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 TomAliciaJosh 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

Mutual links Two students who both name each other form a mutual link A4A1

Person Contact 1 Contact 2 Out- Degree In- degree Mutual Links AliciaJuliaKen221 AndrewJuliaKen222 JohannJuliaTom211 JoshJuliaAlicia221 JuliaAndrewJosh252 KenAndrewAlicia232 RobertoJuliaKen200 TomJohannJosh211 Julia Ken Andrew TomAliciaJosh 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 Activity

Degree distribution We can plot the degree distribution as a bar chart In-degreeMutual 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 Alicias 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 PersonContact 1Contact 2 AliciaJuliaKen 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 Alicias 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 PersonContact 1Contact 2 AliciaJuliaKen 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. Its 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 As 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: PersonContact1Contact2 AliciaJuliaKen AndrewJuliaKen JohannJuliaTom JoshJuliaAlicia JuliaAndrewJosh KenAndrewAlicia RobertoJuliaKen TomJohannJosh Mutual links

We expect 4.6 mutual contacts, but in fact find 10. PersonContact1Contact2 AliciaJuliaKen AndrewJuliaKen JohannJuliaTom JoshJuliaAlicia JuliaAndrewJosh KenAndrewAlicia RobertoJuliaKen TomJohannJosh 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 Each choose two other members of the group. Write everyones choices in a data table. Make the network: –write each persons 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

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?