Curve fitting and analytical tools--number 7

S. Arlinghaus, to appear, Structural Models in Geography.

An application of graph theory to group relationships in history
(source:  Williams:  Finite Mathematics)

The term transition is often used to describe the return of a system to a
balanced situation following a period in
which it has been out of balance.  One classical transition is the
 "demographic transition (Notestein, 1945; Thompson, 1929, 1944).
Typically this transition
describes a condition of high vital (birth and death) rates, followed by a
drop in the death rate during
a period in which the corresponding drop in the birth rate lags behind,
followed by a drop in the
birth rate and the realignment of a low birth rate with a low death rate
(Bogue, 1969).  The
transition is from high vital rates to low vital rates; if the
intermediate stage does not lead to
eventual low vital rates, then there is no transition.

As we have seen, one need not confine the idea of transition to
demography; it extends naturally
to a variety of real-world realms (Drake, 1992).  For any system to be in
some sort of functional
balance, the inputs and outputs must be fairly close to each other in
number:  if the inputs dominate,
the system explodes.  If the outputs dominate, the system withers.
Abstractly, a transition within
a system occurs when the input/output level starts in balance, at a high
level, experiences a drop
in outputs so that the inputs dominate for a period of time, and then
returns to a balanced state
by a corresponding drop in the input level so that once again the
input/output level is in balance.
Symmetry promotes systemic stability.

The transition is from the high level of input/output to a low level of
input/output.  Because the rates
generally do not drop evenly, one curve has more area under it than does
the other, signifying a
period of "boom."  What happens during the transition, in the boom time,
is critical, as Drake notes (1992),
in determining whether or not the transition is completed and it is in
this intermediate stage that so many
complexities often arise.  Drake (1992) notes this situation in a variety
of contexts:  from forestry, to
 education, to environmental toxicity, to a host of others.  We consider
it here in an historical
context:  in succession to the British throne.

In the peiord of British history from William the Conqueror (about 1066)
to Richard the II (about 1399),
the pattern of hereditary succession to the British throne was clear.
When Henry IV overthrew Richard II
in 1399, the Wars of the Roses, involving issues between the House of York
and the House of Lancaster,
concerning succession to the British throone, were the result.  In 1485,
when Henry VII of the House of Lancaster
married Elizabeth of the Hourse of York, the pattern of succession once
again became clear.  An historical
transition was achieved, but, in the "boom" time from 1399 to 1485, what
was the pattern of the dispute?
Structural models, or graphs, offer a way to resolve historical complexity
(Luce and Perry, 1968; Williams, 1979).

Generally speaking, a graph is a collection of nodes together with edges
linking pairs of nodes.
There may be more than one component to a graph.  Some graphs are trees.
Some graphs have
directed edges indicating the direction of flow (digraphs).  The subject
of graph theory is a very
broad one with numerous applications.  The classical text in graph theory
is by Frank Harary, entitled
Graph Theory, published in 1969.

A partial genealogical table of the famiily relationships, for both the
Houses of Lancaster an York, 
is shown below.  This genealogical table can be made into a digraph.  Let
the relationship                                                                                                                        
linking people be "is the father/mother of."  Let each person represent a
node.  Thus, if person P                                                                                                                        
is related to person Q, draw an edge from P to Q, with the direction
pointing from P to Q.  Thus, Q is                                                                                                                       
adjacen within the structural model from P (Harary, Norman, and
Cartwright, 1965).                                                                                                                      

We can code this sort of adjacency in a binary matrix:  the entry from P
to Q described above would                                                                                                                      
be a 1; the entry from Q to P would be a 0.  Using this idea of adjacency,
based on "is the father/mother of,"                                                                                                                     
the structural model of the genealogical table can be expressed as an
adjacency matrix focusing only on                                                                                                                       
the set of 17 people noted in the family tree.  In this case, adjacency is
based on certain key relationships                                                                                                                      
gleaned from historical evidence.                                                                                                                       
                        1       2       3       4       5       6       7
8       9       10      11      12      13
1.  Edward III                          1
2.  Lionel, Duke of Clarence                                    1                                                                               
3.  Philippa                                            1                                                                       
4.  Roger Mortimer                                                      1                                                               
5.  Anne                                                                1                                                       
6.  Richard, Duke of York
7.  Edward IV
8.  Elizabeth                                                                                                                   
9.  John of Gaunt, Duke of Lancaster
1       1               
10. Henry IV                                                                                                                    
11.John Beaufort, Earl of Somerset
12. John, Duke of Somerset
13. Margaret Beaufort                                                                                                                   
14. Henry VII                                                                                                                   
15. Henry VIII                                                                                                                  
16. Edmund, Duke of York                                                                                                                        
17. Richard, Earl of Cambridge                                                                                                                  

