Finding the Paul Brown Number

Introduction

American football is an ever-evolving game where coaches use different tactics and philosophies to create a system with the singular goal of winning football games. Because of this, the sport has a tradition of tracing coaches’ linage; and in the modern NFL, head coaches linages can be traced back to coach Paul Brown:

Figure 1: Head Coaches Genealogy

Figure 1 shows the NFL’s teams’ head coaches at the beginning of the 2024 season, and they all lead back to Paul Brown. Merging football with my other interest in mathematics, I decided to measure the scope’s of Paul Brown influence on the NFL via “collaborative distance”.

Measuring Collaborative Distance

The mathematician Paul Erdos was the most prolific mathematician of the 20th century, publishing around 1500 mathematical papers. To measure his influence on mathematics, Erdos’ peers defined the Erdos number to measure the collaborative distance between a mathematician and Erdos. For example:

Erdos number 0: Erdos himself.
Erdos number 1: Someone who wrote an article with Erdos.
Erdos number 2: Someone who wrote an article with a person who has Erdos number 1.
Erdos number n: Someone who wrote an article with a person who has Erdos number n-1.

Methodology

Gather data from pro-football-reference. The data consists of all the people who coached from 1946, the year that Paul Brown started coaching professional American football, to 2023.
Refine the data to only relevant information and store it in a CSV file which looks like this:
```
name,position,record,team,year
Bill Belichick,Head Coach,8-9-0,nwe,2022
```
Use Pandas to perform data analysis.

Findings

There are 1227 people who coached in the NFL between 1946 and 2023.
Out of 1227, here are the statistics about the Paul Brown’s number:
- 13 coaches with Paul Brown’s number 1.
- 99 coaches with Paul Brown’s number 2.
- 530 coaches with Paul Brown’s number 3.
- 520 coaches with Paul Brown’s number 4.
- 49 coaches with Paul Brown’s number 5.
- 2 coaches with Paul Brown’s number 6.
- 14 coaches where Paul Brown’s number is undefined.
You can find the full JSON output here.

Final Thoughts

Paul Brown coached in the NFL from 1946 to 1975, and he only worked directly with 13 other coaches. The most notable of this group is Bill Walsh. Despite this, Paul Brown’s influence is widespread, as there are only 14 coaches with no direct connection. If you are interested to see the code behind this, you can find the code repository I use to calculate these values here.