{"id":1487,"date":"2023-05-30T17:29:10","date_gmt":"2023-05-30T17:29:10","guid":{"rendered":"https:\/\/www.ethanepperly.com\/?p=1487"},"modified":"2024-03-03T02:24:15","modified_gmt":"2024-03-03T02:24:15","slug":"big-ideas-in-applied-math-markov-chains","status":"publish","type":"post","link":"https:\/\/www.ethanepperly.com\/index.php\/2023\/05\/30\/big-ideas-in-applied-math-markov-chains\/","title":{"rendered":"Big Ideas in Applied Math: Markov Chains"},"content":{"rendered":"\n

In this post, we’ll talk about Markov chains<\/a>, a useful and general model of a random system evolving in time.<\/p>\n\n\n\n

PageRank<\/h2>\n\n\n\n

To see how Markov chains can be useful in practice, we begin our discussion with the famous PageRank problem<\/a>\ufffc. The goal is assign a numerical ranking to each website on the internet measuring how important it is. To do this, we form a mathematical model of an internet user randomly surfing the web. The importance of each website will be measured by the amount of times this user visits each page.<\/p>\n\n\n\n

The PageRank model of an internet user is as follows: Start the user at an arbitrary initial website \"x_0\". At each step, the user makes one of two choices:<\/p>\n\n\n\n