- The chess-board is the world; the pieces are the phenomena of the universe; the rules of the game are what we call the Law of Nature.
T. H. Huxley
In this chapter, we will cover the bare essentials to get us started. There are a lot of aspects of the nonlinear sciences that we will not cover, such as fractals, solitons, cellular automata, and a host of other subjects. Not because they are uninteresting or unimportant, but because we can get there without them. On the other hand, in the notes are some good sources to read to learn more. (1) Our approach will be to briefly contrast linearity and nonlinearity, and then concentrate on the characteristics of complex adaptive systems, which is key to grasping the rest of the material.
We are essentially linear creatures. Whether this is the native mode of humanity or whether it is primarily the result of acculturation is open to question. It can be observed that nonlinearity is more prevalent in contemporary non-Western cultures, and indeed in Western culture itself, prior to the modern era. However, it is unarguable that our society fosters and rewards linear behavior and performance from kindergarten on. Our educational system teaches it and grades on it, our workplaces hire, fire, and promote on it, our governmental and social programs are designed and executed on it, and it drives national security policy and military strategy and operations. Often associated with the name of Sir Isaac Newton, Newtonian, or linear, science became a powerful philosophy to both describe and ultimately control nature, which has proved to be largely illusory.
But what is it? The features of linearity include proportionality, additivity, replication, and demonstrability of causes and effects. With proportionality, small inputs lead to small outputs, greater inputs to larger consequences in an environment where these causes and effects are demonstrably and effectively measurable. Like the linear mathematical equation, only one valid answer is possible, permitted, or expected.
Further, the linear principle of additivity provides that the whole is equal to the sum of its parts. This promotes and legitimizes reductionism, the practice of taking a complicated and large problem and breaking it into more manageable pieces, analyzing the constituent parts, and arriving at a conclusion. The assumption, of course, is that the cumulative analytic product represents a valid derivative of the original whole, faithful and complete. Replication means that the same action or experiment under the same conditions will come out the same way; that results are repeatable, and therefore, independently verifiable. Finally, cause and effect are demonstrable. This can happen in a number of ways: observed, inferred, extrapolated, statistically validated, and so on. Therefore, the nature of linear systems is that if you know a little about their behavior, you know a lot. You can extrapolate, change scales, and make projections with confidence. Unlike nonlinearity, in which 2+2 may yield oranges, in linearity you can rely on the 4.
Two historical factors have reinforced a linear mindset within the U.S. military establishment. The first is the result of the Cold War, in which we lived and struggled for 40 years in a bipolar world, dominated by the USSR and the United States. Two-body problems lie generally in the linear to mildly nonlinear range. In other words, the Cold War marked by the interactions of two world powers habituated participants to an essentially linear environment. The second factor is America's historical industrial and technological prowess which has favored a military strategy of attrition through the use of overwhelming force wherever it could be brought to bear. Overwhelming force can, in effect, significantly linearize conflict. If the odds are big enough, the inherently nonlinear characteristics of warfare don't count as much.
Nonlinearity, which covers such concepts as chaos theory and complexity theory, does not conform to those qualities found in linearity. It is not proportional, additive, or replicable, and the demonstrability of causes and effects are ambiguous. Inputs and outputs are not proportional. The whole is not quantitatively equal to its parts, or even qualitatively recognizable in its constituent components. Results cannot be assumed to be repeatable; the same experiment may not come out exactly the same way twice. A contributing cause to this condition is the phenomenon of nonlinear dynamics, whereby outcomes are arbitrarily sensitive to tiny changes in initial conditions.
As a result, if you know a little about a nonlinear system, you don't know a lot. We cannot extrapolate, change scale, or project. The lack of predictability frustrates planning and control, as we use the terms. Yet, the vastness of the nonlinear world dwarfs the linear. So we must learn to deal with it. Alan Beyerchen, of Ohio State University, addresses this need in practical national security terms.
- Why harp on nonlinearity? Why does it matter? One reason for emphasizing nonlinearity is that it constitutes the well-established mathematical property underlying and making coherent all the faddish-sounding new sciences: deterministic chaos, fractals, self-organizing systems far from thermodynamic equilibrium, complexity and complex adaptive systems, self-organizing criticality, cellular automata, solitons, and so forth. It was in various ways sensed by the ancient Greeks....Yet no one before the late twentieth century could solve the interesting problems posed by nonlinear equations. There are no analytical techniques that work well, and numerical methods were just too cumbersome and time-consuming. Most scientists just bracketed out the nonlinear elements of their equations and went with the idealized linear approximation. Now computers allow us to go after many formerly intractable problems using the computer to pursue numerical solutions.
