Part of this project requires understanding something about how whale size (mass) and shape (length) are related. This is because in some cases, it's possible to get a notion of how long a whale is (for example, a long dead one buried in Miocene sediments), but it's generally very hard to estimate how heavy it is. [3]

This goes back to an old question in animal morphology, which is whether size and shape are related geometrically or elastically. That is, if I were to double the mass of an animal, would it change its body shape in all directions at once (geometric) or mainly in one direction (elastic)? For some species, like snakes and cloven-hoofed animals (like cows), change is mostly elastic; they mainly get longer (snakes) or wider (bovids, and, some would argue, humans) as they get bigger.

About a decade ago, Marina Silva [4], building on earlier work [5], tackled this question quantitatively for about 30% of all mammal species and, unsurprisingly I think, showed that mammals tend grow geometrically as they change size. In short, yes, mammals are generally spheroids, and L = (const.) x M^(1/3). This model is supposed to be even better for whales: because they're basically neutrally buoyant in water, gravity plays very little role in constraining their shape, and thus there's less reason for them to deviate from the geometric model [6].

Collecting data from primary sources on the length and mass of living whale species, I decided to reproduce Silva's analysis [7]. In this case, I'm using about 2.5 times as much data as Silva had (n=31 species versus n=77 species), so presumably my results are more accurate. Here's a plot of log mass versus log length, which shows a pretty nice allometric scaling relationship between mass (in grams) and length (in meters):

Aside from the fact that mass and length relate very closely, the most interesting thing here is that the estimated scaling exponent is less than 3. If we take the geometric model at face value, then we'd expect the mass of a whole whale to simply be its volume times its density, or

where k_1 and k_2 scale the lengths of the two minor axes (its widths, front-to-back and left-to-right) relative to the major axis (its length L, nose-to-tail), and the trailing constant is the density of whale flesh (here, assumed to be the density of water) [8].

If the constants k_1 and k_2 are the same for all whales (the simplest geometric model), then we'd expect a cubic relation: M = (const.) x L^3. But, our measured exponent is less than 3. So, this implies that k_1 and k_2 cannot be constants, and must instead increase slightly with greater length L. Thus, as a whale gets longer, it gets wider less quickly than we expect from simple geometric scaling. But, that being said, we can't completely rule out the hypothesis that the scatter around the regression line is obscuring a beautifully simple cubic relation, since the 95% confidence intervals around the scaling exponent do actually include 3, but just barely: (2.64, 3.01).

So, the evidence is definitely in the direction of a geometric relationship between a whale's mass and length. That is, to a large extent, a blue whale, which can be 80 feet long (25m), is just a really(!) big bottlenose dolphin, which are usually only 9 feet long (2.9m). That being said, the support for the most simplistic model, i.e., strict geometric scaling with constant k_1 and k_2, is marginal. Instead, something slightly more complicated happens, with a whale's circumference growing more slowly than we'd expect. This kind of behavior could be caused by a mild pressure toward more hydrodynamic forms over the simple geometric forms, since the drag on a longer body should be slightly lower than the drag on a wider body.

Figuring out if that's really the case, though, is beyond me (since I don't know anything about hydrodynamics and drag) and the scope of the project. Instead, it's enough to be able to make a relatively accurate estimation of body mass M from an estimate of body length L. Plus, it's fun to know that big whales are mostly just scaled up versions of little ones.

More about why exactly I need estimates of body mass for will have to wait for another day.

Update 17 Nov. 2009: Changed the 95% CI to 3 significant digits; tip to Cosma.

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[0] The pictures are, left-to-right, top-to-bottom: blue whale, bottlenose dolphin, humpback whale, sperm whale, beluga, narwhal, Amazon river dolphin, and killer whale.

[1] Actually, I mean Cetaceans, but to keep things simple, I'll refer to whales, dolphins, and porpoises as "whales".

[2] Thankfully, the project doesn't involve networks of whales... If that sounds exciting, try this: D. Lusseau et al. "The bottlenose dolphin community of Doubtful Sound features a large proportion of long-lasting associations. Can geographic isolation explain this unique trait?" Behavioral Ecology and Sociobiology 54(4): 396-405 (2003).

[3] For a terrestrial mammal, it's possible to estimate body size from the shape of its teeth. Basically, mammalian teeth (unlike reptilian teeth) are highly differentiated and certain aspects of their shape correlate strongly with body mass. So, if you happen to find the first molar of a long dead terrestrial mammal, there's a biologist somewhere out there who can tell you both how much it weighed and what it probably ate, even if the tooth is the only thing you have. Much of what we know about mammals from the Jurassic and Triassic, when dinosaurs were dominant, is derived from fossilized teeth rather than full skeletons.

