PROPORTIONAL ENTROPIES :

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Abstract
            The primary factors in the stasis, sustenance and growth of organizations-which-process {energy, information, skills, commodities, etc} are 'input availability' and 'load capacity'. Back pressure and competition mechanisms play roles but only when resources and processing abilities are in place first, which enable and drive the dynamics. It is important to distinguish between generating-rules and result-patterns - which can be mimicked by simple mathematical functions and thus 'seem' like generating rules. Power Laws analog some patterns and seem like predictive rules, but should be utilized with caution   (e.g., phototropism, which in the surface looks like a direct correlation between light and subsequent plant growth, is in fact a result of metabolic actions reliant on durations of darkness and shade.)
           Complex systems are essentially "nested" organizations and assemblies-of- assemblies (Hebb 1949; Scott 1995). Certain rules pertain to energy/information distributions within assembly layers , while other rules govern production of next-level assemblies. These diverse applications indicate the probability of distinct classes of Power Laws.

Keywords : assemblies, proportional entropies, Integrity, distribution relations, Power Law classes.

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1. Power Law Classes - Defining the Venues

Power Laws are relatives of Gaussian curve functions and statistical distributions. The success of Fractal mathematics and Complexity Theory has elevated the notion that simple recursive equations seem to generate geometric - and even behavioral - patterns which are not limited by focus, resolution or scale. That is, patterns - interpreted as "relationships" - remain constant no matter where they are applied. These patterns seem to correspond with generative rules that produce the world we see and experience. This notion is generally valid, so I won't pursue arguments debating that point.

There are additional interpretations however, that are more pertinent to Systems processes. For all practical purposes current power law applications are confined to specific treatments or system levels. A power law can be applied to behavior of people in groups, or the behavior of stars in a galaxy, or the molecules in a container, but there is no current formulation which gets you from one domain to any of the others. In that sense a single power law may be isolatedly applied, but is partitioned, and is in a sense, quantized. Power Laws (such as 1/f ... inverse frequency) behaviors can be lifted whole and applied in any of several different subject areas ... educational test score performances, behaviors of gasses and fluids, sigmoidal growth and maturation curves, economic interactions, etc. as long as the equations are applied within a topic. Current laws aren't applicable say, if trying to establish a correlation between one person's health and how fast a conference room fills up, even though you can find activity analogs of hormone distribution in a person’s body versus the mixing of bodies filling an auditorium. This compartmenting is tantamount to the ages old concern about quantum vs continuum, merely dressed in 20th mathematics and perceptions. The closest that modern computational techniques can come in bridging separate but somehow linked systems is using what are popularly known in the investment markets as "derivatives". These are mathematical models stemming from physics and fractal theories which indicate a correlation presence and hedge against their joint behaviors. For example, there are purchasable contracts which can depend on specific weather occurrances ("A Calculus of Risk" Sci. Am., May 1998 pp92-97), which investment may mitigate an investment in orange or coffee or sugar futures. Such techniques don’t have to specific any exact cause and effect mechanisms involved, they merely have to recognize that some connection exists and thus create investment opportunities that bear on both event streams.

A second factor of weakness in current models is that fractal/complexity equations produce repetitive patternings no matter how fine or coarse grained the field. That is, if you examine a large cross section, or microscope in as small as possible, you find the exact same imagery of relationships. Now, the physical world operates slightly differently. Atoms don't spawn super-atoms, they generate molecules, which generate metabolisms, which aggregate into organisms, and then communities, and so on. Power Laws are useful in monitoring and even matching dynamics within any separate domains, but integrated systems are "assemblies of assemblies" Hebb(1949), Scott (1995), and current Power Laws miss relational considerations that exist in the up and down links, between those assemblies and layers.

It is important to remember that Systems are continuum streams, even if they can be analyzed and mapped using piecemeal quantum or statistical computations. Pixelled and digitized information still go through final smoothing processes, in order to become visually, audibly or conceptually meaningful and useful. Patterns are translated or transduced - to use a term more in line with behavioral or systems vocabularies - between nested assemblies. This transduction establishes communication among assemblies, where each assembly utilizes energy and information in locally appropriate ways. One person's health may indeed affect the activity of companions - contributing to momentary variations in a particular regime - but also having the functional option-ability of possibly-or-not affecting nested regimes and systemic processes.