- The connotations of linearity still drive a great deal of our thinking, especially in mechanics and the many social scientific disciplines that implicitly try to copy the success of mechanics. Linearity offers structural stability and emphasis on equilibria. It legitimates simple extrapolations of known developments, scaling and compartmentalization. It promises prediction and thus control-very powerful attractions indeed. But linear systems are often restrictive, narrow and brittle. They are seldom very adaptive under significant changes in their environment. Bureaucracy is the quintessential linearization technique in social affairs.
- The connotations of nonlinearity are a mix of threat and opportunity. Nonlinearity can generate instabilities, discontinuities, synergisms and unpredictability. But it also places a premium on flexibility, adaptability, dynamic change, innovation, and responsiveness....
- What is the utility of thinking about war-for our potential opponents and ourselves-in nonlinear terms, especially in the high-tech, research-forefront metaphorical terms from the new sciences? For our opponents the usefulness may be the same as it was for Clausewitz. The Germans were underdogs to the French, and Clausewitz wanted to understand and use against the French their blindspots. He also needed to be the champion of disproportionate effects and unpredictability, for in a linear, predictable world Prussian resistance to Napoleon after 1807 was futile. The opponents of the United States will be looking for our blindspots in an effort to seize opportunities to surprise and shock us. They may also be able to compensate for their disadvantage in a military confrontation such as the Gulf War by consciously striving to affect the political context in order to change the conduct of warfare. An understanding of the porousness of the boundaries between politics and war can be a real weapon against those who envision those boundaries to be impermeable.
- We need for our own sake to understand the limitations our imagination places upon us. Linearity is excellent for the systems we design to behave predictably, but offers a narrow window on most natural and social systems. That narrowness sets blinders on our perception of reality and offers a weakness for an opponent to exploit. But if we know our limits, we can minimize the extent and duration of our surprise, reducing its value. And an expanded sense of the complexity of reality can help us to be more successfully adaptive amid changing circumstances. By thinking more constructively about nonlinearity, we might be able to design more robust systems when we need them. A new form of modeling that takes such concepts as self-organization to heart allows structures to bubble up from below rather than be imposed from above. With such tools we might come to understand better the biological and historical processes with which we must deal. And we may come to realize how conventional, analytical predictive techniques can themselves stimulate a self-defeating, unfulfillable desire to control more of the real world around us than is truly possible. (2)
Fundamental to an understanding of nonlinearity is an understanding of complex adaptive systems, or cas, which are the "engines" that drive nonlinearity.
- Complex adaptive systems are quite different from most systems that have been studied scientifically. They exhibit coherence under change, via conditional action and anticipation, and they do so without central direction. At the same time, it would appear that cas have lever points, wherein small amounts of input produce large, directed changes. It should be easier to discover these lever points if we can uncover general principles that govern cas dynamics. Knowing more about lever points would, in turn, provide us with guidelines for effective approaches to cas-based problems, such as immune diseases, inner-city decay, industrial innovation, and the like....We are only at the beginning of the search for general principles, but we do have some hints as to what those principles might be. (3)
Complex adaptive systems, or cas, contain seven basic attributes. These consist of four properties (aggregation, nonlinearity, flows, and diversity), and three mechanisms (tagging, internal models, and building blocks.)
The first property of cas is aggregation, which "concerns the emergence of complex large-scale behaviors from the aggregate interactions of less complex agents... Aggregates so formed can in turn act as agents at a higher level-meta-agents. Holland cites examples:
- Gross National Product (GNP) as an emergent aggregate property of the aggregate of the economy consisting of firms; individual identity from the immune system composed of anti-bodies, and behavior from the nervous system comprised of neurons. (4)
Emergent behavior, that is, activities that could not be predicted from the system's parts, is a feature of non-additivity-that the sum of the parts of cas is not equal to the whole. Another way to view emergence is provided by the following hierarchy:
Universe
Earth
Ecosystems
Organisms
Cells
Molecules
Atoms
Particles
Note that the structure is hierarchical, but not in the bureaucratic sense. For all of the talk one hears today about the demise of "tyrannical" hierarchies and the rise of "networked" organizations such as the Internet, nature doesn't work that way.