Note that there are three 1s in the first row of the matrix A since Edward
III was the father of Lionel, John of Gaunt, and Edmund.                                                                                                                        
Taking powers of the adjacency matrix, A, counts the number of paths
through the genealogical hierarchy.  The power
A^2 counts the number of paths of length two--grandparent/grandchildren
relationships.  In A^2 there are 1s in the
third, thenth, eleventh, and seventeenth columns, indicating that Edward
III was the grandfather of Philippa, 
Henry IV, John Beaufort, and Richard.  These observations tally with the
family tree.

The fifth power of the adjacency matrix, A^5, has a value of 1 in the
(1,14) position of the matrix (first row,
fourteenth column), showing that there is one path of length five between
Edward III and Henry VIII; that
Henry VIII was descended directly from Edward III through five Lancaster
generations.  Because a value
of 1 is recorded in this position for the first time in the fifth power
matrix, we know that it is exactly five
(and not fewer) generations for the descent.

The seventh power of the adjacency matrix shows an entry of 1 in the (1,8)
position reflecting the
fact that Elizabeth is desceded over eight York generations from Edward
The sixth power matrix and the eighth power matrix both have entries of 1
in the (1,15) entry--for
Henry VIII, the son of Hnery VII or Lancaster and Elizabeth of York.  Had
both Elizabeth and Henry
been descended from Richard III over the same number of generations, there
would have been
an entry of 2 in the (1,15) position at its first appearance in the matrix
sequence--a 1 from each line
a descent.  Thus, Henry VIII, the son of Henry VII and Elizabeth combined
the claim of both the Houses of
Lancaster and York to the British throne.  Historical complexity, that can
occur in the "boom" time
within a transition--between symmetric periods of stability, is resolved
easily using adjacency matrices
of structural models.

R. Duncan Luce and Albert D. Perry, "A Method of Matrix Analysis of Group
Readings in Mathematical Social Science, ed. Paul F. Lazarsfeld and Neil
W. Henry
Cambridge, MIT Press, 1968.

Fractal Geometry

Consider the two lines below--both have Euclidean dimension of one;
intutively, one "fills" more space than does the other although of course
it does not fill
a two-dimensional piece of space.  Hence the idea of a "fractional
dimension" or "fractal."

Benoit Mandelbrot, a computer scientist/mathematician captured this notion
(which is prevalent
much earlier in the history of mathematics--in finding a curve that is
continuous but nowhere
differentiable--kind of an infinite number of absolute value curves) in
his work in the 20th century.
(Mandelbrot, The Fractal Geometry of Nature--is one standard reference.)

Fractional dimension can be calculated as suggested in the following
visual examples:

Example 1:

One large hexagon                                               Four
smaller hexagons
                                                generated from the larger
                                                one using a three sided

The shape of "hexagon" remains constant--its scale changes.  The figures
are said to be

self-similar--the choice of generator was critical in bringing about this
scale transformation.
When the generator is scaled-down again, and applied inside/outside to
each of the four smaller
hexagons, and even more complicted figure of sixteen smaller hexagons
appears.  Carry out
this process infinitely--what is constant is the number four, as a factor
of the increase in
complexity, and the number three and the number of generator sides causing
this increase in
shape complexity.

Let N represent the number of sides in the generator
Let K represent the number of self-similar regions.

In this example, N=3, and K=4.

The fractional dimension D is calculated as:

D = ln N / (ln K^0.5)           

In this case, D =               1.58496


The previous example dealt with a bounded, closed figure.  One can deal
with other shapes, too          
Consider a straight line--a coastline viewed from high above--as one zooms
in, one sees more bays.         

In this case, the straight line is the constant shape and in moving from
one scale to another,   
the generator used to increase coastal complexity has four sides and the
number of self  
similar regions produced is also four.  

When this sequence is carried out infinitely, K=4, N=4, 

D=      2

so that this procedure will cause the lines to bend back and forth
sufficiently to fill a piece of the     
plane.  If cutting bays in a lakeshore were to follow this process, in
order to maximize lakefront views       
and minimize length of linear coastal damage, it might therefore be
prudent to stop the sequence
after some fairly small number of stages.  

References to particular studies on request.