sn- lecture 11

Contagion On Social Networks

Lecture 11

To understand:

Aims Lecture 11

The functioning of contagion in networks

Some properties of real life social networks

How does network structure impact behavior?

Networks & Behavior

In this lecture we will take the networks as given

We are going to see the effect of a network on behavior

Simple infections, contagion, diffusion (1 or 0)

Choices, decisions - games on networks (strategic interaction )

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Next Lecture

However

Keep in mind

The relation between the outcomes we can get for a given network will affect which links we form

There is a co-determination between structure & behavior

The macro-micro-macro link

For this course, we will look at them separately

At the end of the course we will say something more about it

Example

Contagion

In a girls dormitory in college, he asked students about their friendship: specifically in the dinning table

Jacob Moreno (1960)

Two choices (number 1 and number 2 friends you dine with)Who do you dine with?

He put all the data into a network

Example

Contagion

1

1

Cora

2

Jean

Hellen

Robin2

11

Ada

2

1

2 2

Louise Lena

Marion

21

2

Eva

2

1

Martha

21

1

2

1

2

2

Adele

Maxine

Frances1

12

Anna1

Alice Laura

Ella

2

1

Ellen2 1

Edna

1 21 2

Mary

Jane

Hazel

Betty

Hilda

Ruth

Irene

2

11

22

1

21

2

12

1

21

1

2

Contagion

1

1

Cora

2

Jean

Hellen

Robin2

11

Ada

2

1

2 2

Louise Lena

Marion

21

2

Eva

2

1

Martha

21

1

2

1

2

2

Adele

Maxine

Frances1

12

Anna1

Alice Laura

Ella

2

1

Ellen2 1

Edna

1 21 2

Mary

Jane

Hazel

Betty

Hilda

Ruth

Irene

2

11

22

1

21

2

12

1

21

1

2

Ada is my first choice and Jean my

second

Direction in the connections

Contagion

1

1

Cora

2

Jean

Hellen

Robin2

11

Ada

2

1

2 2

Louise Lena

Marion

21

2

Eva

2

1

Martha

21

1

2

1

2

2

Adele

Maxine

Frances1

12

Anna1

Alice Laura

Ella

2

1

Ellen2 1

Edna

1 21 2

Mary

Jane

Hazel

Betty

Hilda

Ruth

Irene

2

11

22

1

21

2

12

1

21

1

2

Ada is my first choice and Jean my

second

Hellen & Robin are my choices

Direction in the connections

Who is the most popular girl?

Contagion

1

1

Cora

2

Jean

Hellen

Robin2

11

Ada

2

1

2 2

Louise Lena

Marion

21

2

Eva

2

1

Martha

21

1

2

1

2

2

Adele

Maxine

Frances1

12

Anna1

Alice Laura

Ella

2

1

Ellen2 1

Edna

1 21 2

Mary

Jane

Hazel

Betty

Hilda

Ruth

Irene

2

11

22

1

21

2

12

1

21

1

2

In directed networks

Variants of degree

In-degree

Number of links from others to me

Out-degree

Number of links from me to others

Reciprocity

I choose a person who also chooses me

Isolated Group

Contagion

1

1

Cora

2

Jean

Hellen

Robin2

11

Ada

2

1

2 2

Louise Lena

Marion

21

2

Eva

2

1

Martha

21

1

2

1

2

2

Adele

Maxine

Frances1

12

Anna1

Alice Laura

Ella

2

1

Ellen2 1

Edna

1 21 2

Mary

Jane

Hazel

Betty

Hilda

Ruth

Irene

2

11

22

1

21

2

12

1

21

1

2

No reciprocityout degree>in degree

Size of these isolated groups

Important

If you are in an isolated group, and a disease outbreaks, it might get

stucked in those isolated locations

Measures of connectivity & navigation in the network are fundamental for problems of

