The Creativity Code. Marcus du Sautoy

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      This means that if we split the balls in a 2:1:2 distribution we see that this weighting is stable. Distribute the balls using our previous game and the sites still have a 2:1:2 distribution.

      Eigenvectors of matrices are an incredibly potent tool in mathematics and the sciences more generally. They are the secret to working out the energy levels of particles in quantum physics. They can tell you the stability of a rotating fluid like a spinning star or the reproduction rate of a virus. They may even be key to understanding how prime numbers are distributed throughout all numbers.

      By calculating the eigenvector of the network’s connectivity we see that websites A and C should be ranked equally. Although website A is linked to by only one site (website C), the fact that website C is highly valued and links only to website A means that its link bestows high value to website A.

      This is the basic core of the algorithm. There are a few extra subtleties that need to be introduced to get the algorithm working in its full glory. For example, the algorithm needs to take into account anomalies like websites that don’t link to any other websites and become sinks for the balls being redistributed. But at its heart is this simple idea.

      Although the basic engine is very public, there are parameters inside the algorithm that are kept secret and change over time, and which make the algorithm a little harder to hack. But the fascinating thing is the robustness of the Google algorithm and its imperviousness to being gamed. It is very difficult for a website to do anything on its own site that will increase its rank. It must rely on others to boost its position. If you look at the websites that Google’s page rank algorithm scores highly, you will see a lot of major news sources and university websites like Oxford and Harvard. This is because many outside websites will link to findings and opinions on university websites, because the research we do is valued by many people across the world.

      Interestingly this means that when anyone with a website within the Oxford network links to an external site, the link will cause a boost to the external website’s page rank, as Oxford is sharing a bit of its huge prestige (or cache of balls) with that website. This is why I often get requests to link from my website in the maths department at Oxford to external websites. The link will help increase the external website’s rank and, it is hoped, make it appear on the first page of a Google search, the ultimate holy grail for any website.

      But the algorithm isn’t immune to clever attacks by those who understand how the mathematics works. For a short period in the summer of 2018, if you googled ‘idiot’ the first image that appeared was that of Donald Trump. Activists had understood how to exploit the powerful position that the website Reddit has on the internet. By getting people to vote for a post on the site containing the words ‘idiot’ and an image of Trump, the connection between the two shot to the top of the Google ranking. The spike was smoothed out over time by the algorithm rather than by manual intervention. Google does not like to play God but trusts in the long run in the power of its mathematics.

      The internet is of course a dynamic beast, with new websites emerging every nanosecond and new links being made as existing sites are shut down or updated. This means that page ranks need to change dynamically. In order for Google to keep pace with the constant evolution of the internet, it must regularly trawl through the network and update its count of the links between sites using what it rather endearingly calls ‘Google spiders’.

      Tech junkies and sports coaches have discovered that this way of evaluating the nodes in a network can also be applied to other networks. One of the most intriguing external applications has been in the realm of football (of the European kind, which Americans think of as soccer). When sizing up the opposition, it can be important to identify a key player who will control the way the team plays or be the hub through which all play seems to pass. If you can identify this player and neutralise them early on in the game, then you can effectively close down the team’s strategy.

      Two London-based mathematicians, Javier López Peña and Hugo Touchette, both football fanatics, decided to see whether Google’s algorithm might help analyse the teams gearing up for the World Cup. If you think of each player as a website and a pass from one player to another as a link from one website to another, then the passes made over the course of a game can be thought of as a network. A pass to a teammate is a mark of the trust you put in that player – players generally avoid passing to a weak teammate who might easily lose the ball, and you will only be passed to if you make yourself available. A static player will rarely be available for a pass.

      They decided to use passing data made available by FIFA during the 2010 World Cup to see which players ranked most highly. The results were fascinating. If you analysed England’s style of play, two players, Steven Gerrard and Frank Lampard, emerged with a markedly higher rank than others. This reflects the fact that the ball very often went through these two midfielders: take them out and England’s game collapses. England did not get very far that year in the World Cup – they were knocked out early by their old nemesis, Germany.

      Contrast this with the eventual winners: Spain. The algorithm shared the rank uniformly around the whole team, indicating that there was no clear hub through which the game was being played. This is a reflection of the very successful ‘total football’ or ‘tiki-taka’ style played by Spain, in which players constantly pass the ball around, a strategy that contributed to Spain’s ultimate success.

      Unlike many sports in America that thrive on data, it has taken some time for football to take advantage of the mathematics and statistics bubbling underneath the game. But by the 2018 World Cup in Russia many teams boasted a scientist on board to crunch the numbers to understand the strengths and weaknesses of the opposition, including how the network of each team behaves.

      A network analysis has even been applied to literature. Andrew Beveridge and Jie Shan took the epic saga A Song of Ice and Fire by George R. R. Martin, otherwise known as Game of Thrones. Anyone who knows the story will be aware that predicting which characters will make it through to the next volume, let alone the next chapter, is notoriously tricky, as Martin is ruthless at killing off even the best characters he has created.

      Beveridge and Shan decided to create a network between characters in the books. They identified 107 key people who became the nodes of the network. The characters were then connected with weighted edges according to the strength of the relationship. But how can an algorithm assess the importance of a connection? The algorithm was simply asked to count the number of times the two names appeared in the text within fifteen words of each other. This doesn’t measure friendship – it indicates some measure of interaction or connection between them.

      They decided to analyse the third volume in the series, A Storm of Swords, as the narrative had settled down by this point, and began by constructing a page rank analysis of the nodes or characters in the network. Three characters quickly stood out as important to the plot: Tyrion, Jon Snow and Sansa Stark. Anyone who has read the books or seen the series would not be surprised by this revelation. What is striking is that a computer algorithm which does not understand what it is reading achieved this same insight. It did so not simply by counting how many times a character’s name appears – that would pull out other names. It turned out that a subtler analysis of the network revealed the true protagonists.

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