Pharyngula

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Thursday, October 27, 2005

λ >> µ, or why I haven't answered you yet

I'm certainly no Darwin nor Einstein, but I am a 21st century wired American, so these data interested me:

During their lifetimes, Darwin sent at least 7,591 letters and received 6,530; Einstein sent more than 14,500 and received more than 16,200. We start from a record containing the sender, recipient and the date of each letter sent or received by the two scientists. Their correspondence exploded after their rise to fame, and reached a highly fluctuating pattern afterwards. Although, on average, they wrote 0.59 (Darwin) and 1.02 (Einstein) letters a day during the last 30 years of their lives, these averages hide significant daily fluctuations. For example, Darwin wrote 12 letters on New Year's Day in 1874 and Einstein received 120 letters on 14 March 1949, his 70th birthday.

Not counting any spam (which gets automatically dumped by my filters, or manually trashed if it gets through), I have received approximately 18,000 email messages in the last 3 months. Over that same span, I've sent out about 800. Email tends to be far briefer than the kinds of letters Darwin or Einstein sent out, but still…we've experienced a bit of an escalation in volume as a consequence of our technology, I think.

Technology also helps manage the input a bit, too. My software automatically sorts the mail, giving a high priority to my students and to friends and family, and also does a little discrimination on the basis of content. As you can see from the asymmetry between my inbox and outbox, though, it isn't any help at all with the outgoing mail.

This is from Nature, so of course the authors did an extensive analysis of the mail habits of Darwin and Einstein.

image
a, Historical record of the number of letters sent (Darwin, black; Einstein, green) and received (Darwin, red; Einstein, blue) each year by the two scientists. An anomalous drop in Einstein's correspondence marks the Second World War period (1939–45, boxed). Arrows, birth dates of Darwin (left) and Einstein (right). b, c, Distribution of response times to letters by Darwin and Einstein, respectively. Note that both distributions are well approximated with a power-law tail that has an exponent α=3/2, the best fit over the whole data for Darwin giving α=1.45±0.1 and for Einstein α=1.47±0.1.

And once you've got all that data, you absolutely must describe it with a mathematical formula.

To understand the origin of the observed scaling behaviour, we have to realize that, given the wide range of response times, both Darwin and Einstein must have prioritized correspondence in need of a response. Thus, a simple model of their correspondence assumes that letters arrive at a rate λ and are answered at a rate µ. Each letter is assigned a priority, with high-priority letters being answered soon after their arrival, and others having to wait.

The waiting-time distribution of this simple model follows P(τ)≈τ-3/2exp(-τ/τ0), which predicts a power-law waiting time for the critical regime λ=µ, when τ0=∞. Given that Darwin and Einstein answered only a fraction of letters they received (their overall response rate being 0.32 and 0.24, respectively), we have λ>µ. This places the model in the supercritical regime, where a finite fraction of letters are never answered. Numerical simulations indicate that in this supercritical regime the waiting-time distribution of the responded letters also follows a power law with exponent α=3/2, which is different from the α=1 obtained for e-mail communications. Therefore, although the response times in e-mail and mail communications follow the same scaling law, they belong to different universality classes.

I highlighted one wonderfully useful part of this analysis. If you've sent me mail, and I haven't replied, my apologies…but my λ>>µ.


Oliveira JG, Barabásil A-L (2005) Darwin and Einstein correspondence patterns. Nature 437:1251


Cosma is unimpressed. What? You mean you can't just plug a pile of numbers into a spiffy formula and get The Answer?


Trackback url: http://pharyngula.org/index/trackback/3239/e3SwNOat/

Comments:
#45655: — 10/27  at  11:57 AM
This reminds, I have a couple of emails I really ought to answer...



#45658: Jake — 10/27  at  12:29 PM
This makes complete sense, but I do have one question. Why are you writing "lambda >> myu", rather than just "lambda > myu"?

(sorry, too lazy to look up the html for lambda and myu(sp?))



#45668: Ithika — 10/27  at  01:32 PM
Jake: >> is "much greater than"



#45780: — 10/28  at  04:34 AM
And once you've got all that data, you absolutely must describe it with a mathematical formula.

Of course, if you've done pointless research and got lots of pointless data, it's surely compulsory to do some pointless regression to fit a pointless model.

Wow, this certainly makes me feel much better about my own research.



#45787: Jonathan Badger — 10/28  at  05:30 AM
"Wow, this certainly makes me feel much better about my own research."

This got in _Nature_, believe it or not. I get the strong feeling that applying power laws to data is what the _Nature_ editors see as cutting-edge research, and the peer reviewers were probably historians of science who were blown away by the math.



#45884: jre — 10/28  at  12:41 PM
Cosma Shalizi has a relevant post with a wonderful title. (See October 28th)



#45932: — 10/28  at  04:18 PM
John Kennedy had one woman who answered all correspondence for him. She would handle 60,000 letters per year.

When I staffed the Senate in the early-mid 1980s, a senator from a small state would get 100,000 letters a month. A senator from a large state might get 1 million letters a month.

Not only did Einstein and Darwin write and get mail prolifically, their stuff was probably much higher quality.

(I think Jefferson's totals are higher than Einstein's, by the way -- 150 years earlier; Jefferson invented an ingenious machine to duplicate letters as he wrote. We have almost all of them.)



Trackback: Kritik kring power law-forskning Tracked on: hakank.blogg (82.182.97.188) at 2005 10 29 02:06:52
Ngra korta lnkningar till en intressant kritik av power law-forskningen och speciellt av Albert-Lszl Barabasi (som jag skrivit om en del tidigare). Joao Gama Oliveira och Albert-Lszl Barabsi skrev nyligen Darwin and Einstein correspondence patterns (PDF, Complementary Materials). New Scientist...



Trackback: How We Collaborate Tracked on: Quædam cuiusdam (216.194.64.152) at 2005 11 01 12:47:55
Here are some thoughts on a part of the history of intellectual collaboration, presented with some fully-populated COinS (generated with the COinS Generator), and with some applications to the present day. Grab a COinS resolver bookmarklet to see the ...



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