Hi again Lucas,
I see and appreciate your point. As Newman suggested, we should be perhaps
more parsimonious and say a degree distribution possesses a ¡®power-law
tail¡¯, instead of saying it follows a power-law over its entire domain.
This will dispel all ambiguities concerning the ¡®observation¡¯ of so-called
power-law distributions.
But I¡¯m not really concerned with whether we observe a power-law
distribution in realities. Rather what seems curious to me is the ¡®idea¡¯
of power-law which people posit as a kind of ¡®normative rule¡¯ to objectify
observation and aggregation process. Numerous geographers have pointed out
that the power-law or the rank-size rule holds better the more carefully
we define, by way of aggregation, an elemental unit (e.g. a ¡®city¡¯) on
which size is recorded. For another example, the CASA¡¯s report on ¡®town
centre definition¡¯ seems to suggest that an individual town centre can be
delimitated in such a way that it conforms to the allometric law at the
national level. In general, in morphology as well as in geography, I think
there is a strong tendency to utilise the idea of power-law in this way.
So, if applying this ¡®rule¡¯ and rephrasing my question, can we say that a
little droop in the lower-tail of an observed degree distribution may
reflect the incompleteness of the data in keeping a consistent mapping
resolution? Or consequently, if continuity aggregation, or any other,
makes the power-law hold better, can we say it is therefore a better way
of representing a street network? Or else, is a ¡®street¡¯ perhaps any
linear aggregation of spaces that entails a power-law degree distribution
at the higher-order level?
By the way, have you also read Lun Li et al.(2005)¡¯s paper ¡°Towards a
Theory of Scale-Free Graphs: Definition, Properties, and Implications¡±? If
not, I highly recommend it for further discussions.
Best wishes,
Hoon
On Fri, 1 Jun 2007 11:01:26 +0100, Lucas Figueiredo
<[log in to unmask]> wrote:
>Hi Hoon,
>
>That is good research question. Before, my point is: get a common
>ruler and put over most 'power laws' you see. It is not an easy
>exercise. Many cases you find a cut-off point and clear tail. Other
>cases you find 2 power-laws regimes and so on.
>
>See:
>Power laws, Pareto distributions and Zipf's law -
>MEJ Newman - Contemporary Physics, 2005 - Taylor & Francis
>
>However, many cases you do not find anything... there is no section in
>the curve that is clearly straight, and any chosen 'cut-off' point
>generates a different exponent, sometimes very different... are they
>power-laws?
>
>About your question, the power law behaviour is more clearly observed
>in the tail, 'top values'. Therefore, it is not the small chunks of
>space that will change it, but the generalisation process that creates
>the large objects (long lines in this case). If different
>generalisation processes generate different or closely related
>power-laws regimes it is a question to be addressed.
>
>In my opinion, the results can be a lot affected by the artificial
>boundary choosen in the map, the quality of the data and the
>generalisation process.
>
>Best Regards,
>Lucas Figueiredo
>
>On 01/06/07, Hoon Park <[log in to unmask]> wrote:
>> Lucas
>>
>> A question: If we include, following 'TeleAtlas' for instance, to map
such
>> countless bits of space as 'The ramp to the car park at 1-19 Torrington
>> Plc' or 'The entrance to the emergency services at UCH' into a street
>> network, could it change its degree distribution from a 'log-normal' to
>> a 'power-law'? If so, can we really claim that the degree distribution
of
>> street networks does not follow a power law but a log-normal?
>>
>> Regards,
>> Hoon
>>
>>
>> >On Thu, 31 May 2007 19:24:49 +0100, Lucas Figueiredo
>> ><[log in to unmask]> wrote:
>> >
>> >>On 31/05/07, Rui Carvalho <[log in to unmask]> wrote:
>> >>> Pitty no one knows what a street is...
>> >>
>> >>Certainly it is not a segment (or route) between two junctions.
>> >>Otherwise we would have things like "Oxford Street sector A, B, C" and
>> >>so on...
>> >
>> >The ramp to the car park at 1-19 Torrington Plc. This comes in GPS car
>> >navigation systems as a decision point? is it a street?
>> >
>> >The entrance to the emergency services at UCH (that's UCL Hospital for
>> non
>> >Londoners). Is it a street?
>> >
>> >This question appears when you process data from services like
TeleAtlas -
>> >the most accurate data available on street networks...
>> >
>> >Looks like Alan should organize that 'mass observation' on what a
street
>> >is after all...
>> >
>> >Comments welcome!
>> >
>> >Rui
>> >
>> >
>> >>
>> >>Regards,
>> >>
>> >>Lucas Figueiredo
>> >>http://www.flickr.com/photos/lucasfigueiredo/
>> >>
>> >>Mindwalk
>> >>http://www.mindwalk.com.br
>>
>========================================================================
>>
>
>
>--
>Lucas Figueiredo
>http://www.flickr.com/photos/lucasfigueiredo/
>
>Mindwalk
>http://www.mindwalk.com.br
>=========================================================================
|