The renewed interest in Power Laws harken back to the earlier (pre-quantum dynamic, pre-fractal chaotic) understandings of systems ... harmonic thermodynamics. In all of these modeling methods, concerns focus primarily on predictive ability, using some mathematical schemata or another which can literally "make decisions" for us that will conform with arbitrary goals ... especially in the race to secure financial gains and acquisition superiority. Models are pattern mappers. Fractal and similar algorithms have shown merit re pattern application in many subjects, but their subtlety and behaviors don’t relate to everyday human experience. Wave functions, inverse frequencies, do. And these are finding prominence again in such diverse areas as economics and the neuro-physiology of consciousness studies.

Again, the two areas which systems perceptions are most concerned with and which conventional mathematics has yet to deal with are: (1) discovering the systemic mechanisms which create and link diverse assembly levels, and so shed light on how power laws can even recur meaningfully in different functional areas, and (2) illuminating what the mechanisms may be as the mathematical "rules" integrate and function in concert with others. I won’t dwell on the intricacies or applications of conventional Power Laws per se, because the literature is fairly comlpete there. What we’ll examine instead is a Power Law variation that binds behavioral levels.

2. A Power Law that defines Complexity instead of analoging it

In a primal model, we can define the distribution of people, physically - (where we live, reside, work), and also in regard to their activity space (the number of hours of time spent eating, conversing, teaching). These distinct distributions are identifiable but definitely not static or fixed. They quantifiably vary all the time, and so enact distribution changes, which we can label as entropy values, with concurrent rates of moving toward or away from other benchmark configurations. These behaviors can in turn be tracked by noting changes in other factors, such as mutually shared commodities, services and wealth. For example, think of two ancient nations separated by a great ocean. In the past, any commodities, services or wealth circulated strictly within each homeland with local focus. In the current world, however, commodities, services and wealth are broadly distributed in coordination. If an "entropy" is defined as the spatial region where you can typically expect to find a thing, then shipping the thing out beyond regional borders, to another country, effectively increases the "entropy" of the thing, because now it’s locatable in a much larger area. If goods or currency comes from the other country in exchange, that item too has an "increased relative entropy". In behavior-space, each country has devoted x-percent of its energy to accomplish that exchange, in preclusion of anything else that could have been done with that energy. This activity physically and behaviorally binds distanced nations into a next level next assembly system-entity called global commerce and international trade.

You might find a recyclable bottle of Coca-Cola somewhere in the factory being filled, or, as I did one day, in a Chechwa hut high in the Andes along the Inca Trail near Machu Picchu in Peru. Or, most likely, you’d find bottles in the stores in metropolitan cities, or somewhere in the more active distribution market and recycling network of an industrialized nation. Correspondingly you’ll find more company delivery and reclamation personnel clustered in conjuntion with the central part of such distribution networks. These various distributions - product, delivery personnel, revenues, et al. - are interactive dynamic groupings, each sensitive to alterations and changes in the other - symbiotically, if you will.

Each of these distribution groupings in turn is affected in structure and therefore function by other local factors such as ... fuel, delivery vehicles in working order, packaging materials, workers on the job, weather, road conditions, and timed coordinated with advertising campaigns. Assemblies and distributions extend causal-process affects, and, simultaneously remain sensitive to the affects generated by other members of the broader system. Cyclic feedback and sensitivities contribute to changes in local and even broad generic system rates, pressures and gradients - influencing them positively, negatively and neutrally.

Some components even act as "buffering" mechanisms. Warehouses, for example, accumulate and store products, therefore prevent unrestricted flooding of networks, and allow controlled and healthy distribution of commodities. Sometimes, as in the case of the rapid-redistribution network designed by Federal Express, instead of having many local hubs handle and re-route packages for delivery, they determined that such redundancy of effort, though good for maximizing the number of jobs created by a massive delivery service, could be done faster and more economically another way. They recognized that it was more cost effective and efficient, to bring all packages to one or two central regional re-routing hubs. In the long run, amortized across all possible origins and destinations of packages, it was wiser to jet transport a package from 121 Main St, Anytown USA, to Memphis, Tennesee, and back again to 221 Main St., Anytown USA, rather than hire extra employees, maintain vehicles, etc. in each Anytown, even to deliver a package just across the street or up the block.

