Spontaneously becoming content Infected by a contact Becoming Neutral Again Spontaneously becoming discontent Infected by a contact Becoming Neutral Again Hill, A. L., Rand, D. G., Nowak, M. A., & Christakis, N. A. (2010). Emotions as infectious diseases in a large social network: the SISa model. Proceedings of the Royal Society B: Biological Sciences, 277(1701),

Questions about the work Tie strength un-accounted for Is there selection bias? Do people tend to be friends with individuals that are prone to become more (dis-)content? 39

A Question 40

Scale Free models Reminder: In scale free networks some nodes have extremely high degree distributions This has two effects: 1.High degree nodes get infected more often than other nodes (more incoming links for infection to travel on). 2.If high-degree nodes get infected they infect many others. (reminder: item 1. is the friendship paradox at work – it is most likely that the friends of a uniformly selected node have more friends than the node itself) 41

Scale Free models 42 *Pastor-Satorras, Romualdo, and Alessandro Vespignani. "Epidemic spreading in scale-free networks." Physical review letters (2001):

This effect initially observed with regards to computer viruses, that persisted far longer than would be expected (via analysis of messages on a virus alert bulletin board) 43 *Pastor-Satorras, Romualdo, and Alessandro Vespignani. "Epidemic spreading in scale-free networks." Physical review letters (2001):

Immunization 44 Dezső, Zoltán, and Albert-László Barabási. "Halting viruses in scale-free networks." Physical Review E 65.5 (2002):

A Question 45

Answer Yes. Below: d=5 What should the infection prob. be to get an epidemic? What about an SIS process on this graph? 46

Infection in Kleinberg’s Small World Model Consider an n-by-n grid of nodes. How quickly does an infection travel on the grid? What if we added uniformly distributed long- distance outgoing links? What do you think happens in the threshold infection model? 47

SmartPhone Viruses Ways for viruses to spread BlueTooth Apps Connection to a PC MMS attachments Web and other files being sent ( , etc.) Malware on your smartphone can potentially: Have access to credit cards (app store purchases) Can call and text See your contacts and spread further through them. (anything else PC malware can do – e.g. identity theft) Very few phones have anti-virus software. 48

(pre smartphone era analysis but still highly relevant) 49 Wang, P., González, M. C., Hidalgo, C. A., & Barabási, A. L. (2009). Understanding the spreading patterns of mobile phone viruses. Science, 324(5930),

Basic Prediction: Cell phone (smartphone) viruses will be more common when market share grows. 50 Wang, P., González, M. C., Hidalgo, C. A., & Barabási, A. L. (2009). Understanding the spreading patterns of mobile phone viruses. Science, 324(5930),

Market share.01 vs.15 Blue tooth eventually reaches all users via spatial infection, but is relatively slow MMS spreads to connected component of the fragmented network almost immediately but stops there Hybrid virus enjoys best of both worlds 51

52 Wang, P., González, M. C., Hidalgo, C. A., & Barabási, A. L. (2009). Understanding the spreading patterns of mobile phone viruses. Science, 324(5930),