The Virtuosi (Posts by DTC)

Special Brain

DTC — Thu, 20 Mar 2014 20:00:00 GMT

A colleague (and friend) of mine (hereafter referred to as Katie Mack the Physics Hack) produced a fun video last year that tried to show how people sometimes react when she tells them that she studies physics:

I loved this video because I've had a number of experiences like this. My favorite reaction that I've ever gotten happened in 2007. I was on a trip during college with other college kids, and I was placed in a hotel room with some guys who went to another school. We met for the first time while unpacking, and naturally we asked each other what we were studying. Turns out my new roommate was majoring in international business, something I knew nothing about. Not wanting to alienate a total stranger I was going to be sleeping in the same room with, I asked him questions and told him that his chosen major sounded interesting and important. I told him that I studied physics, and when he asked me what that meant I told him how I had worked on modeling cell division. My new roommate responded, "You must have a special kind of brain for that."

A special brain. ^[1]

This anecdote has stuck with me for a couple of reasons-- first because "special kind of brain" is a funny turn of phrase, and second because I think it's a perfect example of how an attempt at a flattering response can actually create some uncomfortable social distance between people.

"Special kind of brain" was my roommate's way of expressing how intelligent he thought I was. (Or, as Zach put it in Katie's video, "You must be soooooo smart!") His reaction hinged upon the assumption that what I was interested in doing was so far beyond the understanding of ordinary folk that I could be set apart as a member of an elite group. Instead of being merely complimentary, his comment held me at arms' length. His reaction wasn't something I took offense to, but it made me uncomfortable to hear that he considered me an outsider of sorts based on my professed interests.

Speaking more generally, the notion that scientific professionals are set apart from other people as members of a professional group isn't so ridiculous. After all, these days people's lives are often defined by their careers. (And of course scientists aren't the only profession with associated negative stereotypes-- Anyone know any good lawyer jokes?). But to me, thinking of scientists as some kind of inscrutable cabal of geniuses is an exaggeration. The truth is, not every scientist is a rocket-powered superbrain. Quite the opposite-- scientists make silly mistakes all the time. Being a scientist is a technical profession requiring years of training like law, medicine, or accounting: there are a few practitioners who really are exceptionally smart, while most of the others aren't.

The even more disappointing truth is that being a scientist is actually usually pretty mundane. Don't get me wrong-- the long-term goals of making new discoveries and developing new insights into the world around us are exactly why I like my job. I just mean that the day-to-day labor involved can be as tedious as any other profession. I sit in my cubicle and code (debug) endlessly on my laptop, or I read books and research papers to learn new things about my field. Most days don't get much more action-packed than that. In a lot of ways it's like any other office job. Aside from the end goal of research, working as a scientist is not so special.

I work in a cubicle. Not here.^[2]

Another reaction that I get when I say I study physics is one of apprehensive disappointment. (Zach's pronunciation of 'ohhhhhh...' combining equal parts boredom and distaste was dead on.) I don't think I need to dwell on this too long-- it is undeniably unpleasant for me when I hear this. Upon hearing that I'm a scientist, otherwise polite, kind people will suddenly lose their cool and be unable to hide the fact that my profession conjures up memories of boredom and frustration. ("Oh, man. I HATED physics in high school.") As Katie Mack puts it at the end, "polite interest is the way to go."

"Have you ever seen the Big Bang Theory?
Is that what physicists are really like?
I bet it is. I mean, no offense."

There's another type of off-putting reaction that comes up sometimes, which is commenting (jokingly or not) that I'm similar to familiar caricature of scientists that appear in popular culture. "You're just like Sheldon Cooper!" is a comment I've heard more times than I care to say. I know that The Big Bang Theory is a popular show, but frankly I dislike being associated I with characters that are cartoonishly depicted as condescending and socially tone-deaf ^[3]. Now, I appreciate that some people, when meeting others for the first time, like to demonstrate familiarity with others' jobs, but to me it just seems that making a pop culture references to another person's profession is just a bad way to go. I find this to be a safe bet when meeting anyone, not just scientists, simply because popular culture isn't a great way to learn about anyone else's job. Try telling the next lawyer you meet that they remind you of Saul Goodman, and see how they react.

So, what is there for physicists (and other scientists) to do when this happens? The most facile answer to this question is for us to grow a thicker skin and suck it up. Just ignore it when people have disparaging reactions upon first meeting us, and find a way to get past this in conversation. The thing is, I personally am not good enough at hiding my own negative reaction upon hearing these kinds of obnoxious remarks. Ideally, I'd like to make conversation easier by finding a way to avoid them altogether.

