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

How does Google rank pages? PageRank on a small website

AI HLAA HL Ambitious Also in: Probability, Pure maths, Matrices & Markov

Research question

What are the PageRank scores of the pages in [a small website such as my school's], found as the steady state of a Markov chain, and how sensitive are they to the damping factor?

Make it yours, then agree it with your teacher.

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Why it makes a good exploration

PageRank turned the web into a Markov chain. Mapping the links of a real small website and computing its ranking with matrices is advanced, authentic AI HL mathematics.

The mathematics you'll need

Adjacency and transition matrices
Steady-state vectors and eigenvectors with eigenvalue 1
The damping factor and why it is needed
Matrix powers with technology

Where the data comes from

Map the links between 8–15 pages of a small website by hand.

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

A possible outline

Explain the random-surfer model.
Build the transition matrix.
Find the steady state (powers and eigenvectors).
Vary the damping factor.
Reflect on dead ends, link farms and what “importance” means.

Pitfalls that cost marks

Columns not summing to 1.
Dead-end pages not handled.
Presenting numbers without interpretation.

Showing personal engagement

Use your school's or club's website.
Suggest a link change to raise an important page and test it.
Compare the ranking with page visit counts if available.

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	Good fit	Adjacency and transition matrices; Steady-state vectors and eigenvectors with eigenvalue 1
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Good fit	Adjacency and transition matrices; Steady-state vectors and eigenvectors with eigenvalue 1

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. 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: columns not summing to 1; and dead-end pages not handled. Say how each one affects your answer.

Use of mathematics (E)

HL: adjacency and transition matrices; steady-state vectors and eigenvectors with eigenvalue 1; the damping factor and why it is needed; matrix powers with technology, 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

Prove that the damped matrix has a unique steady state (outline the idea) or compare with simple link counting.

Extending it for HL

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

See a complete IA, marked

How long does a game of Snakes and Ladders last on my grandmother's board? (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 AA HL. We rate it Ambitious: it suits confident students, who should expect to learn some mathematics on their own.

What mathematics does this IA use?

Adjacency and transition matrices, steady-state vectors and eigenvectors with eigenvalue 1, the damping factor and why it is needed and matrix powers with technology.

Where can I get data for this IA?

Map the links between 8–15 pages of a small website by hand. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “How does Google rank pages? PageRank on a small website” filled in: the question, the maths and the data sources.

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