I highly recommend the soon-to-appear book, Networked Life: 20 Questions and Answers by Mung Chiang (pre-order at Amazon) for anyone who wants to know more about the technologies -- technical philosophies, in some cases -- underlying the businesses of high-tech superstars like mobile phone companies, Google, Netflix, Amazon, Wikipedia, Skype, and so on. If you like Wired magazine but want a bit more meat, and are not intimidated by a few equations, then you will like this book.
Networked Life is written for an introductory-level college course, and it’s written in the textbook-lite style. Readers need to know basic mathematical language but the book does not give full mathematical proofs. What appeals to me is the three-step exposition: each chapter starts with a posed question (e.g. “How does Google rank webpages?”), and then gives firstly a general explanation emphasizing big concepts, and secondly, a deeper answer containing more mathematical details, and finally, advanced topics and references to technical materials. As my readers will notice, I used a similar strategy, placing the denser materials in the Conclusion chapter of my book.
Mung is also offering a free online course via Coursera, based on this book. (link)
For disclosure, Mung and I have known each other since high school. He’s currently Professor of Electrical Engineering at Princeton, where I took many courses in my undergraduate years. I thoroughly enjoyed his book, and provided feedback on several chapters, but would have highly recommended it even if he's not a friend. His academic home page is here, and the course web page is here.
The 20 topics in the book cover a wide variety of areas in applied mathematics. Throughout, Mung points out the big concepts that underly many apparently dissimilar applications. A book like this tends to get bogged down in mathematical details but this one always looks at the forests, more so than at the trees.
One of the chapters I reviewed concerns the topic of six degrees, the idea that anyone is linked to anyone else in the world within six connections. The starting point is Milgram’s experiment, in which he sent letters to about 300 people, asking them to forward to a stockbroker with an address in Boston, Mass., or to someone who might know the stockbroker. The median number of steps from sender to recipient was 6, hence six degrees. Mung quickly takes readers to the two key issues: the obvious one of whether short paths exist within these networks, and the deeper question of how one might discover such paths, given the large number of longer paths that exist, or one’s knowledge of only the local topology of the network.
This discussion leads to an introduction to the modern study of constructive "models" that assume random jumps from one point to another, demonstrating how these simple models provide insights into the difficult problem (or not, depending on your perspective!)
I not only enjoy that discussion but it also guided me to think about Milgram's experiment in a different light. The average of 6 is computed only over the 60 or so paths that were successful. About 70 percent of the letters never reached the stockbroker. So the original analysis suffers from survivorship bias. It is only accurate to describe those relationships in which the two parties could eventually discover each other. But the parties that could not discover each other are likely to be separated by more than 6 connections, and if they were accounted for, the median steps would surely be more than 6.
One of the pleasures of Mung's book is its tendency to provoke one's thinking. Hope you'll enjoy it as much as I did.