contagion

Always mutual

Contact

Cora

Jean

Hellen

Robin

Ada

Louise Lena

Marion

Eva

Martha

Adele

Maxine

Anna

Alice Laura

Ella

Ellen

Edna

Mary

Jane

Hazel

Betty

Hilda

Ruth

Irene

Frances

Who are the people in more danger?Contagion

Cora

Jean

Hellen

Robin

Ada

Louise Lena

Marion

Eva

Martha

Adele

Anna

Alice Laura

Ella

Ellen

Edna

Mary

Jane

Hazel

Betty

Hilda

Ruth

Irene

Frances

Maxine

I am sick

& after that?Contagion

Cora

Jean

Hellen

Robin

Ada

Louise Lena

Marion

Eva

Martha

Adele

Anna

Alice Laura

Ella

Ellen

Edna

Mary

Jane

Hazel

Betty

Hilda

Ruth

Irene

Frances

Maxine

Contagion

The previous were just some questions you can answer using networks

More to come up, but first an example

Transmission NetworkExample

https://www.youtube.com/watch?v=VZGHGVIedzA



What’s the extent of diffusion?

Other Questions

How does it depend on the process as well as the network?

Is everyone infected?

Are some network architechtures more suitable for contagion?

Low density - no contagion

Some main results

Part of the population infected

Degree affects who is infected & when

Middles density - some probability of contagion

High density - sure infection & all infected

Network structure matters

This is only one side of the problem

How do they look like?Real life networks

It is important to know what kind of networks allow transmission to flow better/worse

But, how do real life networks relate to this?Are there universal structural properties?

Every network is unique microscopically, but with appropriate definitions, stricking macroscopic commonalities emerge

Large scale networksProperties

Main claim:Typical large scale networks exhibit:

Heavy-tailed degree distributions

Small diameter

High clustering

Hubs or connectors

Six degrees of separation?

Friends of friends are friends

degree distributionsHeavy-tailed

Lots of nodes with small degree and few nodes with very high degree

Degree

Number of nodes

degree distributionsHeavy-tailed

Erdös Number Projecthttp://www.oakland.edu/enp/

Paul Erdös1913-1996

Collaboration network between mathematicians

Nodes are mathematiciansLink if they coautor a research paper togetherPaul Erdös is in the centerThe number of a node is her distance to P.E.P.E. has an Erdös-number = 0A coauthor of P.E. has Erdös-number = 1

Their coauthors = 2, and so on...

http://www.oakland.edu/enp/

http://www.oakland.edu/enp/

Network of P.E.’s coauthorsErdös-number

Degree distributions

Details:

Size (N)

410,000 authorsSize (g)

676,000 links

Average (di)

3.36

Erdös number 0 --- 1 personErdös number 1 --- 504 peopleErdös number 2 --- 6593 peopleErdös number 3 --- 33605 peopleErdös number 4 --- 83642 peopleErdös number 5 --- 87760 peopleErdös number 6 --- 40014 peopleErdös number 7 --- 11591 peopleErdös number 8 --- 3146 peopleErdös number 9 --- 819 peopleErdös number 10 --- 244 peopleErdös number 11 --- 68 peopleErdös number 12 --- 23 peopleErdös number 13 --- 5 people

Diameter (N,g)

7.64

Similar network for acting & Kevin Bacon

Compared to the population sizeSmall Diameter

Arguably, every person in the world is at diameter 6 from anyone else

http://www.youtube.com/watch?v=HLIyuYwbVnA

Think about Milgram’s experiment with the letters

Other Networks:Messenger (Lescovec & Horvitz, 2008)

Diameter = 6.5; N = 180 millionsFacebook (Backstrom et al., 2012)

Diameter = 5; N = 721 millions



Compared to average degreeHigh Clustering

How likely two nodes that share a common neighbor are to be neighbors themselves

Examples Networks:(Watts, 2003)

Movie actor networkC.C.=0.79 ; p=0.00027Neuronal networkC.C.=0.28 ; p=0.05

Checklist

Network structure matters for contagion

Individual degrees affect who is infected and when

Real life networks portray some universal properties

Heavy-tailed degree distribution

Small diameter compared to the size of the population

High clustering compared to the average degree

Questions?

sn- lecture 11

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