PageRank

Introduction

You’ve likely heard of PageRank—it’s the original Google algorithm that revolutionized web search.

PageRank

In this lesson, we’ll apply PageRank to our citation network to find the most influential papers.

At its core, PageRank reveals which nodes matter most by measuring recursive influence.

What You’ll Learn

By the end of this lesson, you’ll be able to:

Quality Over Quantity

PageRank optimizes for quality over quantity. Being cited by an influential paper matters far more than being cited by an obscure one.

A single citation from a foundational paper in your field can outweigh dozens of citations from lesser-known work.

The Random Surfer Model

To understand how PageRank works, imagine a researcher randomly browsing through papers in a library.

Let’s walk through what this researcher does, step by step.

Random Surfer: Step 1

First, our researcher starts at a random paper in the network.

They’ve picked up a paper off the shelf without any particular goal in mind.

Random Surfer: Step 2

Next, most of the time — about 85% of the time with dampingFactor: 0.85 — they follow a citation to another paper.

They see an interesting reference, so they go find that paper and start reading it instead.

Random Surfer: Step 3

But occasionally — about 15% of the time with dampingFactor: 0.85 — they get bored or distracted and jump to a completely random paper instead.

Random Surfer: Step 4

Now imagine this researcher repeats this process millions of times.

Over time, they’ll visit some papers far more often than others.

What PageRank Represents

PageRank equals the probability of landing on each paper during this random browsing process.

Papers that the researcher lands on frequently are considered influential—they’re the papers that naturally attract attention through the network structure.

The Damping Factor

Remember how our random surfer sometimes jumps to a random paper instead of following citations?

The probability of this random jump is controlled by a parameter called the damping factor.

Damping Factor Values

When we set dampingFactor = 0.85, we’re saying that 85% of the time the surfer follows links, and 15% of the time they jump randomly.

The damping factor must be between 0 (inclusive) and 1 (exclusive).

Damping Factor Effects

Higher damping values place more weight on the actual link structure of the graph.

Lower damping values produce a more uniform distribution, where all nodes end up with similar scores.

The default value of 0.85 works well for most graphs, so you rarely need to change it.

Why Do We Need Random Jumps?

The random jump mechanism isn’t just a quirky detail—it actually solves three important problems that can break the algorithm.

Let’s look at each one.

Problem 1: Spider Traps

Spider traps are groups of nodes with no outgoing links to the rest of the graph.

Without random jumps, all the ranking signals would get "trapped" and accumulate here, giving these nodes artificially high scores.

Problem 2: Rank Sinks

Rank sinks are cycles in the graph that accumulate rank infinitely.

The ranking signals keep circling around and around, never escaping to the rest of the network.

Why Not Just Count Citations?

You might wonder: why not just count how many citations each paper has?

Let’s look at an example to see why that approach falls short.

Citation Count vs PageRank

Importance, not popularity

Notice that Paper B has far fewer citations than Paper A, but a much higher PageRank score.

Why? Because Paper B is cited by the right papers—foundational work that itself has high influence.

Now Let’s Run PageRank

With the theory covered, let’s apply PageRank to our citation network.

First, we need to retrieve the graph projection we created in Lesson 5.

Setup: Retrieve the Projection

We use gds.graph.get() to retrieve an existing projection by name.

G = gds.graph.get("cora-graph")

print(f"Retrieved graph: {G.name()}")
print(f"  Nodes: {G.node_count():,}")
print(f"  Relationships: {G.relationship_count():,}")

Key Parameters

Here are the main parameters you can configure when running PageRank:

Understanding Convergence

PageRank is an iterative algorithm. It runs multiple passes over the graph, refining the scores each time.

The algorithm stops when one of two things happens:

Checking for Convergence

If the algorithm used all available iterations, it may not have fully converged.

We can check this by comparing ranIterations to our maxIterations setting.

if PR_result['ranIterations'] == 20:  # maxIterations default
    print("Warning: may not have converged!")
    print("Consider increasing maxIterations")

Top Papers Results

Let’s see which papers came out on top.

print("Top 10 Most Influential Papers:")
display(top_papers)

Subject Influence Results

This tells us which subject areas tend to produce more influential work.

print("Influence by Subject Area:")
display(subject_influence)

Cross-Disciplinary Papers

Some of the most interesting papers are those that bridge different research areas.

These cross-disciplinary papers connect ideas across fields and often have outsized impact.

Interpreting Results

One important thing to remember is that PageRank scores are relative, not absolute.

Citation Network

In our citation network specifically:

Combining with Other Metrics

PageRank tells us an important part of the story, but not the whole story.

For richer insights, combine PageRank with other metrics:

Common pitfalls

Before we wrap up, let’s cover three common mistakes people make when using PageRank.

Pitfall 2: Ignoring convergence

If ranIterations equals maxIterations, the algorithm may not have converged properly.

Always check for this condition, and use stats mode first to verify your parameters before writing results.

Pitfall 3: Damping factor extremes

The damping factor should stay close to the default of 0.85.

Setting it too high — close to 1 — causes the spider trap problems.

Setting it too low makes all scores converge toward uniform values, losing the signal you’re looking for.