I can't change the way other people react to learning about my profession, but I can change how I present myself. Personally, I have given up on telling people that I'm in the physics department. Instead, when asked "what do you study in grad school?" I tell then exactly what I'm up to-- I study how infectious diseases spread through human and animal communities. I've found that I get a much more relaxed reaction when I do this. The same people who may have uncomfortable reactions to physics have enough familiarity with the idea of epidemics to be a little more comfortable. And besides, everyone has some amount of morbid curiosity about the next big plague that's going to kill us all. (I realize that this may not be a viable strategy for some of my colleagues who study nanoscience, magnetic materials, high-energy particles, or other mainstream physics topics. I'm interested to hear if anyone else who works in the sciences has come up with a different technique for breaking through the "I'm a physicist" ice.)

A friend of mine once chastised me for doing this. "Why should you have to hide what you're interested in?" he asked. "If they react badly to your profession, is it really worth getting to know them?" To that I say that I'm still telling them honestly what I'm interested in, I just sidestep the potentially negative associations carried by the word "physics." And besides, just because someone has a bad or obnoxious reaction to finding out that I'm a physicist doesn't mean they aren't worth meeting. The fact remains that I've found this to be a great way to keep the getting-to-know-you conversation light when meeting new people for the first time. I wish I could wave a magic wand and make it so that everyone was comfortable wih the idea of interacting with professional scientists, but I can't. While I'm waiting for Bill Nye and Neil DeGrasse Tyson and others to humanize the profession for the public, this is how I'll be introducing myself.

Katie Mack and Zach's video really got me thinking about how to talk to other people about their jobs with more empathy-- avoiding flattery and stereotyping, and doing my best to hide any negative visceral reactions evoked by the thought of others' jobs and interests. One question that has occurred to me is whether there are people in completely different professions experience similarly frustrating reactions when they say what their jobs are. Programmers, actuaries, office administrators, copy editors, art dealers, karate instructors, gravediggers, lion-tamers, etc.: whoever you are, I want to hear about any difficulties you may have had with telling other people what you do in the comments below.

^ Image from New X-Men #121, written by Grant Morrison with art by Frank Quitely. You can see some more of this particularly trippy story here.
^ All of the imagery of Frankenstein's monster being brought to life with electricity comes from James Whale's Frankenstein from 1931. Mary Shelley's original book contained no mention of electricity, and instead remained eerily vague about the mechanisms for creating life.
^ Here is a really level-headed critique of The Big Bang Theory that I like a lot. There isn't a ton of hand-wringing, and the author does talk about what the show might consider doing differently. It was written three years ago.
Watch what happens when a brain surgeon meets a rocket scientist for the first time . To justify my linking to this skit (outside of the fact that I love it so much), I'll just say that nobody is acting appropriately in this video.
(Sorry, lawyers.)

Quantum Mechanics: Trying to Sort the Physical from the Mystical

DTC — Mon, 23 Sep 2013 00:00:00 GMT

^[1]

A friend of mine (I’ll call him Ron), who knows that I study physics, likes to talk to me about quantum mechanics. He’s an easy-going guy and likes to joke around. “Hey, is it a particle or is it a wave today?” he’ll say, or, “How many dimensions do we have now?” When the conversation turns more serious, he tells me how he believes in the “quantum universe,” which is greater than what we humans are able to ordinarily perceive. He talks about consciousness, immortality, spirits, and the great cosmic grandeur of the universe, all of which he ties together with the label of “quantum.”

These conversations are strange to me. Both of us are using the same two words: quantum mechanics. When Ron thinks about quantum mechanics, he associates it with nonphysical concepts, like spirits. Through my time spent studying physics, I’ve come to understand quantum mechanics as a theory describing the behavior of atoms and subatomic particles.

For example, one day our chatting turned to the topic of medicine and how the human body heals itself. Ron told me that the biggest problem with modern medicine is that doctors think of the body as a physical object only. Healing, he said, was a “quantum” effect. I told him that I could make a pretty strong physical argument for why that wasn’t the case. He responded with this story: Once, when playing football, he severely injured his knee. The injury was so bad that he couldn’t bend it or move it. He didn’t have health insurance and didn’t have the cash on hand to pay for medical treatment. One day, he prayed to the universe that he would get better and a “tornado of light came down” and healed his leg. Since then, Ron says, he’s always believed in and respected the quantum universe.

I can’t tell Ron that what he described in his story didn’t happen, that his experience was wrong or incorrect in some way. I wasn’t there, so I can’t comment on the accuracy of his narrative. And even the story of how his body healed out of the blue isn’t problematic: as far as I can tell, decades after the story took place, Ron is in good shape and his leg is doing fine. What I found objectionable about the story was how, in the end, Ron attributed his healing to the miraculous intervention of quantum mechanics.