A good example of emergent behavior is also demonstrated by a popular simulation known as "boids," in which a few simple rules result in quite complex behaviors akin to the flocking of birds or the schooling of fish. A good Internet site for "boids" is Professor Brakke's web site at Susquehanna University.
The first mechanism of cas is tagging, which
- consistently facilitates the formation of aggregates....The most familiar example is a banner or flag that is used to rally members of an army or people of similar political persuasion. A more operational version of a tag in these days of the Internet is the header on a message that knits together members of a bulletin board or conference group....cas use tags to manipulate symmetries ....The classic example of a full-blown symmetry is a perfect sphere, say the white cue ball in billiards. [C]onsider a set of cue balls in rapid motion on a billiard table....after a strong 'break.' We cannot distinguish the individual cue balls unless we keep a careful record of their trajectories. But again, we can break the symmetry with a tag. If we put a striped cue ball in with the other cue balls, we can easily track it despite its motion....
- [Tags] allow agents to select among agents or objects that would otherwise be indistinguishable [and] provide a sound basis for filtering, specialization, and cooperation. This, in turn, leads to the emergence of meta-agents and organizations that persist even though their components are continually changing. Ultimately tags are the mechanism behind hierarchical organization-the agent/meta-agent/meta-meta-agent/...organization so common in cas. (5)
The second property of cas is nonlinearity. It is not unusual to have a word stand for two meanings when it has both macro and micro relevance, depicting both a global and interior condition: for example, "man" as a species and "man" as an individual. This is the case here, in which nonlinearity stands for a field as a whole as well as a specific property of cas.
- It is little known outside of the world of mathematics that most of our mathematical tools, from simple arithmetic through differential calculus to algebraic topology, rely on the assumption of linearity. Roughly, linearity means that we can get a value for the whole by adding up the values of its parts....Whole branches of mathematics are devoted to finding linear functions that are reasonable approximations when linearity cannot be directly established. Unfortunately, none of this works well for cas. To attempt to study cas with these techniques is much like trying to play chess by collecting statistics on the way pieces move in the game. (6)
One of the most valuable databases known contains the meticulous records kept by the Hudson Bay fur company, which goes back over 150 years. These have been studied in great detail, and a model has been developed which can account for the year-to-year fluctuations in the number of fur pelts acquired. This model deals with the interactions of predator-prey populations, and in this instance, the predator is the lynx and the prey is the hare. It is also one of the simplest illustrations of nonlinearity. The model consists of three factors: (1) a constant which indicates how efficient the predator is based on how much of its territory it searches each day, (2) the number of predators in a given area, and (3) the number of prey in the same area. Let's say that the efficiency of lynx is 50 percent. A wolf pack's might be different. So if there are two lynx per square mile and 10 rabbits, you multiply the three terms, or 50 percent x 2 x 10=10 encounters. If you double the number of lynx and hares, you get 50 percent x 4 x 20=40 encounters. So doubling the number of agents in the system results in quadrupling the number of interactions.
This nonlinearity occurs
- because it entails the product of two distinct variables instead of their sum. That is, the overall predator-prey interaction cannot be obtained merely by adding predator activity to prey activity....even in the simplest situations nonlinearities can interfere with the linear approach to aggregates. That point holds in general: nonlinear interactions almost always make the behavior of the aggregate more complicated than would be predicted by summing or averaging. (7)
The third property of cas is the idea of flows. And they also form the basis for centers of gravity. Pat A. Pentland, a pilot with over 2,400 hours in the A-10, has developed an approach to centers of gravity based upon the principles of nonlinear dynamics, as well as a large dose of ideas from anthropology. His formulation uncannily follows exactly the concept of flows in complex adaptive systems presented here. See Center of Gravity Analysis and Chaos Theory in the Appendix (page 299).
