Google is one of the worlds most valuable companies, with an unquestionable hold over the search engine market, and influence in many other areas. There are many stats that show just how big Google is, but this hardly seems like something that needs to be proven. Everyone knows Google!
When you ask Google a question, it searches an index of web pages and returns results depending onits ranking algorithm, which depends on over 200 parameters. It is likely that many of these are unique to Google, but Googles founders – Larry Page and Sergey Brin – only wrote an academic paper about one: PageRank. And Page and Brins impact has not been limited to search; PageRank has been adapted for many uses beyond the web, from finding the leaders of terrorist networks to optimising engineering based on traffic flow. PageRank gives a measure of the quality of any webpage, or person, or stop-sign – and it can be used in football too, to find the most crucial player in a passing network, in a metric we will call PassRank.
The basic idea of PageRank is that the importance of any web page can be judged by the pages that link to it. If there are many pages that link to a specific page, it is likely that the page is both relevant and important, and that it is a trustworthy source of information. This is even truer if the pages that are linking to the site are themselves respected – an important site has a reputation to uphold, which would be damaged if it were linking to poor quality sites, or sites that did not add any useful information. As a result, a score can be assigned to every page, based off the scores of the pages that link to it.
Likewise, if many players pass to a central figure, then it is likely that he is pretty good, especially if the players passing to him are themselves good – and so more likely to have made the right decision when distributing the ball.
How does this work in practice? We can represent the passes between players in a team by a directed graph, with edges for passes and weights representing the number of times each pass is made. This graph can be summarised by a transition matrix, T, whose entries give the proportion of passes that a given player made to a particular recipient. To illustrate this, consider the reduced network of Arsenal attackers below, taken from the Arsenal v West Ham match on the opening day of the 2015-16 season.
Let us now follow the ball in a random series of passes. It can start at any player with equal probability – say it starts with Cazorla. Of the 23 passes Cazorla made (to another attacker) all game, 15 went to O, giving a probability of 15/23 that the ball will be at Özil after one pass; likewise, it will be at Oxlade-Chamberlain or Giroud with probabilities 5/23 and 3/23 respectively. So if we represent the probabilities of each player having the ball by a vector, we get:
After two pass
