Where else could you buy this?

Or this?

(Via Clauset and Shalizi, naturally.)

I have a confession to make: Once, when I had to give a talk on network theory to a seminar full of management people, I wrote a genetic algorithm to optimize the Newman-Girvan Q index and divide the Zachary Karate Club network into modules before their very eyes. I made Movie Science happen in the real world; peccavi.

(Via symmetry breaking.)

Meanwhile, why not help science bloggers build something a little more lasting than usual, and support The Open Laboratory?

<a href=”http://openlab.wufoo.com/forms/submission-form/”><img src=”http://scienceblogs.com/clock/Open_Lab_2010%20button100×67.png”></a>

<a href=”http://openlab.wufoo.com/forms/submission-form/”><img src=”http://scienceblogs.com/clock/Open_Lab_2010_150×100_b.png”></a>

<a href=”http://openlab.wufoo.com/forms/submission-form/”><img src=”http://scienceblogs.com/clock/Open_Lab_2010_300×200_b.png”></a>

Or watch Ed Witten on TV?

I am a hard rock geologist and a geochemist. As a geologist, I know about compasses, maps, GPS units, minerals, and hammers. As a geochemist, I know about acids, being paranoid about contamination, mass spectrometers, the periodic table, and the table of the nuclides. I know very little about ocean water and oil, and I know even less about deep-sea drilling rigs. Yet, over the past 101 days since the Deepwater Horizon Oil Spill began, I have been asked numerous times about the oil spill and its implications. As soon as people know I work at WHOI, they presume I am an expert about everything related to the oil spill.

I suppose that I should count myself lucky. Most of the people who know that I was trained in science also know that I studied theoretical physics, so I have nothing to say about anything important, ever, including the major disasters of our day.

I have sometimes been asked questions like, “So, what do you think of that surfer dude’s Theory of Everything?” This is where oceanographic eschatologists have a bit of an advantage: their subject is (all too) tangible. The difficulties with their work, or at least some of them, admit comparably easy explanations, such as, It’s hard to tell how much oil got spilled, because the gunk that spewed out is a mix of different stuff, and a lot of it is still beneath the surface. By comparison, when something hits the news which sounds like I would know about it, the issues have been . . . what’s the word? . . . esoteric. (Not to belittle the complexities of anybody else’s scientific field — the issue here is what questions get asked from outside, not what an individual scientist does on any working day.) By the time a story gets from the arXiv, through the physics blogs and into a newspaper, the juice can be sucked right out of it. When there’s nothing left but vague analogies for metaphors for speculations, when the residuals are more metaphysical than physical, what’s there to talk about?

Sometimes, I’ve tried to play Asimov: “Well, the mathematical structure the guy tried to use to fit all the different subatomic particles into a pattern didn’t have enough room to hold them all. And, when you try and squeeze the pattern to make them fit, it gets even worse: your mathematics says that new kinds of particles must show up, which don’t exist in the real world.” I always have a little paranoia about doing this, since invariably, the physics making the headlines hails from a subfield which I’ve never specialized in. So, while I might know more about it than the vast majority of the human population does, and I might be moderately well-equipped to learn more should the need arise, I can smack awfully hard into my limitations.

I’ve also tried to go for general background principles. “Most new ideas on the frontier turn out to be wrong. That’s only natural. If science were easy, we’d be done by now.” Or, getting into a bit more depth, I’ve tried to explain what physicists mean by symmetry, or by some other concept lurking behind the controversy du jour, on the logic that understanding a “basic concept” — that which one must know in order to know lots of other things — is a more lasting and valuable achievement than being able to repeat a soundbyte which will date itself in six months’ time.

(Via Ed Yong.)

I can’t say I think much of the music, tho’. At least they didn’t try to “enhance” the security camera footage during the montage.

(Via Jennifer Ouellette and Sean M. Carroll.)

Some things I’ve enjoyed reading recently:

• Two posts by Tom Levenson.
• The most recent from Brian Switek, describing a neat bit of fossil analysis which feeds into a wider discussion of the role palaeontology plays in evolutionary theory.
• Physicist John Baez has a new blog.
• Silvanus P. Thompson’s classic Calculus Made Easy is available ontube and is free for the downloading. This is old news, but it makes me happy.
• Five Things That Are More Important to Rage About Than Twilight.

And this picture has been sitting on my laptop for, like, three generations of meme-time.

A month or so after I was born, my parents bought an Atari 400 game console. It plugged into the television set, and it had a keyboard with no moving keys, intended to be child- and spill-proof. Thanks to the box of cartridges we had beside it, Asteroids and Centipede were burnt into my brain at a fundamental level. The hours I lost blowing up all my own bases in Star Raiders — for which accomplishment the game awarded you the new rank of “garbage scow captain” — I hesitate to reckon. We also had a Basic XL cartridge and an SIO cassette deck, so you could punch in a few TV screens’ worth of code to make, say, the light-cycle game from TRON, and then save your work to an audio cassette tape.