You can,
- [t]hink of flows over a network of nodes and connectors. The modes may be factories, and the connectors transport routes for the flow of goods between the factories. Similar (node, connector, resource) triads exist for other cas (nerve cells, nerve cell interconnections, pulses): for the central nervous system; (species, foodweb interconnections, biochemicals) for ecosystems; (computer stations, cables, messages) for the electronic Internet; and so on....In general terms, the nodes are processors-agents-and the connectors designate the possible interactions. In cas the flows through these networks vary over time; moreover, nodes and connections can appear and disappear as the agents adapt or fail to adapt. Thus neither the flows nor the networks are fixed in time. They are patterns that reflect changing adaptations as time elapses and experience accumulates. (8)
There are two attributes of flows that confound linear analysis. One is the multiplier effect, which is a major feature of networks and flows, whether money, information, or goods. To illustrate, community economic development specialists estimate, as a rule of thumb, that say for every $75,000 a business generates in gross receipts, one direct job is created. In addition, however, another indirect job is also created. The latter is the multiplier effect. But that job has no association with the creating business; it is a disembodied derivative-a connector or interaction-based ingredient. Actually, the rule of thumb does not work well at the level of the individual firm, or even industry. The jobs show up in counts at the macro-level, say the county or state level, but no more precise cause and effect relationships can be established because they are hidden in the interactions. Once again, the sum is not equal to the parts.
The other attribute of flows which confounds linear input/output is the effect of recycling, whereby the "aggregate behavior of a diverse array of agents is much more than the sum of the individual agents," (9) and is thus a source of nonlinearity. This recycling, however, is also "hidden" within the interactions of the system(s). Once again, as with the multiplier effect, the micro effects are masked, but evident at macro levels.
The role of tags is that they
- almost always define the network by delimiting the critical interactions, the major connections. Tags acquire this role because the adaptive processes that modify cas select for tags that mediate useful interactions and against tags that cause malfunctions. That is, agents with useful tags spread, while agents with malfunctioning tags cease to exist. (10)
It would be fair to say that tags indicating multiplier effects and recycling behavior would tend to favor agents carrying them.
The fourth property of cas is the diversity of agents within the system(s). The agents within cas comprise a diverse community marked by perpetual novelty. You might think that cas, for the sake of economy and efficiency, would favor the evolution of a few kinds of "general purpose" agents which are highly adapted, or optimized, to take advantage of a large range of opportunities. But cas does not do that, because the inevitable stagnation of equilibrium would result. Instead, in a process which is neither accidental nor random, cas seems to consist of hierarchies of "slots," occupied by agents. When an agent is removed from the system by losing its stability with the environment, which includes other agents, it leaves a hole. This is filled in a cascade effect by another agent similar to the former inhabitant, but different enough to achieve the stability to occupy the slot. Other things being equal, the agents with recycling capability to conserve resources, or possessing a multiplier effect, are favored.
This seems to argue, at the micro-nonlinear level of human affairs, for a population of agents which are, within bounds, mildly heterogeneous yet sufficiently differentiated, competitive enough to produce multiplier effects, and cooperative enough to recycle resources. Students are quick to point out that the unique characteristics of the people of America and its turbulent culture have historically fostered these qualities. In fact, America's success may be, in no small measure, due to the fact that Americans make for a pretty darn good natural cas. However, it is more than likely that recent movements which define "diversity" in terms of political correctness, for example, actually decrease that essential diversity so essential for cas. If the factor is significant, we will either bear the consequences or have to make some hard choices.
A second mechanism of cas are internal models that give them the power to anticipate.
- [T]he agent must select patterns in the torrent of input it receives and then must convert those patterns into changes in its internal structure. Finally, the changes in structure, the model, must enable the agent to anticipate the consequences that follow when that pattern (or one like it) is again encountered. How does an agent distill experience into an internal model? How does an agent unfold the model's temporal consequences to anticipate future events?
- To make a start on these questions, let's take a closer look at models as predictors. We usually ascribe prediction only to "higher" mammals, rather than taking it as a property of all organisms. Still, a bacterium moves in the direction of a chemical gradient implicitly predicting that food lies in that direction. The mimic survives because it implicitly forecasts that a certain pattern discourages predators. When we get to the so-called higher mammals, the models do depend more directly on the agent's sensory experience. A wolf bases its movements on anticipations generated by a mental map that incorporates landmarks and scents. Early humans built Stonehenge as an explicit, external model that helped predict the equinoxes. Now we use computer simulations to maker predictions ranging from the flight characteristics of untried aircraft to the future gross domestic product. In all these cases prediction is involved, and in the last two cases external models augment internal models.