Quantum mechanics, in all of its glorious strangeness, is only relevant on inconceivably small scales and at very, very low temperatures. One of the reasons it took humans so long to develop the theory of quantum mechanics is that quantum effects don’t readily appear in everyday life. My colleagues who work to observe quantum mechanics in their experiments use lasers to manipulate atoms (objects that are 1/10,000,000,000th of a meter in size and weigh around 1/10,000,000,000,000,000,000,000,000th of a kilogram) at temperatures less than 1 Kelvin (about -459 degrees Fahrenheit). At larger sizes and temperatures quantum effects are negligible. The human body is more than a meter long, usually weighs around 50-100 kilograms and, if healthy, maintains a toasty 98 degrees Fahrenheit. Quantum mechanics is important at the atomic level, but on the scales at which people interact with the world it hardly shows up at all. So, even if Ron's leg heals as he said it did, I wouldn't give credit to quantum mechanics.^[2]

I have been studying physics for years and still quantum mechanics remains utterly baffling to me. The fact that such an abstract theory can tell us so much about the world feels a little bit like a miracle. Quantum mechanics carries with it a number of counterintuitive ideas like the uncertainty principle ^[3] entanglement, or parallel universes. These ideas are so abstracted from every day life that the subject begins to take on a supernatural quality. Physics no longer seems like physics- it starts to sound like mysticism.

So it makes sense that contemporary culture has seized upon quantum mechanics as a possible explanation for inexplicable things. The theory has so many surprising results that it seems natural to extend it to encompass other things that confuse us, like questions of consciousness. Furthermore, “quantum mechanics” is a term that carries with it the weight of scientific legitimacy. If Ron had said that he had been healed through witchcraft, laying on of hands, or alchemy he would have sounded ridiculous, but attributing his experience to quantum effects allows the story to borrow from the credible reputation of fact-based 20th century science. What my friend doesn’t realize is that terminology is not what makes quantum theory powerful: the scientific methodology supporting quantum mechanics is what matters.

It’s important to keep separate quantum mechanics the physical theory and quantum mechanics as a mystical cosmic principle. Despite how confusing it is, quantum mechanics is an empirically motivated and mature theory that gives us a framework to understand physical phenomena like radiation and chemical bonding. This is fundamentally different from applying quantum mechanical concepts to the nature of reality or consciousness. To do so may be a fun philosophical parlor game, but it is baseless speculation without any evidence to motivate the connection between quantum mechanics and the supernatural that it begins with. This confusion is not just restricted to scientific laymen: there are trained researchers working at well-respected research institutions who also make the same mistake my friend Ron does.

At the end of the day, speculation that the soul, the afterlife, ESP, or whatever else are quantum effects is unscientific, but at least it isn’t dangerous or harmful in the same way as climate change denial or refusing to vaccinate your children. It’s closer to something like intelligent design, which is fundamentally confused about what science is. (We physicists are truly thankful that there is no noisy political movement to teach Deepak Chopra alongside physics in high school classrooms.) So I won’t object to my friends’ stories of sudden, unexpected recoveries from illness, but I will react skeptically when I hear that healing has anything to do with quantum mechanics.

^ Credit where credit is due: I took this from XKCD. This may be one of my favorite comics Randall Munroe has ever done. That it came out while I was thinking about this piece was a great coincidence. (A cosmic coincidence explainable through quantum entanglement? Probably not.)
^ Quantum mechanics may be just as mundane as any other materialistic physical theory, but that doesn’t make it any less amazing. My favorite example is how quantum mechanics allows us to understand why the sun works.
^ In case you didn't take the time to click on the link: Seriously, do yourself a favor and click on the link . It's an essay from The Stone that very elegantly describes how the uncertainty principle is far less cosmically mind-blowing than you may have come to believe. It does a beautiful job bringing us back down to earth and carefully explaining the scope of the principle. I must give it credit for having inspired this piece in no small way.

Re-evaluating the values of the tiles in Scrabble™

DTC — Sun, 20 Jan 2013 22:52:00 GMT

Recently I have seen quite a few blog posts written about re-evaluating the points values assigned to the different letter tiles in the Scrabble™ brand Crossword Game. The premise behind these posts is that the creator and designer of the game assigned point values to the different tiles according to their relative frequencies of occurrence in words in English text, supplemented by information gathered while playtesting the game. The points assigned to different letters reflected how difficult it was to play those letters: common letters like E, A, and R were assigned 1 point, while rarer letters like J and Q were assigned 8 and 10 points, respectively. These point values were based on the English lexicon of the late 1930’s. Now, some 70 years later, that lexicon has changed considerably, having gained many new words (e.g.: EMAIL) and lost a few old ones. So, if one were to repeat the analysis of the game designer in the present day, would one come to different conclusions regarding how points should be assigned to various letters?

