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.

Firstly, in graph theory there is the notion of planar graphs, that is graphs that can be embeddd in the plane without any crossings. If a graph is not planar, it has a genus, that describes its basic topological properties. It is known that such graphs of higher genus can be embedded in compact closed surfaces in 3 dimensions. That is dependign on their genus number. For example torus, brezel, triple torus, .... -> embedding graphs in surfaces. This is one topological feat of graphs at the level of abstraction comparable to connectivity.

Furthermore, in the 80s there was one famous question called the "Four color theorem" ->Wikipedia that asked about how to color countries in maps, such that non-neighbouring countries don't have the same color.

Structural questions on graphs also involve "Clique Partition", which is a computationally hard problem, but maybe can be adapted to work locally.

There are many more questions you can ask about a graph. Most of them are computationally hard, but for small instances or locally they can maybe provide nice insight. Maybe it would be possible to consider such questions locally and then extending topologically by using sheaf theory.

http://en.wikipedia.org/wiki/Four_color_theorem

http://en.wikipedia.org/wiki/Clique_cover_problem

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

Bernard!

As you described your question, maybe you talk about making the graph structure coarse until it reduces to "the core" that is of interest to you. Mathematically, the tool you are searching for is computing a graph factorization by an equivalence relation. In the case of some proofs of planarity of graphs one argument is used to reduce the original graph by successively collapsing its nodes together, i.e. joining two nodes and treating them as one. This reduction is applied repeatedly until one arrives in a simple case that tells whether the graph is planar or not. Those two famous graphs are: K_5 (the complete graph of 5 nodes) and K_3,3 (the bipartite graph of 3 nodes on each side). One could use similar reduction procedure to extract structural feats from graphs.

Another approach would be to extend your mathematical model to graphs that are embedded in the plane or in some 3-dimensional landscape, that have distances between the nodes and so on. Maybe such kind of model would be better suited to develop further ideas.

Greetings, Thomas.

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).

Hi Yan,

That is indeed another interesting approach to the problem. Thanks for the link as well.

Best Regards