Who links the play? Passing networks as matrices

Write down who passes to whom and you have a matrix. Its rows and columns count passes made and received, and multiplying it by itself shows how the ball travels in two passes, who links defence to attack, and who plays the one-twos.

28 September 2026 Intermediate Part 10 of Linear Algebra Through Football

Contents

The football question
The concept
Passes made and received
From counts to probabilities
Two passes: multiply the matrix by itself
The diagonal: one-twos
Why it matters
Limitations
Try it yourself
Further reading

The football question

Commentators talk about the player who "makes the team tick" or "links the play". Some players touch the ball constantly; others are the bridge between defence and attack. How do you find the link player from the numbers, rather than by eye?

Start by writing down who passes to whom. That table is a matrix, and the linear algebra from earlier in this series does the rest.

The concept

To keep the numbers small enough to read, take a made-up five-a-side match: a keeper, a defender, a left-sided and a right-sided player, and a striker. Count every completed pass between them. Put the passer in the row and the receiver in the column:

From \ To	K	D	L	R	S
K	–	8	4	3	1
D	5	–	12	10	3
L	1	9	–	6	8
R	1	7	5	–	7
S	0	2	4	3	–

In plain football

Read along a row for who a player passes to. The defender (row D) found the left-sided player 12 times and the striker only 3.
Read down a column for who a player receives from. The striker (column S) got the ball 8 times from the left, 7 from the right.
The diagonal is empty: you can't pass to yourself.
The matrix isn't symmetric. Keeper to defender is 8, defender to keeper 5. Direction matters.

This is called a passing network, and the matrix is its adjacency matrix. Drawn on the pitch, with each player at his average position, it looks like this:

Made-up five-a-side match, attacking left to right. Line thickness is passes between the pair, both ways (labelled where 10 or more); dot size is each player's passes made plus received.

Passes made and received

Add up each row and each column:

Player	Made	Received
Keeper	16	7
Defender	30	26
Left	24	25
Right	20	22
Striker	9	19

Adding up a row is multiplying the matrix by a vector of ones, the same row-times-column step as player ratings, with every weight equal to 1. The defender is the busiest passer. The striker receives twice as many as he makes, because his possessions often end with a shot, or with losing the ball.

From counts to probabilities

Divide each row by its total and every entry becomes a probability: when this player has the ball, how likely is he to pass to each team-mate? Call the new matrix $P$. The defender's row becomes

To	K	L	R	S
From D	0.17	0.40	0.33	0.10

So when the defender passes, he finds the striker directly only 10% of the time.

Two passes: multiply the matrix by itself

How often does the defender's ball reach the striker within two passes? It can go through any team-mate: defender to keeper to striker, defender to left to striker, and so on. For each route, multiply the two probabilities; then add up the routes:

$$\begin{aligned} &(0.17 \times 0.06) + (0.40 \times 0.33) \\ &\quad + (0.33 \times 0.35) \\ &= 0.010 + 0.133 + 0.117 \\ &\approx 0.26 \end{aligned}$$

In plain football

Each bracket is one route: via the keeper, via the left-sided player, via the right-sided player.
Via the left is the biggest: the defender goes left 40% of the time, and the left-sided player finds the striker a third of the time.
26% in two passes, against 10% direct. The defender rarely finds the striker himself, but he often sets it up.
This assumes each pass follows the team's usual habits, whatever happened on the pass before.

That sum of row-times-column products is exactly matrix multiplication. Doing it for every pair of players at once gives $P^2$: the chance of getting from any player to any other in exactly two passes. Here, the left-sided player carries more than half of the defender's two-pass route to the striker. He's the link.

The diagonal: one-twos

The diagonal of $P^2$ is the chance that the ball comes back to the same player after two passes: a one-two, or a return pass.

Player	Back within two passes
Defender	37%
Left	37%
Right	31%
Striker	29%
Keeper	10%

The defender and the left-sided player swap the ball most. The keeper's passes rarely come straight back.

Show the mathsMatrix powers, Markov chains and where the ball spends its time. Optional.

With pass counts $A_{ij}$ (from $i$ to $j$), the probability matrix is

$$P_{ij} = \frac{A_{ij}}{\sum_k A_{ik}}$$

Each row adds up to 1. The chance of going from $i$ to $j$ in exactly two passes is

$$(P^2)_{ij} = \sum_k P_{ik} P_{kj}$$

and in $n$ passes, $(P^n)_{ij}$. A matrix like $P$ describes a Markov chain: the next pass depends only on who has the ball now.

Keep multiplying by $P$ and the shares of the ball settle down to a fixed row vector $\pi$ with

$$\pi P = \pi$$

an eigenvector of $P$, with eigenvalue 1, as in PCA. It's the long-run share of passes each player would be involved in if the ball kept moving by these habits. Here $\pi$ is about keeper 6%, defender 25%, left 26%, right 23%, striker 20%. The same idea, applied to links between web pages, is Google's original PageRank. In football, versions of it rank players by how central they are to their team's passing.

Why it matters

Finding the link player. Two-pass and three-pass routes show who connects the units, not just who has the most touches.
Planning a press. If most of a team's route to its striker runs through one player, that's who to stop. Without the route through the left-sided player, the defender's two-pass chance of finding the striker drops from 26% to 13%.
Comparing matches. The same matrix for a win and a loss shows whether the team's shape of passing changed, and where.
Scouting. Passing networks from event data show how a player fits a team's patterns, not just how many passes he completes.

Limitations

A made-up match. Real passing networks come from event data that records every pass; our football-data results files don't include it.
Counts, not quality. A sideways pass and a defence-splitting one count the same. Weighting passes by how much closer to goal they move the ball (as in the passing part) changes the picture.
No memory. $P^2$ assumes each pass depends only on who has the ball, not on what happened before. Real moves have patterns, like rehearsed routines, that this misses.
Possessions end. Shots, fouls and lost balls aren't in the matrix. The striker's short row is partly about his job, not his passing.
Average positions mislead. Players move. The dots on the diagram are averages, and a player's average spot can be somewhere he rarely stood.
One match is a small sample. A few passes either way change the probabilities a lot.

Try it yourself

Using the pass counts above, work out the keeper's chance of reaching the striker within two passes. Which team-mate is the best route? (Answer: about 20%, against 6% direct, and the best route is via the left-sided player.)