The Slope of the Rank Size Distribution Some insight into the problem of which index is more valid can be gained by examination of the factor of slope in the linear log-log graph. The simple rank size rule is usually regarded as a special case of the more general rank-size rule r_i((p_i)^q) = K (6) where q is an exponent with a value of 1 in the simple rank size rule. In rank size research the exponent of 1 generates a slope of -1 on a log log graph. Usually this slope of unity is seen as coincidental, an accompanying but not necessarily theoretically important accessory characteristic of the rank size relationship, despite the frequency with which an exponent near unity occurs when empirical data are examined. A slope near unity is commonly found in studies of the urban system of the United States. The research in this paper suggests that the slope of -1 (an exponent of 1.0 and an angle of 45 degrees in the slope of the function on log log graph paper) is not coincidental, but rather derives from the fact that Fibonacci sequences and the (phi) function when plotted on a log log graph, both approach a slope of -1 (Figure 8). In fact the difference in the higher values between the two indices may be responsible for the frequently noted "concavity" in the empirical data. This issue of slope is thus worth exploring at some length. Frequently the line which best approximates the slope of graphed rank-size data is close to unity and the slope is simply left out of the equation by omission of the exponent q. In fact the phrase "simple rank size rule" means the rank size formula is used but the exponent, q, is ignored. Inclusion of the exponent allows the formula to be adapted to rank size distributions which are log linear but which do not exhibit a slope of -1. Because the rank size rule can be adapted in this fashion, it is tempting, but incorrect, to assume that the simple rank size rule is a special or odd case of a more general, more useful formula. Beguin, for example, suggests that "The case where the general slope is 1 is not particularly significant. . ." and that it "appears as some empiric average observation" (Beguin 1984, 754). Rosen and Resnick (1980,166) state that the rank size rule is a special case of the Pareto distribution where the exponent equals 1. Instead, this research argues that the simple rank size rule is the general formula and other distributions should be explained as variations of the general principle. The situation is complicated by the fact that the ranked empirical data never exactly equal a slope of 1.000. It does not necessarily follow that this slope, thus, is coincidental; indeed it would be startling, given the myriad forces impacting upon urban systems and the inaccuracies inherent in the collection of data, if the slope of the empirical data of any rank size hierarchy exactly equaled unity, although it does come close for aggregate United States data (Rosen and Resnick 1980,171). Almost every rank size researcher has considered, in some fashion, the issue of the slope of unity. Z. K. Zipf, (1949, 131) in his classic work, Human Behavior and the Principle of Least Effort noted that "it is only natural to ask why the value, p = 1 (the slope) should be of critical significance..." He interpreted the slope as a result of the equation of the generalized harmonic series and tried to explain the slope in a partially developed argument concerning competing forces of unity (clustering and homogenizing forces) and disunity (scattering and differentiating forces) (Zipf 1949, 130 31). This is not unlike Coffey's definition (1981, 227) that in general, "... a hierarchical spatial system is a dynamic equilibrium state or, if you prefer, a compromise between the opposing tendencies of organization and disorganization, between complete agglomeration and complete disagglomeration" or Sahal's (1978, 1374 75) description of accumulative growth and differential growth. Although mainly dealing with business firms rather than urban areas, Steindl (1965) sees the slope of unity in a rank size distribution as analogous to a center of gravity about a mean. He argues that given an initial distribution of different sizes of firms, concentration takes place which results in the elimination of large numbers of smaller firms and the creation of a few larger ones. Similar principles obviously apply to urban areas. Some firms continue to grow while numerous smaller firms attempt to get started. In theory, a variety of mechanisms operate to maintain equilibrium. For example, continuous development of larger firms implies a successful industry and thus more small firms enter. If the growth rate of firms declines, mortality of firms is increased and the process is reversed. The significance of the slope of unity is that, in theory, at this value, growth of firms can continue indefinitely. Steindl states that the tendency to concentrate is to some extent endemic, which explains why the Pareto coefficient hovers round a level of 1.1 or so in most cases. The growth of industry, even while equilibrium-in terms of the model- lasts and the mean size of the firm is stable, leads to tension owing to the fact that some firms become very large in relation to the rest of the industry. This change is connected with a statistical measure of "concentration" in a different sense--the share of the largest one, two, or three, etc., firms; it is obvious that concentration in this sense of the term proceeds continuously in a growing