PowerLaws

There are many phenomena which exhibit the following property :

• you have a large population of rival things

• in some way each thing has a score

You notice that a few of the things have high scores, and many of the things have lower scores. In fact the lower the score, the more things have it, and vice versa. Sound complex and abstract?

OK, consider the distribution of words used in language (this is Zipf's Law)

• a few words, like "the", "and", "it" etc. have very high frequencies.

• the majority of the words have very low frequencies.

• in fact there's a relation ship of something like this f ~ 1/N where f is frequency, N is number of words. (I'm using ~ to mean proportional to) So for a very small number of words (N is low) f is very high; for the majority of words (N being high) f is very low.

When you see an example of this distribution, you are seeing a power law distribution in action. Networks with power law distributions of links are also called ScaleFreeNetworks. And are typically evidence that the network has grown organically.

More specifically power laws are often the result of a process which adds links, but rewards well linked nodes with yet more nodes.

What else gets power law distributions?

• Wealth : a few people have most of it /WealthDistribution

• Health : see http://www.guardian.co.uk/life/lastword/story/0,13228,1265949,00.html

• website readership : a few get most readers

• web referrals (links to) : most readers come from a few links

ShirkysLaw

In fact there's a whole class of networks where nodes are linked according to a power law distribution. Ie. a few nodes have most links etc. Not all networks have this architecture, but let's think about those that do

• ClayShirky on Power Laws governing weblogs : http://www.shirky.com/writings/powerlaw_weblog.html
  • maybe this really is bad : https://burningbird.net/steve-levy-dave-sifry-and-nz-bear-you-are-hurting-us/

2006 "Dance Mix" : http://many.corante.com/archives/2006/02/14/powerlaws2006dance_remix.php (Good distinction between the hierarchy of the org-chart : formal, official, Shirky calls "structural" but I don't like the terminology; and the emergent hierarchy of top-100, A-lists etc.)

Counter to Shirkey's application of Power Laws to weblogs

Winners Don't Take All

David M. Pennock, Gary W. Flake, Steve Lawrence, Eric J. Glover, and C. Lee Giles : (Via http://www.pnas.org/cgi/content/full/99/8/5207 (Via http://modelingtheweb.com/))

PDF : http://modelingtheweb.com/pennock-pnas-2002-weblinks.pdf

Asserts that Although the connectivity distribution over the entire web is close to a pure power law, we find that the distribution within specific categories is typically unimodal on a log scale, with the location of the mode, and thus the extent of the rich get richer phenomenon, varying across different categories.

Network Types

RossMayfield argues (RossMayfieldsThreeScalesOfNetwork) that there are different types of networks, with different scales :

• The Political network is large, and follows a Power law distribution.

• The Social network (consiting of our friends and acquantances) is rooted in human psychology and usually contains about 150 people. Mayfield suggests that links here follow more a bell curve distribution.

• The Creative network is your network of close friends and collaborators on projects. This is around 12 people and is densely interconnected.

If this is true, then we have to understand that certain networks have interesting and relevant substructure. It raises several questions, does Shirky's Power Law hypothesis not hold for weblogs at all because of this substructure; or does it hold when looking

at a large scale; but fail to work at the smaller scales.

Or, thinking about OneWayLinks, perhaps we're talking about different networks here. Shirkey is refering to the large scale, 1-way linked traditional blogosphere; while Ross is refering to smallish groups of friends who've discovered each other through blogging and already have a more equal (2-way linked) relationship.

Further :

• Areas of the network are like "cells" (structuration of the network can be divided into discrete components) : (Kind http://alex.halavais.net/news/archives/000268.html (Kind of goes against the idea of an internet (ie. a collection of subnets which blur into a single whole ... but if there are discrete subnets eg. subnets which are separated by type of links ...

• http://ming.tv/flemming2.php/d10/v10/__showday/w2003-02-14#000010-000568

Closer Detail

• Kevin Marks "the many lightly linked weblogs outnumber the few heavily linked ones" : http://homepage.mac.com/kevinmarks/powerlaws.html

And further discussions : http://radio.weblogs.com/0114726/2003/02/06.html

CosmaShalizi on why power-laws interest physicists : http://cscs.umich.edu/~crshalizi/notebooks/power-laws.html

(Interesting that he distinguishes power-laws from exponentials.)

And more http://cscs.umich.edu/~crshalizi/weblog/232.html

Clearly I'm ignorant of a lot of this stuff. How I wish there was really a way to LookMathsUp. :-(

Networks will organize themselves into power-law distributions of links as long as

• a) the probability of the next link is somehow proportional to the number of links you already have

But power-laws distributions of links will be reduced if there is an AbsoluteLinkCarryingCapacity

See also SnowballEffect, TheCourtOfKingBlog, MappingTheBlogosphere, SocialNetworkAnalysis, NetoCracy

CategoryNetworks, CategoryNetworkSociety