I’ve decided to add my own analysis to the recent development because I have found most of the other blog posts to be unsatisfactory for a variety of reasons^[1].
One article calculated letters’ relative frequencies by counting the number of times each letter appeared in each word in the Scrabble™ dictionary. But this analysis is faulty, since it ignores the probability with which different words actually appear in the game. One is far less likely to draw QI than AE during a Scrabble™ game (since there’s only one Q in the bag, but many A's and E's). Similarly, very long words like ZOOGEOGRAPHICAL have a vanishingly small probability of appearing in the game: the A’s in the long words and the A’s in the short words cannot be treated equally. A second article I saw calculated letter frequencies based on their occurrence in the Scrabble™ dictionary and did attempt to weight frequencies based on word length. The author of this second article also claimed to have quantified the extent to which a letter could “fit well” with the other tiles given to a player. Unfortunately, some of the steps in the analysis of this second article were only vaguely explained, so it isn’t clear how one could replicate the article’s conclusions. In addition, as far as I can tell, neither of these articles explicitly included the distribution of letters (how many A’s, how many B’s, etc) included in a Scrabble™ game. Also, neither of these articles accounted for the fact that there are blank tiles (that act as wild cards and can stand in for any letter) that appear in the game.

So, what does one need to do to improve upon the analyses already performed? We’re given the Scrabble™ dictionary and bag of 100 tiles with a set distribution, and we’re going to try to determine what a good pointing system would be for each letter in the alphabet. We’re also armed with the knowledge that each player is given 7 letters at a time in the game, making words longer than 8 letters very rare indeed. Let’s say for the sake of simplicity that words 9 letters long or shorter account for the vast majority of words that are possible to play in a normal game.

Based on these constraints, how can one best decide what points to assign the different tiles? As stated above, the game is designed to reward players for playing words that include letters that are more difficult to use. So, what makes an easy letter easy, and what makes a difficult letter difficult? Sure, the number of times the letter appears in the dictionary is important, but this does not account for whether or not, on a given rack of tiles (a rack of tiles is to Scrabble™ as a hand of cards is to poker), that letter actually can be used. The letter needs to combine with other tiles available either on the rack or on the board in order to form words. The letter Q is difficult to play not only because it is used relatively few times in the dictionary, but also because the majority of Q-words require the player to use the letter U in conjunction with it.

So, what criterion can one use to say how useful a particular tile is? Let’s say that letters that are useful have more potential to be used in the game: they provide more options for the players who draw them. Given a rack of tiles, one can generate a list of all of the words that are possible for the player to play. Then, one can count the number of times that each letter appears in that list. Useful letters, by this criterion, will combine more readily with other letters to form words and so appear more often in the list than un-useful letters.

(I would also like to take a moment to preempt criticism from the competitive Scrabble™ community by saying that strategic decisions made by the players need not be brought into consideration here. The point values of tiles are an engineering constraint of the game. Strategic decisions are made by the players, given the engineering constraints of the game. Words that are “available to be played” are different from “words that actually do get played.” The potential usefulness of individual letter tiles should reflect whether or not it is even possible to play them, not whether or not a player decides that using a particular group of tiles constitutes an optimal move.)

To give an example, suppose I draw the rack BEHIWXY. I can generate^[2] the full list of words available to be played given this rack: BE, BEY, BI, BY, BYE, EH, EX, HE, HEW, HEX, HEY, HI, HIE, IBEX, WE, WEB, WHEY, WHY, WYE, XI, YE, YEH, YEW. Counting the number of occurrences of each letter, I see that the letter E appears 18 times, while the letter W only appears 7 times. This example tells me that the letter E is probably much more potentially useful than the letter W.

The example above is only one of the many, many possible racks that one can see in a game of Scrabble™. I can use a Monte Carlo-type simulation to estimate the average usefulness of the different letters by drawing many example racks. Monte Carlo is a technique used to estimate numerical properties of complicated things without explicit calculation. For example, suppose I want to know the probability of drawing a straight flush in poker.^[3] I can calculate that probability explicitly by using combinatorics, or I can use a Monte Carlo method to deal a large number of hypothetical possible poker hands and count the number of straight flushes that appear. If I deal a large enough number of hands, the fraction of hands that are straight flushes will converge upon the correct analytic value. Similarly here, instead of explicitly calculating the usefulness of each letter, I use Monte Carlo to draw a large number of hypothetical racks and use them to count the number of times each letter can be used. Comparing the number of times that each tile is used over many, many possible racks will give a good approximation of how relatively useful each tile is on average. Note that this process accounts for the words acceptable in the Scrabble™ dictionary, the number of available tiles in the bag, as well as the probability of any given word appearing.

In my simulation, I draw 10,000,000 racks, each with 9 tiles (representing the 7 letters the player actually draws plus two tiles available to be played through to form longer words). I perform the calculation two different ways: once with a 98-tile pool with no blanks, and once with a 100-tile pool that does include blanks. In the latter case, I make sure to not count the blanks used to stand in for different letters as instances of those letters appearing in the game. The results are summarized in the table below.

There are two key observations to be made here. First, it does not seem to matter whether or not there are blanks in the bag! The results are very similar in both cases. Second, it would be completely reasonable to keep the tile point values as they are. Only the Z, H, and U appear out of order. It’s only if one looks very carefully at the differences between the usefulness of these different tiles that one might reasonably justify re-pointing the different letters.

For fun, I have included in the table my own suggestions for what these tiles’ values might be changed to based on the simulation results. (Note: here's where any pretensions of scientific rigor go out the window.) I have kept the scale of points between 1 and 10, as in the current pointing system. I have assigned groups of letters the same number of points based on whether they have a similar usefulness score. Here are the significant changes: L and U, which are significantly less useful than the other 1-point tiles may be bumped up to 2 points, comparable to the D and G. The letter V is clearly less useful than any of the other three 4-point tiles (W, Y, and F, all of which may be used to form 2-letter words while the V forms no 2-letter words), and so is undervalued. The H is comparable to the 3-point tiles, and so is currently overvalued. Similarly, the Z is overvalued when one considers how close to the J it is. Unlike in the previous two articles that I mentioned, I don't find any strong reason to change the value of the letter X compared to the other 8 point tiles. I suppose one could lower its value from 8 points to 7, but I have (somewhat arbitrarily) chosen not to do so.

One may also ask the question whether or not the fact that a letter forms 2- or 3-letter words is unfairly biasing that letter. In particular, is the low usefulness of the C and V compared to comparably-pointed tiles due to the fact that they form no 2-letter words? Performing the simulation again without 2-letter words, I found no changes in the results in any of the letters except for C, which increased in usefulness above the B and the H. The letter V's ranking, however, did not change at all, indicating that unlike the C the V is difficult to use even when combining with letters to make longer words. Repeating the simulation yet again without 2- or 3-letter words yielded the same results.

As a final note, I would like to respond directly to to Stefan Fatsis's excellent article about the so-called controversy surrounding re-calculating tile values and say that I am fully aware that this is indeed a "statistical exercise," motivated mostly by my desire to do the calculation made by others in a way that made sense in the context of the game of Scrabble. Similarly, I realize that these recommendations are unlikely to actually change anything. Given that the original points values of the tiles are still justifiably appropriate by my analysis, it's not like anybody at Hasbro is going to jump to "fix" the game. Lastly, my calculations have nothing to do with the strategy of the game whatsoever, and cannot be used to learn how to play the game any better. (If anything, I've only confirmed some things that many experienced Scrabble players already know about the game, such as that the V is a tricky tile, or that the H, X, and Z tiles, in spite of their high point values, are quite flexible.)

Notes

1. ^ To state my own credentials, I have played Scrabble™competitively for 4 years, and am quite familiar with the mechanics of the game, as well as contemporary strategy.

2. ^ Credit where credit is due: Alemi provided the code used to generate the list of available words given any set of tiles. Thanks Alemi!

3. ^ Monte Carlo has a long history of being used to estimate the properties of games. As recounted by George Dyson in Turing’s Cathedral, in 1948 while at Los Alamos the mathematician Stanislaw Ulam suffered a severe bout of encephalitis that resulted in an emergency trepanation. While recovering in the hospital, he played many games of solitaire and was intrigued by the question of how to calculate the probability that a given deal could result in a winnable game. The combinatorics required to answer this question proved staggeringly complex, so Ulam proposed the idea of generating many possible solitaire deals and merely counting how many of them resulted in victory. This proved to be much simpler than an explicit calculation, and the rest is history: Monte Carlo is used today in a wide variety of applications.

Additional References:

The photo at top of a Scrabble™ board was taken during the 2012 National Scrabble™ Championship. Check out the 9-letter double-blank BINOCULAR.

For anyone interested in learning more about the fascinating world of competitive Scrabble™, check out Word Freak, also by Stefan Fatsis. This book has become more or less the definitive documentation upon this subculture. If you don't have enough time to read, check out Word Wars, a documentary that follows many of the same people as Fatsis's book. (It still may be available streaming on Netflix if you hurry.)

The Skeleton Supporting Search Engine Ranking Systems

DTC — Tue, 01 Jan 2013 19:09:00 GMT

A lot of the research I’m interested in relates to networks – measuring the properties of networks and figuring out what those properties mean. While doing some background reading, I stumbled upon some discussion of the algorithm that search engines use to rank search results. The automatic ranking of the results that come up when you search for something online is a great example of how understanding networks (in this case, the World Wide Web) can be used to turn a very complicated problem into something simple.

Ranking search results relies on the assumption that there is some underlying pattern to how information is organized on the WWW- there are a few core websites containing the bulk of the sought-after information surrounded by a group of peripheral websites that reference the core. Recognizing that the WWW is a network representation of how information is organized and using the properties of the network to detect where that information is centered are the key components to figuring out what websites belong at the top of the search page.

Suppose you look something up on Google (looking for YouTube videos of your favorite band, looking for edifying science writing, tips on octopus pet care, etc): the search service returns a whole spate of results. Usually, the pages that Google recommends first end up being the most useful. How on earth does the search engine get it right?

First I’ll tell you exactly how Google does not work. When you type in something into the search bar and hit enter, a message is not sent to a guy who works for Google about your query. That guy does not then look up all of the websites matching your search, does not visit each website to figure out which ones are most relevant to you, and does not rank the pages accordingly before sending a ranked list back to you. That would be a very silly way to make a search engine work! It relies on an individual human ranking the search results by hand with each search that’s made. Maybe we can get around having to hire thousands of people by finding a clever way to automate this process.

So here’s how a search engine does work. Search engines use robots that crawl around the World Wide Web (sometimes these robots are referred to as “spiders”) finding websites, cataloguing key words that appear on those webpages, and keeping track of all the other sites that link into or away from them. The search engine then stores all of these websites and lists of their keywords and neighbors in a big database.

Knowing which websites contain which keywords allows a search engine to return a list of websites matching a particular search. But simply knowing which websites contain which keywords is not enough to know how to order the websites according to their relevance or importance. Suppose I type “octopus pet care” into Google. The search yields 413,000 results- far too many for me to comb through at random looking for the web pages that best describe what I’m interested in.

Knowing the ways that different websites connect to one another through hyperlinks is the key to how search engine rankings work. Thinking of a collection of websites as an ordinary list doesn’t say anything about how those websites relate to one another. It is more useful to think of the collection of websites as a network, where each website is a node and each hyperlink between two pages is a directed edge in the network. In a way, these networks are maps that can show us how to get from one website to another by clicking through links.

Here is an example of what a network visualization of a website map of a large portion of the WWW looks like. (Original full-size image here .)

Here is a site map for a group of websites that connect to the main page of English Wikipedia. (Original image from here.) This smaller site map is closer to the type of site map used when making a search using a search engine.

So, how does knowing the underlying network of the search results help one to find the best website on octopus care (or any other topic)? The search engine assumes that behind the seemingly random, hodgepodge collection of files on the WWW, there is some organization in the way they connect to one another. Specifically, the search engine assumes that finding the websites most central to the network of search results is the same as finding the search results with the best information. Think of a well-known, trusted source of information, like the New York Times. The NY Times website will have many other websites referencing it by linking to it. In addition, the NY Times website, being a trusted news source, is likely to refer to the best references for other sources that it wants to refer to, such as Reuters. High-quality references will also probably have many incoming links from websites that cite them. So not only does a website like the NY Times sit at the center of many other websites that link to it, but it also frequently connects to other websites that themselves are at the center of many other websites. It is these most central websites that are probably the best ones to look at when searching for information.

When I search for “octopus pet care” using Google I am necessarily assuming that the search results are organized according to this core-periphery structure, with a group of important core websites central to the network surrounded by many less important peripheral websites that link to the core nodes. The core websites may also connect to one another. There may also be websites disconnected from the rest, but these will probably be less important to the search simply because of the disconnection. Armed with the knowledge of the connections between the different relevant websites and the core-periphery network structure assumption, we may now actually find which of the websites are most central to the network (in the core), and therefore determine which websites to rank highly.

Let’s begin by assigning a quantitative “centrality” score to each of the nodes (websites) in the network, initially guessing that all of the search results are equally important. (This, of course, is probably not true. It’s just an initial guess.) Each node then transfers all of its centrality score to its neighbors, dividing it evenly between them^[1]. (Starting with a centrality score of 1 with three neighbors, each of those neighbors receives 1/3.) Each node also receives a some centrality from each neighbor that links in to it. Following this first step, we find that nodes with many incoming edges will have higher centrality than nodes with few incoming edges. We can repeat this process of dividing and transferring centrality again. Nodes with many incoming links will have more centrality to share with their neighbors, and nodes with many incoming links will themselves also receive more centrality.

After repeating this process many times, we begin to see a difference between which nodes have the highest centrality scores: nodes with high centrality are the ones that have many incoming links, or have links to other central nodes, or both. This algorithm therefore differentiates between the periphery and the core of the network. Core nodes receive lots of centrality because they link to one another and because they have lots of incoming links from the periphery. Peripheral nodes have fewer incoming links and so receive less centrality than the nodes in the core. Knowing the centrality scores of search results on the WWW makes it pretty straightforward for us to quantitatively rank which of those websites belongs at the top of the list.

