Just for the sake of clarity...

Betweenness centrality doesn't 'assume' anything about densities of  
'sources and sinks'. It is just a measure of the relational structures  
within a graph. If you take the kinds of graph we use in this  
mailbase's domain then the densities vary radically from location to  
location. In urban areas there are more nodes, in rural areas there  
are fewer (per metric area). It doesn't matter whether we are talking  
about streets, axial lines, segments, continuity lines - whatever -  
densities vary radically from area to area.

In fact the notion of 'source and sink' is not particularly meaningful  
in this context either as every node in the graph is treated as both  
source and sink in calculating betweeness measures. It is confusing to  
try to equate 'sources' to houses and 'sinks' to offices, or vice  
versa. That kind of modelling is not generally what this field does.  
What it tends to do is take structural measures of morphology (such as  
betweeness) and use these to test what proportion of the variance in  
some other aspect of spatial behaviour can be  
'explained' (statistically) by that measure.

Now, on the issue of 2. What may or may not constitute a 'shortest  
path' depends on what metric you choose to use. Currently most people  
are using the segment map and are considering the metric to be angular  
deviation from segment to segment. When you do this the M25 comes out  
as high on betweenness, as do a series of long straight radials  
through the centre of London, but by no means all of central London,  
whether you mean the West End or the City.

On the issue of capacity and traffic flows, I suggest you try a  
thought experiment. Imagine building a new bridge across the Thames.  
Now this single additional segment in a large map will generally have  
quite a substantial effect on shortest paths in the system - the size  
of the effect will vary depending on exactly where it connects to at  
each end. This effect will be seen in measures of betweenness (amongst  
many other measures). Now the assertion that 'traffic does not flow  
along topological shortest paths -it flows along the routes with the  
highest capacity' suggests that what will matter for this bridge is  
just its width, not what it connects to.

Try another thought experiment. Lets take a segment of Tottenham Court  
Road and just for that segment double its capacity from three lanes to  
six. Will this result in substantially greater flows? I suspect not.

In fact there tends to be a correlation between capacity and being on  
topological shortest paths which I have argued comes from a historical  
process of allocation of road space to satisfy demand. The demand  
comes from the topology of the system, the supply is the capacity. It  
would therefore be a mistake to attribute causality to the correlation  
of traffic flows with capacity - both are a result of system  
configuration.

Alan


On 18 Mar 2009, at 11:27, Rui Carvalho wrote:

> 1. Betweenness centrality assumes an equal density of sources and  
> sinks,
> independently of spatial aggregation effects.
>
> In layman's words: there are as many different offices as there are  
> households.
>
> 2. The shortest path between north and south greater London would go  
> through
> the centre of London, not through the M25. But traffic does not flow  
> along
> topological shortest paths -it flows along the routes with the  
> highest capacity.
>
> Are these obvious points of any relevance to this community?
>
> That's not up to me to decide. But I'm obviously skeptical that this  
> line of
> research will have an impact beyond the space syntax symposium  
> series, which
> is why my research has shifted to other urban problems.
>
> All the best,
> Rui
>
> ______________________________________
> Recent work on spatial networks, GIS datasets and cities:
> 1. Robustness of Trans-European Gas Networks: The Hot Backbone
> Rui Carvalho, Lubos Buzna, Flavio Bono, Eugenio Gutierrez, Wolfram  
> Just,
> David Arrowsmith
> http://arxiv.org/abs/0903.0195
>
> 2. LivingScience -where Science is Happening:
> http://www.livingscience.ethz.ch/
>
>
> Dr. Rui Carvalho
> School of Mathematical Sciences
> Queen Mary, University of London
> Mile End Road, London E1 4NS, UK
> http://www.ruicarvalho.org/
>