If a graph depicts connectivity, what does it tell us about the space it segments?
Theory: Land is segmented into "uses" by the marketplace. In equilibrium, the price of land will be equal on the boundary between land uses. There are boundary effects in the sense of prices that reflect preferences of the uses for proximity to the use across the boundary. For example, if a use is injured by proximity, then price falls near the boundary. Alternatively, if a use is helped by proximity, then price rises near the boundary.
Empiricism: A way to estimate these sorts of connections is to observe land price topography via baricentric coordinates and interpolation:
“A Primer on Piecewise Parabolic Multiple Regression Analysis via Estimations of Chicago CBD Land Prices,” The Journal of Real Estate Finance and Economics , Special Issue on Spatial Econometrics, 1998 Vol. 17:1, 87-97.
Dear Peter,
Thanks for the input, pointers and leads. As always your answers are packed with insights!
Best Regards,
Bernard
I should add that zoning of some kinds can constrain these uses so that market equilibrium cannot be achieved. That is prices on the two sides of the boundary will not be equal.
Dear Thomas,
Though I'm aware of the four color theorem, the genus aspects that you brought up really helped in my thinking of what kinds of graphs may be suitable for the kind of problem I am facing.
I am trying to get to the inverse of graph-based solutions. What I mean by "inverse" is akin to veins in leaves (I take this to be graphs in this analogy but perhaps they are more "tree-structures"). The question is how to investigate what is left if I were to take out all the veins in a tree leaf? The remaining spaces of chlorophyll is what I am interested in. Which I think is a kind of segmentation- therefore am not sure whether this is a graph-related problem or should be another mathematical domain altogether.
Thanks,
Bernard
Hi Thomas,
Thanks for the leads and pointers, will look into it.
Cheers,
Bernard
Hi Horst,
Thanks for the care and detailed treatment you gave to the question, I think I am beginning to see the light!
Cheers,
Bernard
Your question makes me think about mathematical morphology (the mathematical framework is based on lattices, but it is applied on grids, graphs). Connectivity is important, there are some notions as watersheds, it ls related with segmentation (more preciselly it is a tool for some segmentation technics) and so on... I give you the wikipedia link but there exists of course many academic sources (http://en.wikipedia.org/wiki/Mathematical_morphology).
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