Wednesday, 19 October 2011

Geo-Tagged Tweeting and Power Laws for Dummies


Tweets over America, tweets cluster in a power law

Above is the distribution of tweets over the United States for a three hour period. Notice how the tweets tend to cluster in to hubs. We say that at this scale the tweets cluster according to a power law. Below you see a graphic illustration of a random group of circles where there size is governed by a power law


Circles randomly arranged on a map, their sizes are distributed by a power law.

The above graph shows circles randomly placed on a grid.  The sizes of the circles are distributed with a power law.  This means that most of the circles are small while a very few are large.  But those few large ones are so large they dominate the spaces.  So though this graph has many times more small images than large images most of the space that is covered by circles is covered with large circles.

For most views of twitter geo-tagging from the city level to the global level this is precisely what you see. A few spots have the most concentrated in a few locations.


Tweets over London, you can see in the graphic above that a few hubs in the city have the most concentrated tweets.  Each red tweet bird shows a tweeting location, the brighter red an area is the more tweets are from that location.  

Geo-tagged tweets over London on a black background, notice how a few areas have heavy density of tweets.


Tweets during the Occupy Wall Street protests in New York
Opposed to a power law distribution would be a normal distribution as you see below.  Here the circles are all the same size and distributed randomly.  Because they are distributed randomly there is a tendency for them to cluster together, so some areas have more circles then others, but the overall distribution is more even than in the power law distribution.


Random distribution of equal size circles, though there is clustering it is not a power law.

This type of random distribution now following a power law is what you see when you zoom in to a map of geo-tag tweeting within a few hundred meter. So there is a scale variance.  As you zoom in closer and closer to a cluster of tweets you see less and less of a power law.  Look at the sample geo-tagged tweet distributions below from some major cities:


London



New York



Tokyo



Moscow

This would make sense.  Within a small area all the buildings tend to be the same size, maybe with a few f high rises.  You can only fit so many people in a spot unless they are in a very tall building.  The limits of space tend to spread people out in small scale.  If people try to crowd in to a single place they are confronted with lines.  Nature limits how high you can normally stack people. 

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