[4] Silva, "Allometric scaling of body length: Elastic or geometric similarity in mammalian design." J. Mammology 79, 20-32 (1998).

[5] Economos, "Elastic and/or geometric similarity in mammalian design?" J. Theoretical Biology 103, 167-172 (1983).

[6] Of course, fully aquatic species may face other biomechanical constraints, due to the drag that water exerts on whales as they move.

[7] Actually, I did the analysis first and then stumbled across her paper, discovering that I'd been scooped more than a decade ago. Still, it's nice to know that this has been looked at before, and that similar conclusions were arrived at.

[8] Blubber has a slight positive buoyancy, and bone has a negative buoyancy, so this value should be pretty close to the true average density of a whale.

A. Clauset, C. R. Shalizi and M. E. J. Newman. "Power-law distributions in empirical data." SIAM Review 51(4), 661-703 (2009). (Download the code.)

Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution -- the part of the distribution representing large but rare events -- and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov–Smirnov (KS) statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data, while in others the power law is ruled out.

Here's a brief summary of the 24 data sets we looked at, and our conclusions as to how much statistical support there is in the data for them to follow a power-law distribution:

Good:
frequency of words (Zipf's law)

Moderate:
frequency of bird sightings
size of blackouts
book sales
population of US cities
size of religions
severity of inter-state wars
number of citations
papers authored
protein-interaction degree distribution
severity of terrorist attacks

With an exponential cut-off:
size of forest fires
intensity of solar flares
intensity of earthquakes (Gutenberg-Richter law)
popularity of surnames
number of web hits
number of web links, with cut-off
Internet (AS) degree distribution
number of phone calls
size of email address book
number of species per genus

None:
HTTP session sizes
wealth
metabolite degree distribution

The answer, according to the Facebook Data Team, is that while it depends on how you define "real," with access to the underlying data, you can pretty clearly see how much interaction actually flows across the different links. One neat thing they found (within a lot of interesting analysis) is that the amount of interaction across all your connections scales up with the number of connections you have. That is, the more friends you have, the more friends you interact with. (It can't be a linear relationship, though, since otherwise, people with 1000s of friends would be spending all of their free time on Facebook... oh wait, some people actually do that.)

2.
A related point that I've found myself discussing several times recently with my elders (some of whom I think are, at some level, alienated and befuddled by computer and Web technology), is whether Facebook (or, technology in general) increases social isolation, and thus is leading to some kind of collapse of civil society. I've argued passionately that it's human nature to be social and thus extremely unlikely that technology alone is having this effect, and that technology instead actually facilitates social interactions, allowing people to be even more social overall (even if they may spend slightly less time face-to-face) than before. Mobile phones are my favorite example of social facilitation, since they allow people to interact with their friends in situations when previously they could not (e.g., standing in line at the bank, walking around town, etc.), even if occasionally it leads to ridiculous situations like two people sitting next to each other, but each texting or talking on their phones with people elsewhere.

And, just in time to bolster my arguments, The Pew Internet and American Life Project released a study this week (also discussed in the NYTimes) showing that technology users are more social than non-technology users, and that other, non-technological trends are to blame for the apparent decrease in the size of (non-technology using) Americans' social circles over the past 20 years. Of course, access to and use of technology often correlates with affluence, so what really might be going on is that, like with nutrition, the affluent are better positioned to lead physically and socially healthy lives than the poor.

3.
Recently, for a project on evolution, I've been reading pretty deeply in the paleontology and marine mammal literature (more on that in the next post). The first thing that I noticed is how easy it is now to access vast amounts of scientific literature from the comfort of your office. Occasionally, I had to get up to see Margaret, our librarian, but most of the time I could get what I needed through electronic access. But, sometimes I would encounter a pay wall that my institutional access wouldn't allow me to circumvent.

At first, it was extremely irritating and induced open-access revolutionary spirits in me. Then, I did what I suspect many of you have done, too, which is to ask my friends at other universities to try to get access to the paper using their institutional access, and to send me a copy. On a small scale, this is like asking your friends to share individual musical tracks with you. So, naturally, the logical solution to the problem is to make a P2P sharing system for scientific papers, right? Exactly. There's apparently already such a system for mainly medical papers, but I think the time is ripe for something more ambitious. Given what's been learned about how to run a good P2P system for music, it should be pretty simple to develop a good system (distributed, searchable, scalable) for sharing PDFs of journal papers, right? I can't wait until the academic publishing industry starts suing researchers for sharing papers...

