Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

How connected is my year group? Analysing a friendship network

AI HLAI SL Solid Also in: Statistics

Research question

In an anonymised friendship network of my year group, what is the average shortest path between two students, and is the “friendship paradox” (your friends have more friends than you) true?

Make it yours, then agree it with your teacher.

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Why it makes a good exploration

Network scientists study “small worlds” and the friendship paradox. With careful ethics, a year-group network gives unusual primary data and some surprising mathematics.

The mathematics you'll need

Graphs, degree and degree distribution
Adjacency matrices and powers to count paths
Shortest path lengths
Mean degree vs mean degree of friends

Where the data comes from

With teacher approval and consent, collect anonymised “who do you talk to regularly” data; never publish names.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

A possible outline

Plan the survey ethically and anonymously.
Build the adjacency matrix.
Compute degrees and path lengths.
Test the friendship paradox.
Reflect on survey honesty and one-way friendships.

Pitfalls that cost marks

Any identifiable data in the IA.
Directed vs undirected friendships not defined.
Too small a network to say anything.

Showing personal engagement

Explain why the paradox happens in your own words.
Compare with an online network you use (conceptually).
Discuss what the network means for spreading news or illness.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Not a natural fit	The mathematics is mainly from the AI course; at AA HL the exploration would need an AA-level approach (calculus, proof or probability theory) to reach the top of Criterion E.
AI SL	Good fit	Graphs, degree and degree distribution; Adjacency matrices and powers to count paths
AI HL	Good fit	Graphs, degree and degree distribution; Adjacency matrices and powers to count paths

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: any identifiable data in the IA; and directed vs undirected friendships not defined. Say how each one affects your answer.

Use of mathematics (E)

SL: graphs, degree and degree distribution; adjacency matrices and powers to count paths; shortest path lengths; mean degree vs mean degree of friends, each explained in your own words and carried out correctly.

How networks & graphs explorations reach the top bands in general: Networks, graphs & algorithms IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Compare your network's clustering with a random network of the same size.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL networks & graphs exploration adds.

See a complete IA, marked

What is the quickest way to deliver a leaflet to every house on my estate? (AI HL) is a complete networks & graphs exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AI HL and AI SL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Graphs, degree and degree distribution, adjacency matrices and powers to count paths, shortest path lengths and mean degree vs mean degree of friends.

Where can I get data for this IA?

With teacher approval and consent, collect anonymised “who do you talk to regularly” data; never publish names. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “How connected is my year group? Analysing a friendship network” filled in: the question, the maths and the data sources.

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