Sheep,
OK - so now rephrase that in terms of axial lines not streets (which are
poorly defined entities as we know). To satisfy the small world requirement,
what proportion of the network needs to be involved in these triangle
relations? In organic deformed grid cities there are often a reasonably
large number of what we used to call trivial rings - that is exactly three
way triangles that don't go around building blocks but connect in open
space. These tend to lie along the major route system.
To be pedantic - what proportion of all possible graphs are small world
graphs by your definition? Don't answer that - its obviously a meaningless
question - but it illustrates that this line of argument: 'my graph is
'abnormal' because it comprises a smaller uncountable subset of an
uncountable superset' is pretty pointless... or have I got my logic wrong
here?
Alan
>
> > One point of clarification on your bailing Lucas out - can you
> > expand on : "
> > They are not small world graphs ( these require having a high
> > cluster factor
> > that axial/convex maps cannot)" - can you demonstrate easily that
> > axial and
> > convex maps CANNOT achieve high cluster factor? Or is this an
> > empirical
> > statement about real urban systems - that we tend not to find such
> > graphs in
> > reality?
>
> Watts amd Strpgatz used clustering coefficient of a graph to
> determine if a
> graph is small world or not.
>
> The only time you get a clustering coefficient bigger than zero is
> when 3 or more axial lines
> intersect at a junction. Even then your very dependant on the axial
> lines all being slightly long and precisely how they intersect to
> form lots of mini triangles.
>
> Believe me I programmed in clustering coefficient into webmap@home
> and didn't get anything exciting out of it. Basically clustering
> coefficient works on the basis that If A knows B and B knows C then
> it is likely that C knows A. This is how all the social networking
> stuff works.
>
> For an axial map If street A connects to Street B and Street B
> connects to Street C then it is highly unlikely that street C
> connects to Street A (in fact the reverse is more true).
>
> Same for convex spaces but not for isovist grids.
>
> If you could go out to radius 3 or 4 the case would be different but
> this is not how Watts and Storogatz defined it just degree/connectivity.
>
> Notice we are in an interesting twilight world where axial maps are
> largely 'scale free' (some highly connected hubs, most are low
> connections) but not small world. This is a shame as if they were we
> could use the Derek J. de Solla Price generative mechanism and be
> able to run the growth of cities into the future.
>
> so short of redefining what you mean by small world axial maps are
> not small world and so moderately unique and so abnormal.
>
> sheep
|