There have been requests for an update to my hubness posts.  At long
last, I have made enough progress to compose an initial post.  If
interest is generated in one or more particular area(s), I'll respond
in greater detail.

The top 48 hubs of the United States, from the many MSAs and cities
whose aerial hubnesses I have computed, are the following:

New York, NY
Los Angeles, CA
Dallas, TX
Chicago, IL
Seattle, WA
Miami, FL
Denver, CO
Atlanta, GA
Billings, MT
Phoenix, AZ

Salt Lake City, UT
Minneapolis, MN
San Francisco, CA
Albuquerque, NM
New Orleans, LA
Boise, ID
Lubbock, TX
Saint Louis, MO
Detroit, MI
Kansas City, MO

Las Vegas, NV
El Paso, TX
Spokane, WA
Houston, TX
Charlotte, NC
Rapid City, SD
McAllen, TX
Cincinnati, OH
Nashville, TN
Fargo, ND

Tampa, FL
Washington, DC
Memphis, TN
Boston, MA
Oklahoma City, OK
San Antonio, TX
Bismarck, ND
Grand Junction, CO
Shreveport, LA
Buffalo, NY

Missoula, MT
Omaha, NE
Sioux Falls, SD
Garden City, KS
Fresno, CA
Pocatello, ID
Springfield, MO
Norfolk, VA

(There might yet be a city missing from this list.  In particular,
such a missing city would have to have a hubness higher than 147.31,
the hubness of Norfolk, VA.  Let me know if you find a missing city,
and I'll update my data accordingly.)

Now, we can redraw state lines aroudn these 48 hubs, which I will call
_capitals_.  I will call these new states _estates_, to distinguish
them from current states.

How do we determine the boundaries of these estates, however?  I will
call the most obvious method METHOD 1:

* Given a point X, let H be the capital closest to X.  Then H is the
capital of the estate to which X belongs.

I have actually performed this construction, creating a GIF political
map of the United States with these new 48 estates.  Not having a
place to put it, I posted it to the Kansas City Highways Yahoo group
(hope you don't mind, TEXAS!), at the following URL:

http://groups.yahoo.com/group/kcroads/

You'll have to join the group; then, the map is filed under the Photos
section.  In the meantime, you'll see some other image postings of
mine, most notably some Signmaker fictional gantries and a color-coded
compilation of MoDOT's traffic count data.

If joining Kansas City Highways seems like to much work, e-mail me and
I'll send the GIF as an attachment.

On the map, each crosshair is a capital hub; the house icons are other
high-hubness cities.  Some especially notable features of the map:

* The Ozarks gets its own estate, with Springfield as its hub, and the
Joplin and Fayetteville areas as notable cities.  This is one of
America's fastest-growing regions, especially relative to surrounding
areas.

* Texas becomes what might be called the Texas Coalition, composed of
six estates:  Dallas, Houston, San Antonio, McAllen, Lubbock, and El
Paso.

* California becomes three estates:  San Francisco, Fresno, and Los
Angeles.

* Memphis and Cincinnati become, as they ought to be, the centers of
their respective estates.

* New England is defined by a single estate.

* Shreveport, LA gets its own estate.  For one who may not understand
why I-69 is being planned along its current alignment, the
hubness-based estates make the benefits clear:  McAllen, Houston,
Shreveport, and Memphis are all estate capitals.  If one avoids being
distracted by current state lines, I-69 makes more sense.

* The I-29 corridor benefits, with Fargo, Sioux Falls, Omaha, and
Kansas City becoming centers of estates.

* Garden City, KS becomes the center of an estate that, as far as I
can tell, has no MSAs.

One of the best advantages of using hubness to determine estates and
capitals is that the 48 capitals, many of them large metro areas, each
lie within a single estate.  This eliminates having to plan across
state lines, a problem in many of America's cities.

However, using METHOD 1 for determining estate boundaries is
problematic, since non-capital hubs can still be split across estate
lines (this occurs in Pittsbugh, PA and Columbia, MO).

Ideally, we would like the estate lines to trace through sparesly
populated areas.  I actually have an idea on how to accomplish this!
However, it is somewhat involved; I will denote it METHOD 2.

I'm making some of this up as I go along, so hopefully it will be
coherent/correct.  The algorithm is as follows:

* Let X be a point in the United States (latitude-longitude position).
* Define pop(H) to be the population of hub H.
* Define the _order 1 hub_ of X, H_1, to be the closest hub to X.
* Define the _order 2 hub_ of X, H_2, to be the closest hub to H_1
such that pop(H_2) > pop(H_1).
* Define the _order 3 hub_ of X, H_3, to be the closest hub to H_2
such that pop(H_3) > pop(H_2).

* Eventually, H_n, for some n, will be one of the 48 capitals.  Then X
is defined to belong to the estate with capital H_n.

(Yes, I'm a math/computer science student!)  This algorithm, with a
recursive streak to it, successfully adjusts the boundaries so that
the hinterlands of smaller hubs stay with these hubs, even as parts of
a larger estate.  Thus, estate boundaries under this algorithm follow
sparsely populated regions, a very desirable quality.

The above algorithm may seem very difficult to visualize; however, one
can actually perform the algorithm by hand to get a better idea of its
operation.  This works as follows:  for every non-capital town X,
determine the closest town Y to X such that pop(Y) > pop(X), and draw
a line from X to Y.

If you draw enough lines (it can take quite a few), you'll get an idea
of how my METHOD 2 HUBNESS algorithm above works.  Your map will have
a starburst of lines enamating from each capital to some sub-capitals,
with smaller lines enamating from these sub-capitals to
sub-sub-capitals, and so on.

Around each capital, then, a shrublike tree structure forms, with
leaves branching out to the outermost regions of the estate.  The
estate boundaries will then be clear as the only remaining whitespace
between estates.

If one starts with the hubness method, then, after determining the
major hubs, constructs this branch-out pattern for each hub, one
immediately sees possible applications of this method, whether for
determining phone/postal districts, or for positioning product
distribution centers.

How does METHOD 2 change the boundaries of the US map I posted to
Kansas City Highways (which was based on METHOD 1)?  It's difficult to
say, exactly; it would be a lot of work to get accurate METHOD 2
boundaries, since the boundaries depend on many smaller towns, not
just the capitals themsevles.

However, one can get a general impression of how METHOD 2 boundaries
would be different.  The entire Columbia, MO area would become part of
the Saint Louis estate, at the expense of the Kansas City and
Springfield estates.  The Omaha estate would probably also grow,
fanning out to the Des Moines hinterlands at the expense of other Des
Moines area estates.  This would ultimately be beneficial, uniting
Omaha and Des Moines into a single estate that improves the situation
of Des Moines will still retaining Omaha as the estate capital.

You've finally reached the end of another long-winded hubness post.
This is an amateur analysis; however, the ideas are quite interesting.
 It can difficult to think outside the usual perspective of state
lines, and this helps to do just that.  I certainly welcome comments,
questions, contentions, and suggestions.

--
John Fischer
math major at Mizzou
Columbia, MO