Please note: First, I am not an author here; if at all anything, a reporter or a reviewer or, worse but true, a copier. There are two parts.
• The first part gives the summary of a recent scientific study trying to characterize and model the dynamics of online popularity. One of the main conclusions is that the dynamics of information networks is ‘scale-free’. This is typical in a wide range of “critical” physical, economic and social systems, such as avalanches, earthquakes, stock market crashes and human communication. Having mentioned that, it is impossible to avoid the next part.
• Benoit Mandelbrot, the father of fractal geometry, died on October 14, 2010. The fractal geometry he developed would be used to measure natural phenomena like clouds or coastlines that once were believed to be non-measurable. He applied the theory to physics, biology, finance and many other fields of study.
Part 1: Characterizing and modeling the dynamics of online popularity by Ratkiewicz et al. (Physical Review Letters, Vol. 105, No. 15. (Oct 2010), 158701.) [http://arxiv.org/abs/1005.2704]
It is believed that online popularity has enormous impact on opinions, culture, policy and profits.
The authors study the dynamics of popularity of two information networks: (1) the Wikipedia and (2) the web space of Chile.
As popularity proxies they have chosen the traffic of a document, expressed by the number of clicks to that page generated by a specific population of users, and the number of hyperlinks pointing to a document. Given either of these proxies, they study its relative variation in a time unit, that is, its logarithmic derivative. Note that: if x(t) is the quantity being studied at time t, its logarithmic derivative is [x(t) - x(t-1)]/ x(t).
They find that the dynamics of popularity are characterized by bursts in its relative variation or logarithmic derivative (hereafter, simply labeled as f).
Observations:
• Almost all pages experience a burst in f near the beginning of their life.
• Many pages receive little attention thereafter.
• Some pages maintain a nearly constant positive f indicating an exponential growth.
• A number of pages continue to experience intermittent bursts in f later in their life.
• In all cases, they observe heavy-tail behaviour. Such heavy-tailed burst magnitude distributions suggest a dynamics lacking a characteristic scale.
Model:
Researchers have used various models to describe the heterogeneous statistical properties of the Web (with distributions characterized by fat-tails roughly following power-law behaviour) and some models are based on the rich-get-richer mechanism.
The models based on rich-get-richer mechanism have two main ingredients:
(1) We need a growing network. But growth alone cannot explain. When a new node is created on the network, it is not sufficient to assume that it will link randomly and democratically. Though the senior nodes will have a clear advantage (since these nodes had the longest time to collect links and the poorest node will be the last to join), the distributions that result from such an assumption follow an exponential rather than a fat-tailed power law.
(2) We also need preferential attachment and need to discard any democratic (random) character. We attach a greater probability to link to those nodes which are already heavily linked.
The authors of this paper feel that this is not sufficient. They include a third ingredient: occurrence of exogenous factors that shift the attention of users and suddenly increase the popularity of specific topics because of events such as an actor winning a prize, political elections, rescue missions, scandals, etc.
With their rank-shift model, they introduce a new parameter, by which they can shift forward the ranking/popularity of a node.
As a layperson in a network, at the end of the day, the message to take home is: (a) you are part of a growing network; (b) if you are smart, you will believe in preferential attachment and link to those who are very popular; (c) if you want to be smarter, create exogenous factors that will suddenly attract attention of any sort.
Due to selfish interests, I would like to explore: (i) the dynamics of pages that receive little attention and the proximity factor (do friends of friends (of friends of …) contribute to that initial burst; (ii) dynamics of tightly-knit close or ‘gated’ online communities and the formation of such in open dynamic networks; (iii) a symmetric rank-shift model which allows for exogenous factors that could reduce ranking via a burst of unpopularity due to variants of untouchability, blocking and censorship, unpopular views and opinions, democratic decisions of the majority to ostracize, etc.; and so on and so forth.
Part 2: Obituary of Benoit Mandelbrot
On the 18th of October, in the inside pages of The Hindu, I found this column about Benoît Mandelbrot (mathematician, born 20 November 1924; died 14 October 2010):
Date:18/10/2010 URL: http://www.thehindu.com/2010/10/18/stories/2010101861860900.htm
Mandelbrot, father of fractal geometry, dead
WASHINGTON: Benoit Mandelbrot, a French-American mathematician who explored a new class of mathematical shapes known as “fractals,” has died at age 85 in Cambridge, Massachusetts, The New York Times reported on Saturday. His wife Aliette told the newspaper he died of pancreatic cancer.
His seminal book, “The Fractal Geometry of Nature,” published in 1982, argued that irregular mathematical objects once dismissed as “pathological” were a reflection of nature. The fractal geometry he developed would be used to measure natural phenomena like clouds or coastlines that once were believed to be unmeasurable.
He applied the theory to physics, biology, finance and many other fields of study.
“Fractals are easy to explain, it's like a romanesco cauliflower, which is to say that each small part of it is exactly the same as the entire cauliflower itself,” Catherine Hill, a statistician at the Gustave Roussy Institute, told AFP. “It's a curve that reproduces itself to infinity. Every time you zoom in further, you find the same curve,” she said.
David Mumford, a professor of mathematics at Brown University, told the Times that Mr. Mandelbrot had effectively revolutionised his field. “Applied mathematics had been concentrating for a century on phenomena which were smooth, but many things were not like that: the more you blew them up with a microscope the more complexity you found,” the paper quoted him as saying.
A professor emeritus at Yale University, Mr. Mandelbrot was born in Poland but as a child moved with his family to France where he was educated. — AFP
© Copyright 2000 - 2009 The Hindu
Please refer to the following obituary in The Guardian for more details:
http://www.guardian.co.uk/science/2010/oct/17/benoit-mandelbrot-obituary
Here, I do assume that his work has definitely touched everyone. Can you finish graduate studies or any kind of research without encountering the work of Mandelbrot? I can still remember how thrilled I was when I found a ‘Devil’s staircase’ (or, at least, something that looked exactly like that) in a dynamical system some time in the last millenium. [For a definition of Devil’s staircase, please refer to http://mathworld.wolfram.com/DevilsStaircase.html.]
Or, if you have kids, didn’t they ask you at least once, surely: “Amma/Appa, how does a world with a dimension of 0.63 look like?” or “O gee, Pop/Mom! Look at that…it is made out of itself…every small part looks like the big part…is it a fractal?” Well, if they have not, they should go to a different school.
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