Of course, there are more complex ways that one can add to and improve this procedure. Google’s algorithm PageRank (named for founder Larry Page, not because it is used to rank web pages) and the HITS algorithm developed at Cornell are two examples of more advanced ways of ranking search engine results. We can go even further: a search engine can keep track of the links that users follow whenever a particular search is made. (This is almost the same as the company hiring someone to order sought-after web pages automatically whenever a search is made, except all the company lets the user do it for free.) Over time, search engines can improve their methods for helping us find what we need by learning directly from the way users themselves prioritize which search results they pursue. Still, these different search engine ranking systems may operate using slightly different methods, but all of them depend on understanding the list of search results within the context of a network.

Notes

1. ^ It's not always all - there are other variations where nodes only transfer a fraction of their centrality score at each step.

Sources (and further reading)

I wanted to include no mathematics in this post simply because I cannot explain the mathematics behind these algorithms and their convergence properties better than my sources can. For those of you who want to see the mathematical side of the argument for yourselves (which involves treating the network adjacency matrix as a Markov process and finding its nontrivial steady state eigenvector), do consult the following two textbooks:

Easley, David, and Jon Kleinberg. Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press, 2010 (Chapter 14 in particular)

Newman, Mark. Networks: an Introduction. Oxford University Press, 2010 (Chapter 7 in particular)

A popular book on the early development of network science that contains a lot of information on the structure of the WWW:

Barabasi, Albert-Laszlo. Linked: How Everything is Connected to Everything Else and What It Means. Plume, 2003.

A book on the history of modern computing that contains an interesting passage on how search engines learn adaptively from their users (that deserves a shout-out in this blog post).

Dyson, George. Turing's Cathedral. Pantheon, 2012.

Batman, Helicopters, and Center of Mass

DTC — Tue, 26 Jun 2012 11:10:00 GMT

A couple weeks ago, I came home after a long day at work looking for a break. I thought to myself, "What’s more fun than physics?"

Batman.^[1]

I sat down to play the latest Batman videogame, in which Batman’s current objective was to use his grappling hook to jump onto an enemy helicopter to steal an electronic MacGuffin. As awesome as this was, it occurred to me that something was very wrong about the way the helicopter moved while Batman zipped through the air.

See if you can spot it too. (Watch for about 5 seconds after the video starts. Ignore the commentary. Note: The grunting noises are the sounds that Batman makes if you shoot him with bullets.)

What occurred to me was this: If the helicopter’s rotors provided enough lift to balance the force of gravity, wouldn’t Batman’s sudden additional weight cause the helicopter to fall out of the sky? Also, to get lifted up into the air, the helicopter must be pulling up on Batman: shouldn’t Batman also be pulling down on the helicopter? By how much should we expect to see the helicopter’s altitude change?

To address the first question, let's go to Newton's second law:

$$ \sum \vec{F} = m\vec{a} $$

Let’s assume that the helicopter is hovering stationary, minding its own business, when Batman jumps onto it. Let's also assume the helicopter pilots are totally oblivious to Batman and make no flight corrections after Batman jumps onto it. In order to hover, the lift from the helicopter's rotors exactly matched the pull of gravity.

$$ \sum \vec{F} = \vec{F}{rotors} - \vec{F} = 0 $$

Batman's sudden additional weight would cause the helicopter to start falling, as the forces would no longer balance:

$$ \sum \vec{F} = \vec{F}{rotors} - \vec{F} - \vec{F}_{Batman} < 0 $$

So the helicopter does accelerate (and move) when Batman jumps onto it. How much does it move? Let’s assume there are no crazy winds or other external forces acting on the helicopter or Batman while Batman grapples onto the helicopter. “No external forces” means that momentum of helicopter + Batman does not change during Batman's flight.

Let's make things a little simpler and assume that neither Batman nor the helicopter had any vertical momentum before Batman used his grappling hook. (I can choose to approach this problem from a reference frame where the center of mass is stationary. Choosing a frame where the center of mass moves won't change the results, it will just make the calculation more complicated.) Because the momentum of helicopter + Batman does not change, then the center of mass does not move while Batman zips through the air:

$$ \frac{d}{dt} y_{COM} = \frac{d}{dt} \frac{m_{Bat} y_{Bat} + m_{Copter} y_{Copter}}{m_{Bat} + m_{Copter}} = \frac{p}{m_{Bat} + m_{Copter}} = 0 $$

The center of mass must remain stationary, so we can find how much the helicopter's height changes by if Batman starts on the ground (y = 0) and both end up at the same height with Batman hanging from the helicopter:

$$ y_{COM} = \frac{m_{Copter} y_{Copter} + m_{Bat} (0)}{m_{Bat} + m_{Copter}} = \frac{m_{Copter} y_{final} + m_{Bat} y_{final}}{m_{Bat} + m_{Copter}} $$

$$ \Delta y = y_{Copter} - y_{final} = \frac{m_{Bat}}{m_{Bat} + m_{Copter}} y_{Copter}$$

Now, some numbers: The police helicopters in the game are pretty small, probably about 1500 kg.

