Work and play: Disease spread, social behaviour

and data collection in schools

Dr Jenny Gage, Dr Andrew Conlan, Dr Ken Eames

Interpreting the network

Julia

Ken

Andrew

Tom AliciaJosh

Roberto

Johann

• How is the network different from a random network?• What features of the network are unexpected?

Classifying links

The out-degree is the number of people the student named:

A1 has out-degree 3

The in-degree is the number of people who named the student:

A1 has in-degree 2

A1

A1

Mutual links

Two students who both name each other form a mutual link

A4A1

Person Contact1

Contact2

Out-Degree

In-degree

Mutual Links

Alicia Julia Ken 2 2 1Andrew Julia Ken 2 2 2Johann Julia Tom 2 1 1Josh Julia Alicia 2 2 1Julia Andrew Josh 2 5 2Ken Andrew Alicia 2 3 2Roberto Julia Ken 2 0 0Tom Johann Josh 2 1 1

Julia

Ken

Andrew

Tom AliciaJosh

Roberto

Johann

Mutual links

Using the data table on the previous slide

4. Find the Mutual Degree

1. Find the Out-degrees

2. Find the In-degrees

3. In the table, circle Mutual Links

How variable is the dataset?Are these patterns random?

A1

A1

A1

Activity

Degree distribution

We can plot the degree distribution as a bar chart

In-degree Mutual degree

Some variation is natural; can use statistical tools to tell us how unexpected the observed distributions are.

• 8 people fill in the survey; each names 2 contacts.

• The probability that Alicia’s first contact (Julia) also names Alicia equals 2/7. Why?

• Total number of mutual links expected is therefore

8 x 2 x 2/7 ≈ 4.6

Mutual links

Person Contact 1 Contact 2Alicia Julia Ken

If people choose their contacts at random, how many mutual links would we see?

• 8 people fill in the survey; each names 2 contacts.

• The probability that Alicia’s first contact (Julia) also names Alicia equals 2/7. Why?

• Total number of mutual links expected is therefore

8 x 2 x 2/7 ≈ 4.6

Mutual links

Person Contact 1 Contact 2Alicia Julia Ken

If people choose their contacts at random, how many mutual links would we see?

Actually this is double the number of links, since

each link has two ends. It’s the number of entries ringed in red in the data

table.

• N people fill in the survey; each name k contacts.

• The probability that person A’s first contact names person A equals k / (N - 1).

• Total number of mutual links expected is therefore:

Mutual links

With 8 people, we expect 4.6 mutual contacts:

Person Contact1 Contact2Alicia Julia KenAndrew Julia KenJohann Julia TomJosh Julia AliciaJulia Andrew JoshKen Andrew AliciaRoberto Julia KenTom Johann Josh

Mutual links

We expect 4.6 mutual contacts, but in fact find 10.

Person Contact1 Contact2Alicia Julia KenAndrew Julia KenJohann Julia TomJosh Julia AliciaJulia Andrew JoshKen Andrew AliciaRoberto Julia KenTom Johann Josh

Many more mutual links than a random network.

This is what we would expect if connections represent interactions such as friendships.

Mutual links

• Split into groups of 8-12.• Each choose two other members of the group.• Write everyone’s choices in a data table.• Make the network:

– write each person’s name on a piece of paper– place person with the most connections in the

centre– starting with the second most “popular” arrange

the other names around the centre– work through the table and make connections– move people around to make the network clearer– draw final network onto paper

Activity

• Tabulate the in-degree and out-degree for each person.• Find the actual number of mutual links.• Calculate the predicted number of mutual links, using the

formula:

• Do you think the choices you made were random or not?

Activity

N = no. people

in group

k = no. choices

Example of network data

Primary school network, pupils aged 10-11.

What can you tell from this network?

It is likely that:•green and red distinguish between boys and girls•someone was absent

Cliques where everyone names everyone else

Why is this near-clique unusual?