industry, while the equilibrium of our model is undisturbed and the mean size of the firm remains stable. Woldenberg (1971) and Coffey (1981) also attach significance to the slope of -1. They see it as the differentiating factor between allometric and isometric growth. Both allometric and isometric growth can be expressed as a straight line when plotted on log log graph paper. When two factors reflecting allometric growth are plotted (body weight to surface area, to use an organic example), one grows faster than the other and this is reflected in an exponent and slope other than -1.0. Isometry is a special case of growth which, by definition, is different from allometric growth because both factors grow at the same rate. Visually, this can be seen on the log log graph by the maintenance of a 45 degree angle, or equidistance, between the line marking growth and the two axes reflecting the growth. Woldenberg (1971, 4-5) explains that strong allometric growth, a strong influence in one direction rather than another, cannot continue for long because of size limitations. This correlates with Steindl's observation that growth at a slope of -1, isometric growth, can theoretically continue indefinitely. Coffey (1981, 194) summarizes Bertalanffy's concern with the exponent (slope) as follows: Briefly, the allometric relationship may be viewed as an expression of competition within a given system, with each system component taking its share of the available resources of the total system according to its capacity, as expressed by the exponent. The exponent may thus be regarded as a "growth partition coefficient" that expresses the capacity of a component to seize its share of the resources. An exponent indicating positive allometry, having a value greater than one when the equation is dimensionally balanced, signifies that the component in question captures a proportionately larger share of the resources than either the total system or a second component. Conversely, an exponent less than one indicates that a component captures a share proportionately less than the system or a second component. It follows that an exponent of exactly one means the components are securing resources exactly in proportion to their respective sizes (Gilbrat's Law, or the law of proportionate effect, again). It should also be noted that the slope of unity has significance in physics. In the logarithmic plot of functions which illustrate exponential decay with time, Shire and Weber (1982,27) note that special significance is attached to the time at which the exponent of the function equals unity. (The exponent changes as the process goes through time.) At that instant, approximately 37% of the decay has occurred and this instant is called the one over e time (e for the natural logarithm), the relaxation time, or the time constant of the process. This measure of time, which is system dependent, may be related to Sahal's statement (1979,1375) related to the evolution of a system: "According to the framework provided by the Pareto distribution, the appropriate concept is then one of system specific time; that is, the passage of time measured by successive events which are specific to the system under consideration." Despite the general acknowledgment of some unexplained significance to the slope of unity, rank size researchers usually believe the slope of 1.0 derived from the simple rank size rule to be a special case of a more general rank size process. Carroll (1982, 1) calls the simple rank size rule the restrictive rank size rule and in a discussion of the "misleading rank size versus primate continuum", he states "In particular, the rank size rule with exponent of unity is too restrictive to be the limiting case against which primacy is contrasted" (1982, 3). Berry (1971, 148) is aware of the slope of unity but interprets it as a measure of balance between cities in the system and cities entering the system, that is, a reflection of the Yule distribution: . . . unity is therefore the value of the exponent to be expected if one is dealing with a fixed number of cities (lognormal case) as opposed to the case in which the number of cities is growing. If (the part of growth attributed to the entry of new cities reaching a threshold size) is substantial, as in the case of a system of cities rapidly expanding in numbers above threshold, q (the exponent) will exceed unity. If, on the other hand, the number of centers in the system is shrinking . . . q will be less than 1.0. In general, a value of q exceeding unity implies proportionately more small places, whereas q's of less than one arise with increasing concentration of the urban population in larger cities. In Berry's explanation, as in many of the others, a balance of competing forces is seen as the mechanism generating a slope of unity. His explanation also dovetails with that of Woldenberg, above, in that rapid growth of cities or very slow growth produces allometric rather than isometric growth because growth progresses faster along one axis rather than another. The slope of empirical data which are linear but which do not approximate a slope of -1, has not been a major concern of this research. As (phi) and the simple rank-size formula are essentially the same measure after rank seven, the (phi) formula can be adapted. Near the end of this paper, the application of a systematic bias to (phi) will show how this can be accomplished.
Next Section: The Spiral Constant and Concavity of Rank Size Distributions