Tyler Cowen (where does he ever find the time?) alerts me that the extremely sharp, thoughtful, and witty Clive Crook now has a weblog.

Jeebus. I am in trouble. I used to think that the fact that I was early into an expanding arena meant that for a long time my internet footprint would be far above my intellectual merits. But the competition for web mindshare is becoming really, really stiff...

Apropos of which: from the archives: whether well-known-ness--internet celebrity--trumps excellence is a problem depends, of course, on whether one is an internet celebrity:

http://www.j-bradford-delong.net/movable_type/2003_archives/001185.html: Are the best websites--the most interesting, the most informative, the most authoritative--the easiest to find? We have a world wide web in which we use the link structure to find things. But because we ourselves add what we find to the web's link structure, the number of links to a site depends not just on its quality but also on how easy it is to find. To the extent that services like Google that are in part functions of the web's link structure have become key search tools, these potential positive-feedback mechanisms have been strengthened.

Is there a danger that we are drifting toward a web of celebrity rather than of information--one in which well-known sites are well-known and prominent because of their well-knownness rather than their quality?

This is an interesting problem to try to think about...

Let's start with the simplest possible useful model of how the links to a website evolve over time. At any moment the rate of change of the links L to a website are:

- increasing at a rate b
_{1}L as relatively clueless links are added by people whose websurfing is guided by the existing link structure, or by things like Google that aggregate the existing link structure.- decreasing at a rate b
_{2}L through linkrot.- increasing at a rate Q, where Q is an index of the quality of the website, as the clued-in link to websites that are useful, informative, and authoritative.

This means that the dynamics of links L follow the simple equation:

(1) dL/dt = b

_{1}L - b_{2}L + Q

And our questions are two: First, will the number of links to a website converge to be proportional to the quality Q of the website? Second, how long will this convergence take?

If the website starts at some time 0 with L_{0} links (derived from past history or celebrity or whatever), then the solution to the differential equation (1) above is:

(2) L = L

_{0}e^{-(b2-b1)t}+ (1 - e^{-(b2-b1)t})(Q/(b_{2}-b_{1}))where t is the index of the current time.

If b

_{2}is greater than b_{1}--if (independent of quality) having a lot of links tends to put downward pressure on the number of links to a website, as linkrot removes links faster than the clueless who are just surfing the web's link structure add them--then this equation is well behaved. As t grows larger, e_{-(b2-b1)t}shrinks to zero: the impact of the initial link number L_{0}on the current link number L vanishes. As t grows larger, (1 - e^{-(b2-b1)t}) grows to equal one: the number of links converges to an amount proportional to the site's quality:(3) L = (Q/(b

_{2}-b_{1}))The closer is b

_{1}to b_{2}, the less relevant is this long-run result: it might take eons for convergence to occur...If b

_{2}is less than b_{1}--if (independent of quality) having a lot of links tends to put upward pressure on the number of links to a website, as the clueless who are just surfing the web's link structure add links faster than linkrot removes--then this equation is not so well behaved. It is most illuminating to rewrite (2) as:(4) L = (e

^{(b1-b2)t})(L_{0}+ Q/(b_{1}-b_{2})) - Q/(b_{1}-b_{2})Over time, (e

^{(b1-b2)t}) grows without bound: positive feedback produces rapid exponential growth, after all. Looking across websites, as long as (b_{1}-b_{2})t is relatively large, different sites' relative link numbers are not proportional to their qualities Q, but instead to (L_{0}+ Q/(b_{1}-b_{2})). If Q is large relative to L_{0}(b_{1}-b_{2}), then there is little long-run impact: relative link numbers are nearly proportional to website quality. But if Q is not large relative to L_{0}(b_{1}-b_{2}), then initial conditions--early start, web celebrity, whatever--have a powerful influence on relative links numbers--and thus on effective web footprint--even in the longest of runs.So how relevant is this simple model? I don't know. I'm thinking about it...

Memo:See http://www.shirky.com/writings/powerlaw_weblog.html.

And: