Wednesday, August 22, 2012

Dendritic vs. Reticulated: A Study in Market Forces of Transportation Networks

In a recent post regarding modern transportation planning versus, say, natural, organic city building, I wrote that the dendritic networks planned and built exert a diametrically opposite force on the real estate market than does a more interconnected grid.  The first question is, what is dendritic?

Below is a diagram of a dendritic system, often used to describe watersheds.  With regards to street networks it is a convenient way to categorize a hierarchy of streets in a branching out pattern.  Local roads feed Connectors which feed Arterials which then in turn feed highways, the central "trunks" of the system, so to speak.

As you can notice by the diagram above pick any two various points and they won't be as well interconnected as say, a true grid:

But the fascinating thing about these two separate ways of building the "bones" of a city for which land use provides the flesh, is the diametrically opposite force exerted upon the real estate market.  The transpo network, the connectivity, is the first order system applied to a place.  The "body" or the physical city of uses and buildings is the emergent second order network that is entirely dependent upon the degree of connectivity at the order below.

The rigidly hierarchical dendritic pattern is indicative of a flat or spread out "market."  No place is really any more or less interconnected than other places (thus why studies like intersection density and space syntax are valuable -- they measure interconnectedness).  In cities, the degree to which you are connected to persons, places, and things determines the value of that particular site.  The higher the value, the greater the demand, opportunity, and therefore density.  Because the hierarchical 1st order system produces little hierarchy in the 2nd order system, it suggests an outward or centrifugal force, or outward pressure, on the real estate market.  All places are equitable.  Evenly and poorly interconnected.  And the result is sprawl.  Demand smeared evenly across the entire landscape.

The grid, though it seems "democratic" in that all blocks are treated differently has its own built in hierarchy, which is functionally "nested" within the grid.  Think of the black lines above being streets and blocks, with buildings within.  The usable spaces become increasingly private as you move further inward within those blocks, from street to building entry to corridor to room, etc., each nested further within it.  These is a common characteristic within all complex systems.

Side note: the grid can be more complex or radial, as long as it is highly interconnected, ie "reticulated," in that there are multiple routes between various points.

Because the hierarchy inherent in reticulated systems only appears at the next level of complexity and order, it can be said that reticulated grids exert centripetal force on the real estate market or convergent.  There is greater value at intersections and hubs of networks because they are more interconnected.  How we value the availability and amount of the possible interconnections between people, places, and things drives city form by driving demand, and in turn density.  Inward.

Let's examine it mathematically...

The reason is because (and this is before we get into the local/global hierarchy, which amplifies certain locations, ie hubs within hubs into infinitely complex networks upon networks) different parts of the grid are more interconnected than other parts.

If we take the basic grid for time/simplicity's sake, and put a semi-transparent (10%) circle upon every intersection (call it a neighborhood), we get the following diagram.  Each neighborhood would consist of the same amount of people, places, and things.  Each is equal.  Yet as you can see when overlaid to the entire system, another level of order appears, a gradient with the greatest "density" at the center.

But that assumes all neighborhoods are equal.  Are they?  

Below you can see the individual circles overlaid at different points on the grid.  The number within is a measure of the intersections within, ie how interconnected is each "neighborhood" in relation to its place within the grid:

Below are the numbers (degree of interconnection) factored across the entire grid:

Which then suggests the neighborhoods aren't created equal, but that location does matter.  So I tweaked the transparency levels (density) from a flat 10% to integer relevant to degree of integration, ranging from 8% to 25%.  As you can see, that further distorts the gradient emergent centripetal force.  The greatest value is at the most interconnected places.  On the other hand, disconnect (with hierarchical dendritic networks) begets disintegration begets disinvestment and finally decay.

As you can see, it is both darker at the center and lighter at the edges than the previous every-neighborhood-is-considered-equal model.  On the other hand, where dendritic systems have been overlaid upon cities creates a higher order system where all places have the same degree of interconnection, let's call it '1'.  However, since land is more expensive (say 1.5) toward the center (see above) and cheaper on the edge (.5), it is not profitable to build inward. The pressure is therefore continually outward if all places have the same interconnected value within a gradient of land prices.

This entire course of study emerged from discussions regarding the flaws within transportation modelling, how big should roads be based on proposed development.  The problem is this runs counter to actual city building and what we call organic growth (edited for blog consumption):

With that said, [traffic modelling) is...needless... if not outright pernicious.  Roads rarely need to be widened.  Picture a historic grid.  When a street gets overwhelmed, the demand load spills outward.  Rather than widening roads (b/c only popes and kings and Robert Moses could take private land and knock down buildings for road widening/straightening), the grid expanded and because the historic streets were too crowded, it was evident to investors and developers that demand was significant enough to add floor space and expand the city.  The roads followed.  It was organic.  It was rational.  The grid may have grown outward, but there was still inward, dare I say attractive, pressure on the market.  

Today it is wholly irrational, unnatural, and illogical.  Broken.  Anti-city.