Many of these dynamics and interconnections may all seem very obvious even intuitive and therefore well understood. But the fact remains that all the mathematical systems dynamics tools developed to date do not yet capture the coordinated systemic behavioral relationships involved. They may track local behaviors and derivatively correlate them, but it is still a technique of bean-counting, no matter how massive or number crunching, or Power Law patterned.

Let me offer one last example of the distinctions between the suggested classes of Power Laws.

Extant Power Laws have well defined limits ... ranges of operation ... and so mathematically we are used to dealing with x-information or energy. The alternative class of Power Laws includes "open factors". Not only can we plug any value into equation factors and generate useful products/outputs, but we can also, in the sense that I'm suggesting, allow the freedom to add whole terms into an equation. Such factors are "in the environment" whether we are aware of them or not, and can have a bearing on the behavior of a "system". E.g.. seat 4 people at a table to play cards and have them each bring money and skill. Math can deal with all the card-playing probabilities and gain/loss outcomes. What it is not set up to do is deal with fifth and sixth persons joining the game. It can deal with them, but only as totally new and independent configurations/equations having nothing to do with the 4-person arrangement. E.g., before irretrievably committing to raising the bid in a poker hand - considering only the cash status of the current players - the bet should take into consideration the possibility that a high-roller may step in before the next hand and that it may be wiser not betting the last dollar on the current hand - and possibly lose, and so shut out any chance of participating in subsequent hands and vieing for any "fresh money" brought into the game. Amazingly, current predictive math avoids that kind of scenario/consideration. On the other hand, the new class of Power Laws looks at an equational-relationship that says "here's what happens to the game structure when you change the number of participants (number of parameters, number of dimensions). The entity "card game" remains, but it functions in different ways.

Briefly, the new class Power Laws are exampled by the proportionality that says that "the distribution entropy of a set of components that are in recursive exchange between agents is inversely proportional to the entropy of the behavior-space of the agents" ( 1973,1992). The behavior entropy of one level of assembly varies in opposition to the behavior entropy of the next lower or the next higher assembly level. Additionally, participation probabilities of any of the components within a defined entropy may drop to zero without destroying the proportionality. The proportionality (and thus the larger assembly) ceases to exist only when all recursive communication or exchange ceases. And it is not so much a matter of "extinction" of an assembly when those circumstances arise, as it is returning to a state of pure potential versus some state of enactment. The Olympics are re-enactable cyclic occurrances because every 4 years athletes reassemble to compete for awards.

The concluding picture is that though current Power Laws may efficiently analogue the relationships by which energy gets smoothly distributed within system levels in general, Systems Thinking needs to include attention to specific mechanisms that interconnect the levels, layers and nodes in more realistic behavioral relational ways, since these relationships are the causal parts of systems competencies. Traditional Power Laws would be exampled by the Newtonian Law of Gravitation. Take any two isolated bodies in the universe and their attractive force will always be proportional to their masses and vary inversely with the square of the linear distance between them. Add in a multitude of bodies however, and have them in any manner of direction, momentum, and bound/unbound states, and you have Systems Laws and Behaviors, which are indicative of quite another class of Power Laws. Fractal mathematics is having some success in modeling what is known as the ‘many-bodied problem’. But something else is required. These would be Systems Laws which treat linked nested domains having behaviors which are reversely and inversely connected. They would delineate the mechanisms through which broad robust complexity arises and becomes applicable in panoramas of multi-modal thinking, in ways that are not just utile, but creatively expressive, rich and abundant in functional/behavioral alternatives.

 

References:

Hebb, Donald. 1949. The Organization of Behavior. Wiley, New York.
. 1973. Four Plane Universe Conundrum. SUNY Stonybrook.
. 1992. Understanding the Integral Universe. Ceptual Institute.
Schroeder, Manfred. 1991. Fractals, Chaos, Power Laws. Freeman & Co., New York.
Scott, Alwyn, 1995. Stairway to the Mind. Springer-Verlag, New York.
Stix, Gary, 1998. "A Calculus of Risk", Scientific American, May, pp92-97. NewYork.