From my vantage point in the twenty-first century, it seems so strange: you could push in a cartridge, close the little door, turn on your TV set and be able to program.

Granted, you had to work with a cumbersome programming language whose use might leave you with bad programming habits, but it was so immediate. Nowadays, we have more capable languages, ones which you can begin with as a novice and do serious work in later. (Ones in which you don’t have to put a number at the front of each line.) But I can’t just plug in a Python cartridge and get it on my TV; on many computers, I have to go download it and then fuss through the installation process, not only for the language itself but also for additional packages needed for particular applications. Said process is, of course, different, not just for each operating system, but for each incremental version thereof. I had to go through this for a roomful of people and their laptops several weekends ago, getting SciPy and matplotlib and NetworkX running. It wasn’t pretty, although the end result is quite useful. (And then we have to keep it going. Here in the office, the very next week, we saw all our SciPy/matplotlib/Sage installations break because, as far as we can tell, Mac OS X decided to upgrade itself and thereby clear-cut the whole dependency tree. Software only ever “just works” if you never do anything with it.)

But anyway! In the spirit of those long-forgotten BASIC code listings in algebra books, let’s have a bit of code with pædagogical utility. We’ve already seen how to implement Conway’s Game of Life in a few minutes; how about we try for another classic, the logistic map?

This is a mathematical idealization of a population of organisms, living in an environment which has a carrying capacity: the environment can’t sustain more than some fixed number of individuals. Here, we work in discrete time. That is to say, the population jumps from one size to another according to the rule we define, rather than changing smoothly; no time exists in between the ticks of the clock. We’ll let $x_t$ denote the fraction of the available space which is filled at time $t$. We specify the behaviour of our system by writing the update rule which tells us what $x_{t+1}$ will be, given $x_t$. In this case, we use

$$ x_{t+1} = rx_t(1 - x_t). $$

When $x_t$ is small, then this roughly approximated by ordinary exponential growth, $x_{t+1} = rx_t$, but if $x_t$ is closer to 1, then the second term within the parentheses kicks in, and the growth rate is diminished.

An aside: why is this called “logistic” growth? Nobody knows for sure why Verhulst first decided to call his equation of this sort “la courbe logistique“. He taught at Belgium’s military academy, so he may have picked up the term logistics, which military folks had just recently started to use in the modern sense, referring to how one goes about provisioning armed forces with food and other necessary materiel. This may have suggested logistique as a description for an equation which treats population growth in the case of limited resources. Also, in Verhulst’s day (1840s), logistique referred to mathematical work done through explicit arithmetical computations — what we might call “number-crunching” — so he may have chosen that term to stress the numerical emphasis of his approach to demography.

Anyway! The other day, I hacked together logistic.py, which implements the logistic map and plots the results. I wanted something which demonstrated (for the weekend class I mentioned earlier) a few features of the Python language which I’ve found useful: reading arguments in from the command line, using try/except blocks to handle unexpected circumstances and so forth. The whole thing took less time than I’ve spent figuring out what to say in prefacing it. And that includes making the pretty pictures.

First, let’s look at what the program makes by default. Our Python code follows a standard scheme for exploring the logistic map: pick a value of the growth rate $r$, pick some starting value of the population size $x_0$ and run through the update rule a bunch of times. This is to give the system a chance to settle down. Then, iterate again several times, plotting the population size. If the population size stops bouncing around and settles to a steady, fixed value, then all the dots we plot will lie on top of each other. If the population size bounces between two values, back and forth, cyclically, then we’ll see two dots.

The big picture:

When the growth-rate parameter $r$ is less than 1, the population size drops to zero and stays there. For larger values of $r$, we still have a fixed point, but its value increases with $r$. Then, weirdly, we get a cyclic behaviour pattern: $x$ jumps back and forth between two numbers. The separation between these two numbers grows as we crank up $r$. From two, the cycle then splits into four.

Soon enough, the behaviour of $x$ becomes a chaotic mess!

(The SciPy software provides different options for plotting points, so you can tweak the code to make thinner lines, different colours, etc. My choices in this demo are biased by what I think looks cool.)

More weirdly still, if we zoom into one of those branching points, we get the entire original picture, reproduced in miniature. SciPy offers a zoom feature in its plotting window; by passing different command-line arguments to logistic.py, we can tell it to draw a sub-section of the bifurcation diagram with the same amount of detail we used in the original.

Notice the axes on this plot:

We can continue the process. Zoom again!

The tiny piece of the diagram in between $r = 3.560$ and $r = 3.575$ reproduces the original, in shrunken form. This self-similarity is a defining characteristic of fractal shapes.

Homework: find manually, and/or code a program to determine, the ratios between the values of $r$ at the successive branch points.

(Footnote: yes: somebody did implement the logistic map in BASIC. Some people juggle geese . . .)

Ah, complex networks: manufacturing centre for the textbook cardboard of tomorrow!