Batman is a big guy who works out and probably weighs around 100 kg (220 lb). Plus, he’s wearing body armor (hence surviving when bullets hit him) and a utility belt and all of those other Bat-gadgets, which probably adds about 30 kg ($\sim 30$ lb for the gadgets, $\sim30$ lb for the armor). If Batman has to grapple onto a helicopter 30 meters above him, then the helicopter should drop out of the air by about 2.4 m. This is greater than the height of Batman himself, and would be noticeable if the helicopter physics in the game were perfect. Of course, if the helicopters appearing in the game were the giant army helicopters (they do carry rockets, after all), their mass would be much larger $(\sim 5000-10000~{\rm kg})$ so the effect of Batman’s additional weight would be much smaller. None of these considerations detracted from the fun I had playing the game, but it did seem odd that the helicopters appeared to be nailed to the sky instead of moving freely through the air. I’ll be writing the game developers a strongly-worded letter directly.

Notes

1. ^ The DC superhero, not the city or the fish.

Proofiness: A look into how mathematics relates to American political life

DTC — Tue, 06 Mar 2012 14:37:00 GMT

Dearest readers, This is my first post on The Virtuosi, so I thought I’d take a moment to introduce myself. I’m a first year physics graduate student at Cornell, recently joined after 2 years working as an engineer first at a private firm and then at a national lab. I myself have had lots of fun following the exploits of my estimable colleagues here on The Virtuosi, and I thought I could bring a new angle to the content here. I would like to use this space to discuss how science interacts with everyday life in a cultural sense. How does science appear in popular culture? How do political or social issues relate back to science? Those sorts of questions. (I understand that there are plenty of other resources elsewhere that offer far more intelligent insight into these matters than I can, but in the very least this will give people a chance point them out to me as they yell at me in the forum below.) Enough intro, here begins my very first blog post: Being interested in how science is communicated to the public, I am an avid reader of popular science. While academic types sometimes dismiss this kind of writing as shallow or otherwise uninteresting, I think science writers perform a very important function serving as a way to convey information about conceptually challenging topics to a general audience. At their best, I find that these books serve as examples for how I can communicate my own ideas better, and in addition challenge my understanding of how science relates back to society in general. This being said, I cannot recommend Charles Seife’s Proofiness enough. The basic premise of this book is to explore the way that good mathematics is hijacked, twisted, or ignored in everyday life, and the ugly consequences of the tendency to misunderstand numbers and measurements. Seife gives a number of fascinating examples of the ways in which numbers and math connect to American democracy. American government functions through representation, and so the “enumeration” of citizens and their opinions through the Census and elections is an essential part of the democratic process. This “enumeration” is a counting measurement, subject to errors like any other. And yet, the laws that govern how Censuses and elections are run ignore this fact. Seife’s discussion of elections (and in particular Bush v. Gore) is fascinating, but I won’t spoil that here. Here’s my take on the discussion of the Census that appears in Proofiness: Consider a (vague) physics experiment. I want to know how many particles are inside a box. To figure this out, I have a detector that goes ping every time a particle passes through it. I set up my detector inside the box and count the number of times that it goes ping in a certain amount of time. I can then use that count to guess at the number of particles that I have in my box. My measurement will let me estimate N to within some margin of error. This process is perhaps unnecessary if I have only five particles in my box (in which case, I might just open the box and count what I see inside), but if I have 300 million particles in my box, it would be totally impractical for me to reach into the box 300 million times and count each one individually. We can consider the Census to be just like this physics experiment. I have N inhabitants (particles) living in my country (box), and I can use my detector (census replies) to count a certain number of people. In principle, using well-understood statistical techniques of regression and error analysis, I can estimate to within a very good margin of error how many people live in each region of the country. Instead, what the Census requires is that we reach inside the box (send representatives to every household that doesn’t reply by mail) and count every single person. The whole process ignores the fact that even if we send a representative to every single household there will still be some margin of error in our counting measurements. No such measurement can be made without errors. The consequences of ignoring these errors, says Seife, can be that we waste money in attempting the impossible and trying to count everybody. From a civic-minded perspective, this attitude towards the perfection of the Census can backfire. For example, if undercounting occurs (i.e., certain households do not respond for some reason), the Census has no mechanism for correcting that miscount. Counter-intuitively, the Census laws actually prohibit the use of any statistical techniques to correct miscounting. The result is that those slow to respond are ignored and not taken into account when allotting seats in the legislature to represent them. Proofiness is a fascinating book and a fun read, and I recommend you all look it up. In addition, it serves as an excellent example of science writing that helped me to rethink how scientific ideas relate to everyday life. I hope to invite consideration of these topics here and in future posts. If you want to know more about the inspiration for this post, go here.