Social Networks & Structure

Block 4

This block comprises the remaining second part of the course

Aim Block 4

To understand

The emergence of social networks & their consequences for individual behavior and well-being

How do networks emerge (form)?

How do networks influence behavior?

Lecture 8

B a c k g r o u n d & Fundamenta l s o f Social Networks

Lecture 8

Aim Lecture 8

To learn

Basic ways of navigating through a network

How to represent networks

Some elements of the structure of a network

Nodes, links...

Walks, paths, cycles...

Components, Neighborhoods, clustering

Network Studies

Social Networks

People’s relationships show patterns

SOCIAL NETWORKS

These social networks affect people’s lives considerably

Social NetworksExamples

Western industrialized countries

Marsden & Gorman, 2001

Between 1/3 & 2/3 of the working population found their job through informal social ties

Chances of illness, recovery or dyingPartly depend on people’s network

House, Landis & Umberson, 1988

Participation in political protestAffected by friendship and family networks

Opp & Gern, 1993

study networksMany disciplines

Sociology

In the course we will look at networks in which

Phone networks - Email networks - Marriages - Friendships - co-authorships - collaborations

Decision-making people are the nodes

Economics

Computer Science

Statistical Physics

Mathematics (graphs theory)

Relationships between different people, organizations, countries

ExampleFacebook - Friendship relations

Mary

Ana Tom

Link between two people indicates they are friends (relationship)Ana is friends with Tom & Mary

Centrality - How many paths does a person have between othersIf Mary & Tom want to meet they need to go through Ana

Ana lies in the shortest path between Tom & MaryPopularity

There are different kindsPatterns in Networks

Global patternsHow are the different connectedness of individuals distributed in the society (people well connected, or not)?

How long does it take me to reach one person from another (path lengths)?

Segregation patternsIf people have different characteristics (types), do we see separation between them?

There are different kindsPatterns in Networks

Local patternsDo we see tight clusters of people connected to each other?

Positions in networks

How influential is somebody in the network (centrality)?

Are my friends friends between each other (transitivity)?

We will be looking at networks overall & also zooming-in to the individuals

Macro and Micro levels

Representing NetworksSome notations

N={1,...n}

gij=1

nodes, vertices, players

g in {0,1}nxn adjacency matrix

a b c dabcd

0 1 0 01 0 1 10 1 0 00 1 0 0

a b

cd

link, tie, edge between i and j

ij in g alternative notation

Network (N,g) pair: set of nodes and adj. matrix

g=

Ways of navigating through a networkBasic Definitions

A walk from i1 to ik: a sequence of nodes (i1, i2,...,ik) and a sequence of links (i1i2,i2i3,...,ik-1ik) such that ik-1ik in g for each k

A sequence of links that take you from the first node to the last node

A path is a walk (i1, i2,...,ik) with each node ik distinct

A cycle is a walk where i1 = ik

A geodesic is a shortest path between two nodes

Represented by its nodes (i1, i2,...,ik)

Illustration

1

2 3

4 56

7

A path (and a walk) from 1 to 7:1,2,3,4,5,6,7

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2 3

4 56

7

Walk from 1 to 7 that is not a path1,2,3,4,5,3,7

1

2 3

4 56

7

Simple cycle (and a walk)from 1 to 1: 1,2,3,1

1

2 3

4 56

7

Cycle (and a walk) from 1 to 1:1,2,3,4,5,3,1

walks, paths, cyclesImportance

will help us understand

Centrality

DiffusionTransmission of information

Subgraphs that make up the network

Components

Connectivity

A network (N,g) is connected if there is a path between every two nodes

I can reach any node from any other node

Component

A maximally connected subgraph- (N’,g’) is a subset of (N,g)- (N’,g’) is connected- i in N’ and ij in g, implies j in N’ and ij in g’

Most social networks(even large) have the property that a large portion of nodes are connected

Illustration

Is this a component?

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2

3 4

5 10

6

7 8

9

Is the subset of nodes {3,4,5} and links {3-4,4-5,5-3} maximally connected?

Can we find any player connected to either of the nodes in the subset {3,4,5} who is not in our

selected “component”?

Illustration

This is not a component: player 1 is left out

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2

3 4

5 10

6

7 8

9

What are the components in this network?

Illustration

This is not a component: player 1 is left out

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2

3 4

5 10

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7 8

9

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3 4

5 10

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This is a network with 4 components

Tend to have short path lengths

Real life networks

We connect a large number of nodes using fairly small number of links

Milgram (1967) - Experiment

Ask people to send letters from one part of the US to anotherI have to send a letter to John

Smith, who lives in Massachusetts, who is a lawyer, and I live in

Nebraska

Send the letter to someone you know, tell them to send them to somebody they know, and so forth...

The intent is reaching the person the letter is sent to

FindingsMilgram (1967) - How many steps?

Median 5 for the 25% of the letters that made itQuite small - consider you start from an individual to reach another, in the other side of the country, without knowing her

Coauthorship studies - Reaching one author from anotherGrossman (1999) Math: mean 7.6, max 27Newman (2001) Physics: mean 5.9, max 20

Goyal et al. (2004) Economics: mean 9.5, max 29WWW - Reaching one page from another

Adamik & Pitkow (1999): mean 3.1 (in 85% of 50 million possible pagesFacebook - Reaching one person from anotherBackstrom et al. (2012): mean 4.74 (721 million users)

DefinitionsNeighborhood & Degree

The neighborhood of a node i: Ni(g)={j|ij in g}The set of other nodes that i is linked to in g

The degree of a node i: di=#Ni(g)Count how many neighbors i has in g

Basic ideas that will be very important along the rest of the course

Average degree tells only part of the story

Degree Distributions

Do most nodes have very similar or very different degrees?Does everyone has one or two

If we want to understand some properties of the network

1 2 3 4 5 6 7

Does some have 6 and others 1

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2 3

4 56

7

Different properties (i.e., diffusion)

What fraction of my friends are friends with each other

Clustering

Take a given node i i

k

j

Choose two neighbors j and kWhat is the chance that j and k are friends?What is the frequency of links among the friends of i?

Clustering of i: Cli(g)=#{kj in g|k,j in Ni(g)}/#{kj|k,j in Ni(g)}

Fraction: How many pairs of my friends are connected to each other divided by every pair of friends I have

Average Clustering: Mean of all the individual clusterings

Overall Clustering: Aggregate of all the individual clusterings

Illustration

i

Average tends to 1

Overall tends to 0

Friendship network

In groups all friends are friends

They are not friends between the groups

AssumeWe add, to player i, more groups of 3, all friends with each other

My friends are friends

Transitivityi

k

j

Tendency towards transitivity

The existence of two links ij in g and jk in g often makes it more likely that also the link ik in g exists:

prob(ik|ij,jk in g) > prob(ik in g|ij in g and ik not in g)

Observed in many real-life networks:If people link because they share common attributes (i.e., race), transitivity is one of the consequences. (Simmel, 1908)

In networks of inter-firm alliances transitivity is common, for it helps reduce risk of opportunistic behavior by sharing common partners (Gulati, 1998)

Checklist

Many relationships are networked

i

k

j

Understanding network structure can help us understand behavior & outcomes

Networks are complexBut can be partly described by some characteristics- Degree distribution- Clustering

In the next lecture we will check network characteristics & behavior

Questions?