4.
If you're male, when you use a public restroom, what do you think about for those seconds while your body is busy but your mind is free to wander? Randall Munroe, of xkcd fame, apparently, thinks about the mathematics of restroom awkwardness and minimum awkward-ness packing arrangements for men using urinals. Who knew something so mundane could be so amusing?

5.
Finally, this next bit is already almost a year old, but it's just so good. Remember last year when the media when predictably bonkers over two studies, by Nicholas Christakis and James Fowler, showing that happiness and obesity were (socially) contagious? That is, if you're depressed, you can blame your friends for not cheering you up, and if you're fat, you can blame your friends for making you eat poorly. (Or, wait, maybe it's that misery loves company...?) Shortly after those studies hit the media, a wonderful followup study was published by Cohen-Cole and Fletcher. Their study used the same techniques as Christakis and Fowler and showed that acne, headaches and height are also socially contagious! If only we had the data, I'm sure social network analysis could be show that hair color, IQ and wealth are socially contagious, too. Their concluding thoughts say it all, really:

There is a need for caution when attributing causality to correlations in health outcomes between friends using non-experimental data. Confounding is only one of many empirical challenges to estimating social network effects, but researchers do need to attempt to minimise its impact. Thus, while it will probably not be harmful for policy makers and clinicians to attempt to use social networks to spread the benefits of health interventions and information, the current evidence is not yet strong enough to suggest clear evidence based recommendations. There are many unanswered questions and avenues for future research, including use of more robust empirical methods to assess social network effects, crafting and implementing additional empirical solutions to the many difficulties with this research, and further understanding of how social networks are formed and operate.

E. Cohen-Cole and J. M. Fletcher, "Detecting implausible social network effects in acne, height, and headaches: longitudinal analysis." BMJ 337, a2533 (2008).

Update 10 Nov.: Oh jeez. Olivia Judson, please get a clue.

The short story is that the popular quality function called "modularity" (invented by Mark Newman and Michelle Girvan) admits serious degeneracies that make it somewhat impractical to use in situations where the network is large or has a non-trivial number of communities (a.k.a. modules). At the end of the paper, we briefly survey some ways to potentially mitigate this problem in practical contexts.

The performance of modularity maximization in practical contexts

Benjamin H. Good, Yves-Alexandre de Montjoye, Aaron Clauset, arxiv:0910.0165 (2009).

Although widely used in practice, the behavior and accuracy of the popular module identification technique called modularity maximization is not well understood. Here, we present a broad and systematic characterization of its performance in practical situations. First, we generalize and clarify the recently identified resolution limit phenomenon. Second, we show that the modularity function Q exhibits extreme degeneracies: that is, the modularity landscape admits an exponential number of distinct high-scoring solutions and does not typically exhibit a clear global maximum. Third, we derive the limiting behavior of the maximum modularity Q_max for infinitely modular networks, showing that it depends strongly on the size of the network and the number of module-like subgraphs it contains. Finally, using three real-world examples of metabolic networks, we show that the degenerate solutions can fundamentally disagree on the composition of even the largest modules. Together, these results significantly extend and clarify our understanding of this popular method. In particular, they explain why so many heuristics perform well in practice at finding high-scoring partitions, why these heuristics can disagree on the composition of the identified modules, and how the estimated value of Q_max should be interpreted. Further, they imply that the output of any modularity maximization procedure should be interpreted cautiously in scientific contexts. We conclude by discussing avenues for mitigating these behaviors, such as combining information from many degenerate solutions or using generative models.

(This was my first time using power tools to carve a pumpkin, and I have to say, they make it a lot easier and a lot more fun!)

Luckily, the news media, in the form of the indomitable John Oliver, were there to cover and support the efforts. (And thus these savvy protesters made it on the Oct. 1 Daily Show for about 3 seconds at the 9m11s mark; blink and you'll miss them.)

Tip to Jake Hofman and Arthur Gretton (whose photos these are).

If that's not enough hilarity about chimps vs. orangs, or, if you were really intrigued by the arguments in favor of orangs, read this.

Tip to Jake Hofman.

The ACM, with Microsoft, Google, Intel and some other organizations, managed to persuade the US Congress that Computer Science is a Good Thing(tm) and that it deserves some recognition for driving economic growth (you know, making things like medicine, movies, music, and cars) [1]. To recognize the goodness, Congress passed a resolution (H. RES. 558) to designate the week of December 7 as “National Computer Science Education Week.” [2]

The resolution, H. RES. 558, sponsored by Congressmen Vernon Ehlers (R-MI) and Jared Polis (D-CO) [3], designates the week of December 7 as “National Computer Science Education Week.” Citing the influence of computing technology as a significant contributor to U.S. economic output, the House resolution calls on educators and policymakers to improve computer science learning at all educational levels, and to motivate increased participation in computer science.

