The web seems to be one of many areas of human activity where power laws hold sway. The book by Barabasi gives a very comprehensible account, neatly sidestepping the mathematics.
The web is a network, in fact several different kinds of networks and the number of links that come from each network node can often be expressed by a power law. For example we may be interested to know how the number of incoming links on a web page (the node) varies from page to page. We might expect that most web pages had roughly the same number, not too far from some average, just as most people's height is around the average for the population. But as Barabasi recounts that is not the case for links to web pages. Most web pages have a few incoming links, whilst a few pages have a very large number. If you plot the graph it might look like this:
The mathematical expression of a power law is in fact rather simple as such things go. It is something like
number of web pages = constant * number of links power
The value of power is typically in the range -2 to -3
This idea has been applied by Milovanovic to four quite different sets of data. They all show, with varying degrees of conviction, a power law distribution.
The one which interests us here is a plot of the number of followers of 11.5 million Twitter users. This shows a very clear power law distribution with a power value of -2.25. The power law in this case refers to the whole linked network of these millions of users. That certainly helps us to understand the structure of the whole network. However we may be interested in the nature of the network as experienced by an individual user such as the distribution of their followers. To do this we can plot the number of followers of each of the followers of this individual user - does this also give a power law? Having done this for a few users the answer seems to be that typically as the number of followers becomes large, of the order of 10's of thousands, a power law distribution is approached.
But most users do not have 10's of thousands of followers. What is the distribution of their followers? They can be characterised in part by simple measures such as the proportions of their followers that fit into defined categories, such as the tail of the distribution, set perhaps arbitrarily at 100000 followers.
In networks such as that of Twitter users there would seem to be considerable possibilities to explore deviations from power law patterns.
Barabasi, A (2002), Linked, Penguin Books Ltd, London
Milovanovic, G.S.(2010) Online Contributions to the Global Community: The Power Laws for New Media Available from:
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