When you work in the corner of science where I do, you hear a lot of “sales talk” — claims that, thanks to the innovative research of so-and-so, the paradigms are shifting under the feet of the orthodox. It’s sort of a genre convention. To stay sane, it helps to have an antidote at hand (“The paradigm works fast, Dr. Jones!”).

For example, everybody loves “scale-free networks”: collections of nodes and links in which the probability that a node has $k$ connections falls off as a power-law function of $k$. In the jargon, the “degree” of a node is the number of links it has, so a “scale-free” network has a power-law degree distribution.

So, you take your favourite system and boil it down to a set of nodes and links. You count up the node degrees and plot the curve on a log-log plot, because taking the logarithm of both sides of

$$\displaystyle P \propto k^{-a},$$

you realize that

$$\displaystyle \log P = -a\log k + b,$$

so a power-law dependence will show up as a straight line. Unfortunately, you might already be fooling yourself, as other functions can look to the eye — and to linear regression — like straight lines on such a plot. You can hide a multitude of sins on a log-log graph!

Problem number two: The degree distribution does not by itself characterize a network. Two networks can be quite different but have identical degree distributions. For example, consider the “clustering coefficient”, defined as the probability that two neighbours of a node will themselves be directly connected. (It measures the “cliquishness” of the network, in a way.) One can build networks with indistinguishable power-law degree distributions but arbitrarily different clustering coefficients. The NetworkX Python module has a built-in function to do just that: powerlaw_cluster_graph().

FURTHER READING

• A. Clauset, “People v. The scale-free networks hypothesis” (blog entry, 20 April 2010), which cites
• A. L. G. de Lomana, “Statistical Analysis of Global Connectivity and Activity Distributions in Cellular Networks” Journal of Computational Biology; and also
• R. Tanaka, “Scale-Rich Metabolic Networks” Phys. Rev. Lett. 94 (2005): 168101; not to mention
• A. Clauset et al., “Power-law distributions in empirical data” SIAM Review 51 (2009): 661–703, for which the code is available ontube.
• J. Dall and M. Christensen, “Random Geometric Graphs” Phys. Rev. E 66 (2002): 016121.
• C. Herrmann et al., “Connectivity Distribution of Spatial Networks” Phys. Rev. E 68 (2003): 026128.

By the way, what I have just outlined is what I call a “physicist’s history of physics,” which is never correct. What I am telling you is a sort of conventionalized myth-story that the physicists tell to their students, and those students tell to their students, and is not necessarily related to the actual historical development, which I do not really know!

Richard Feynman

Back when Brian Switek was a college student, he took on the unenviable task of pointing out when his professors were indulging in “scientist’s history of science”: attributing discoveries to the wrong person, oversimplifying the development of an idea, retelling anecdotes which are more amusing than true, and generally chewing on the textbook cardboard. The typical response? “That’s interesting, but I’m still right.”

Now, he’s a palaeontology person, and I’m a physics boffin, so you’d think I could get away with pretending that we don’t have that problem in this Department, but I started this note by quoting Feynman’s QED: The Strange Theory of Light and Matter (1986), so that’s not really a pretence worth keeping up. When it comes to formal education, I only have systematic experience with one field; oh, I took classes in pure mathematics and neuroscience and environmental politics and literature and film studies, but I won’t presume to speak in depth about how those subjects are taught.

So, with all those caveats stated, I can at least sketch what I suspect to be a contributing factor (which other sciences might encounter to a lesser extent or in a different way).

Suppose I want to teach a classful of college sophomores the fundamentals of quantum mechanics. There’s a standard “physicist’s history” which goes along with this, which touches on a familiar litany of famous names: Max Planck, Albert Einstein, Niels Bohr, Louis de Broglie, Werner Heisenberg, Ernst Schrödinger. We like to go back to the early days and follow the development forward, because the science was simpler when it got started, right?

The problem is that all of these men were highly trained, professional physicists who were thoroughly conversant with the knowledge of their time — well, naturally! But this means that any one of them knew more classical physics than a modern college sophomore. They would have known Hamiltonian and Lagrangian mechanics, for example, in addition to techniques of statistical physics (calculating entropy and such). Unless you know what they knew, you can’t really follow their thought processes, and we don’t teach big chunks of what they knew until after we’ve tried to teach what they figured out! For example, if you don’t know thermodynamics and statistical mechanics pretty well, you won’t be able to follow why Max Planck proposed the blackbody radiation law he did, which was a key step in the development of quantum theory.

Consequently, any “historical” treatment at the introductory level will probably end up “conventionalized.” One has to step extremely carefully! Strip the history down to the point that students just starting to learn the science can follow it, and you might not be portraying the way the people actually did their work. That’s not so bad, as far as learning the facts and formulæ is concerned, but you open yourself up to all sorts of troubles when you get to talking about the process of science. Are we doing physics differently than folks did N or 2N years ago? If we are, or if we aren’t, is that a problem? Well, we sure aren’t doing it like they did in chapter 1 of this textbook here. . . .

“On his Huffington Post blog, Deepak Chopra explained how this research proves the quantum nature of bodacious sitar riffs.

“Now this.”