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 demographer's "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 1 2. Lionel, Duke of Clarence 1 3. Philippa 1 4. Roger Mortimer 1 5. Anne 1 6. Richard, Duke of York 1 7. Edward IV 1 8. Elizabeth 9. John of Gaunt, Duke of Lancaster 1 1 10. Henry IV 11.John Beaufort, Earl of Somerset 1 12. John, Duke of Somerset 1 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 III. 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. Reference: R. Duncan Luce and Albert D. Perry, "A Method of Matrix Analysis of Group Structure." 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; however, 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 generator. 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 Example: 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.