“Increasing energy efficiency, advancing healthcare, and improving communication in the digital age are just a few of the national priorities that depend on computer science, which Congress has recognized. Computer science teaches students design, logical reasoning, and problem-solving – all skills that have value well beyond the classroom,” said Rick Rashid, senior vice president of Research for Microsoft.

“Despite serious economic challenges confronting the nation, computer science-related jobs are among the fastest-growing and highest paying over the next decade,” said Alfred Spector, vice president of Research and Special Initiatives at Google, Inc. “These times require an increasing supply of diverse students exposed to rigorous and engaging computing courses at the K-12 level, and National Computer Science Education Week can help to reinforce this effort.”

Good fanfare, and good effort for sure. It's a small gesture really, but I guess it does give organizations like the ACM a hook to hang their public campaigns on. And for sure, education about computers, computer science, and their use (and abuse) in society is something the public could do with some educating on.

Tip to Tanya Berger-Wolf.

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[1] Thankfully, they didn't mention that computer science and computers have also produced massive amounts of wasted time, the estimation of which never ceases to amuse me. (If you'd like to estimate it for yourself, try this.)

[2] In a fit of gender-neutrality (something Computer Science is not known for), the date was chosen to honor Grace Hopper, who wrote the first compiler and helped invent the indispensable COBOL, in addition to being a Rear Admiral in the Navy, and having a Naval destroyer named after her.

[3] Incidentally, I was very happy to discover that Mr. Polis represents the 2nd District of Colorado, and he'll be my representative once I move to Boulder next summer.

We are all connected (ft. Sagan, Feynman, deGrasse Tyson & Bill Nye)

Tip to Cris Moore.

Appointments begin Fall 2010. As a current postdoc, I can tell you that SFI is an exceptionally good place to spend a few years doing research. It's not for everyone, but if you do your best work in unstructured environments, have a good nose for asking fundamental questions, and are comfortable working in a highly interdisciplinary environment, then it's hard to beat the freedom and resources SFI provides.

This year's deadline is November 2.

The Santa Fe Institute (SFI) will be selectively seeking applications for Omidyar Fellows for 2011, starting in the Fall of 2010. Fellows are appointed for up to three years during which they pursue research questions of their own design and are encouraged to transcend disciplinary lines. SFI’s unique structure and resources enable Fellows to collaborate with members of the SFI faculty, other Fellows, and researchers from around the world.

As the leader in multidisciplinary research, SFI has no formal programs or departments and we accept applications from any field. Research topics span the full range of natural and social sciences and often make connections with the humanities. Most research at SFI is theoretical and/or computational in nature, although some research includes an empirical component in collaboration with other institutions. Descriptions of the research themes and interests of the faculty and current Fellows can be found at http://www.santafe.edu/research/.

Benefits: The compensation package includes a competitive salary and excellent health and retirement benefits. As full participants in the SFI community, Fellows are encouraged to invite speakers, organize workshops and working groups, and engage in research outside their fields. Funds are available to support this full range of research activities.

Requirements: SFI is known for its catalytic research environment and applicants must demonstrate the potential to contribute to this community. Candidates must have a Ph.D. (or expect to receive one by September 2010), an exemplary academic record, and a proven ability to work independently. We expect a demonstrated interest in multidisciplinary research and evidence of the ability to think outside traditional paradigms.

Applications:Applications are welcome from candidates in any country. Successful foreign applicants must acquire an acceptable visa (usually a J-1) as a condition of employment. Women and minorities are especially encouraged to apply. SFI is an equal opportunity employer.

TO APPLY: View the full position announcement and application instructions at www.santafe.edu/education/fellowships-postdoctoral.php. Application due by November 2, 2009. For further information, email ofellowshipinfo@santafe.edu.

This is a little extreme, but to be honest, scientists also love to extrapolate, and some of what they say is not much better than this...

(thanks xkcd)

I like this.

(tip to onegoodmove)

The appointment starts in the Fall of 2010, which gives me another year to finish out my postdoc at the Santa Fe Institute, and, more importantly, to finish up a lot of the projects that I've started here on topics like macroevolution and the mathematics of terrorism. So, come Summer 2010, it'll be goodbye postdoc, and hello responsibility! Wish me luck!

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[1] Oh yeah. I'm getting married, too!

The Harvard Political Networks Conference

11 June -- 13 June 2009 at Harvard in Cambridge, MA

Organizers: James H. Fowler and David Lazer