Hoon,

You are quite right - when I said I didn't know why people 'had trouble with
this', I meant with the process of the RA and RRA equations, and the
question of why size of system matters in comparing graph measures across
systems. 

As you quite correctly point out a lot of trouble needs to be taken in
understanding exactly what is going on, and this is a very live area of
investigation at the moment, particularly in terms of how to handle
different axial, segmental, node and VGA representations along with
topological, angular and metric measures of both depth and radius, and
different types of global measure - eg. depth distributions and choice.
Taken together a lot of 'trouble' needs to be taken to understand what is
going on, and a lot of people including you are taking that trouble.
Apologies if I implied anything else - it was not intended.

My answer to your second question: 
> Is it necessary in the first place to relativise mean depth when it is
> already free from size effects?

I think we have to take a step back and ask just what is it we are doing
when we define measures of configurations? In my view everything one does in
measure construction is aimed at trying to isolate and define metrics for
different structural factors in the way that configurations can be
constructed so that these can be used separately in statistical analysis.

In the case of radius measures of depth two things can vary at once - first,
the configuration dictates that within a fixed radius there may be different
numbers of nodes; second, that those nodes may be distributed differently in
terms of depth from the root (the node under consideration). There are of
course several measures one can consider - the number of nodes within that
radius of each root in turn is a perfectly valid measure in its own right.
The total of the depths of those nodes is also a valid measure, but it will
include a component based on the varying number of nodes within the radius.
The mean depth of those nodes 'takes out' the effect of number of nodes
within the radius, but does so in such a way that the same mean depth value
can arise in many ways (eg for a small total depth with small number of
nodes one can get a high mean depth, and vice versa). The RA equation (and
variants of it for radius measures) sets a given mean depth value onto the
scale between the deepest and shallowest a valid configuration (no sub
graphs) can be with that given number of nodes. This is aimed at isolating
the structural factor of 'depth distribution' from the structural factor of
'number of nodes within radius' in such a way that one can distinguish
between two nodes of the same mean depth but different numbers of nodes
within radius and different distributions of depths.

Now my question about all this is as follows. In a network everything
relates to everything else. Any measure of a node that involves anything
more extensive than its immediate neighbours seems to me to necessarily
involve intimate relationships between the different 'structural factors' I
have talked about above - thus number of nodes within a radius and the depth
distribution of those nodes seem to be intimately related. Is it therefore
possible to isolate and define metrics that are in any sense independent, or
is this actually an impossible aim?


Alan


> 
> Dear Bin
> 
> Truly, as Alan pointed out, the objective of using RRA is to "remove the
> effects of number of nodes in urban axial graphs from the average mean
> depth in a system."
> 
> This is achieved by taking into account that, as a system grows, the
> increase of "average mean depth" or "characteristic path length" will be
> very rapidly suppressed, a phenomenon popularly known as "small-world."
> 
> Specifically, the mean depth of a diamond shaped structure we assume
> scales as log N, where N is the number of nodes in a system. So that RRA
> in its simplest form can be expressed as 'Mean Depth / log N.' Compare
> this form with that of RA, i.e. 'Mean Depth / N', which assumes a "large
> world" where mean depth scales linearly with N.
> 
> In Space is the Machine (p105), Bill suggests that RRA defined as such is
> not just one of countless ways of relativising measures but reflects a
> fundamental theory on urban spatial forms: How urban forms achieve
> integration in their evolutionary processes.
> 
> Now when applying RRA to local analysis, two questions arise to me:
> 
> 1) Is the assumption of log N empirically verified? What I find from my
> empirical study seems to suggest that, as radius varies diachronically,
> the mean depth of *a node* does not scale as log N but as some fractional
> power of N. If mean depth is thus relativised by an observed power of N, I
> can simply obtain RRA that is independent of size effects. But with log N
> in its current form, RRA does NOT remove size effects - a misspecified
> model it seems to be.
> 
> 2) When the same radius, say, radius 3, is applied synchronically, mean
> depth becomes homogeneous across *all nodes* in a system, independently of
> N counted differently within the radius.  Sheep finds this as a universal
> phenomenon. If this is the case, it is clear that RRA values of nodes are
> differentiated not by their mean depth but by their neighbourhood sizes,
> in such a way that RRA ~ 1/log N. So we are in fact saying that a node
> is "more asymmetric" or "segregated" BECAUSE N is small. My question is:
> Is it necessary in the first place to relativise mean depth when it is
> already free from size effects?
> 
> In my opinion, there seems to be a good reason to have trouble with all
> these.
> 
> Regards
> Hoon
> 
> On Fri, 3 Mar 2006 14:53:26 -0000, Alan Penn <[log in to unmask]> wrote:
> 
> >Its funny isn't it? I wonder why people have trouble with this one.
> >
> >The RA equation puts mean depth onto a 1-0 scale between the deepest and
> the
> >shallowest you could possibly have given that number of nodes in the
> graph.
> >This is a normalisation.
> >
> >The RRA equation then relativisies this as compared to the mean depth of
> a
> >diamond shaped structure with the given number of nodes. This is an
> >empirical relativisation ie. not particularly 'theory driven' in that it
> >works statistically in removing the effects of number of nodes in urban
> >axial graphs from the average mean depth in a system. There are many
> other
> >ways that this could be done, and since there is no pre-existing
> theoretical
> >assumption built into this process nothing is lost by ding it a different
> >way. However, it does allow you to compare some properties of graphs
> between
> >maps of different sizes on a more or less comparable basis. Something you
> >certainly cannot do for the unrelativised RA values.
> >
> >Alan
> >
> >
> >>
> >> On 03/03/06, Bin Jiang <[log in to unmask]> wrote:
> >> > Usually we compare space syntax measures within a same system, not
> >> > across different systems. This is my perception. Am I wrong?
> >>
> >> Well, if you use local integration for a single axial map you *are*
> >> comparing different systems because the number of nodes involved to
> >> calculate this measure vary for each node. It is the same if you
> >> compare global integration between different maps.
> >>
> >> Therefore, if you do not agree that RRA / Diamond Shapes provides some
> >> help... forget the whole thing about local integration... just does
> >> not work.
> >>
> >> For me it is quite OK.
> >>
> >> > I am not convinced by the popular saying that the local integration
> is a
> >> > good indicator of pedestrian or vehicle flows. Recently I happened to
> >> > get some vehicle observation datasets with pressure-sensed techniques
> >> > (so must be very precise observation). I compared the datasets with
> >> > local integration, and did not end up with a good correlation (R
> square
> >> > value about 0.5).
> >>
> >> I have got the same in this paper:
> >> "Continuity lines: aggregating axial lines to predict vehicular
> >> movement patterns"
> >> http://www.mindwalk.com.br/papers/
> >>
> >> That is a problem. I produced a continuity map that reveals clearly
> >> the main street system of my city (Recife, Brazil). But ... correlate
> >> abstract graph properties with real movement is another issue. There
> >> are many other factors, such as attractors, street width, etc.
> >>
> >> Therefore, this is just a matter what is the number you accept as a
> >> good proof that the urban grid itself (ignoring the other factors) can
> >> organise movement patterns.
> >>
> >> Regards!
> >> Lucas Figueiredo
> >>
> >> CASA - Centre for Advanced Spatial Analysis
> >> University College London
> >> 1-19 Torrington Place
> >> London WC1E 7HB England
> >> E-mail: [log in to unmask]
> >========================================================================