- Taking these models into account we will find it useful to distinguish two kinds of internal models, tacit and overt. A tacit internal model simply prescribes a current action, under an implicit prediction of some desired future state, as in the case of bacterium. [BOIDS is also a good example of a tacit or low-level model.] An overt internal model is used as a basis for explicit, but internal, explorations of alternatives, a process often called lookahead. The quintessential example of lookahead is the mental exploration of possible move sequences in chess prior to moving a piece. Both tacit and overt models are found in cas of all kinds-the actions and identity supplied by an immune system fall at the tacit end of the scale, whereas the internal models of agents in the economy are both tacit and overt....(11)
The third mechanism of cas are building blocks, which you will see again in Chapter 10 where we will cover pattern recognition.
- In realistic situations an internal model must be based on limited samples of a perpetually novel environment. Yet the model can only be useful if there is some kind of repetition of the situations modeled. How can we resolve this paradox?
- We get the beginnings of an answer when we look to a common-place human ability, the ability to decompose a complex scene into parts. When we do this, the component parts are far from arbitrary. They can be used and reused in a great variety of combinations like a child's set of building blocks. Indeed, it is evident that we parse a complex scene by searching for elements already tested for reusability by natural selection and learning....
- Wherever we turn, building blocks serve to impose regularity on a complex world. We need only look at the use of musical notation to transmit the endless variety of music, or the use of a limited range of morphologies to describe the tremendous spectrum of animal structures. The point applies with at least as much force to our everyday encounters. If I encounter a "flat tire while driving a red Saab on the expressway," I immediately come up with a set of plausible actions even though I have never encountered this situation before. I cannot have a prepared list of rules for all possible situations, for the same reason that the immune system cannot keep a list of all possible invaders. So I decompose the situation, evoking rules that deal with "expressways," "cars," "tires," and so on, from my repertoire of everyday building blocks. By now each of these building blocks has been practiced and refined in dozens or hundreds of situations. When a new situation is encountered, I combine relevant, tested building blocks to model the situation in a way that suggests appropriate actions and consequences....(12)
In a fascinating reprise, Holland uses the seven basics we have just covered to describe New York City as a complex adaptive system:
- Agents formed by aggregation are a central feature, typified by firms that range from Citibank and the New York Stock Exchange to the corner deli and the yellow cab. These agents determine virtually every fiscal transaction, so that at one level of abstraction the complex adaptive system that is New York City is well described by the evolving interactions of these agents. We have only to look to advertising, trademarks, and company logos to see how tags facilitate and direct these transactions. The diversity of these tags underscores the variety in the city's firms and activities, and the complex flow of goods into, and out of, and through the city that results. That New York retains both a short-term and a long-term coherence, despite diversity, change, and lack of central direction, is typical of the enigmas posed by cas. As is usual, nonlinearities lie near the center of the enigma. New York's nonlinearities are particularly embedded in the internal models-models internal to the firms-that drive transactions. These models range from spreadsheets to sophisticated corporate plans. There are also continual innovations, such as the steady flux of financial instruments on Wall Street ("Derivatives," the current innovation, have absorbed even more money than their predecessors, "junk bonds"). Trend projection and other linear analyses provide few insights into these activities. New perceptions will surface, I suspect, if we can uncover the building blocks that are combined and recombined to determine the city's outward appearance. The building blocks for this enterprise are less obvious than for some other cas, though contracts, organization charts, permissions, pieces of city infrastructure, and taxes are all obvious candidates. (13)
[J. Holland, Hidden Order, pages 13-40. ©Copyright John H. Holland. Reprinted by permission of Addison Wesley Longman, Inc. ]
In common with other fields, more progress has been made in working with nonlinearity than in defining and measuring it. The problem has been compared to that
- confronting early 19th-century scientists as they tried to get a grip on a mysterious concept called energy. Today, people take energy so much for granted that it is hard to appreciate how abstract the concept really is. 'Many people had a pretty good idea what energy did and how it behaved.... But energy was not really understood....until people came up with a precise definition. The result was the laws of thermodynamics.'
This search for definition and measurement is covered in Researchers on Complexity Ponder What It's All About by George Johnson (14) in the Appendix. The article ends on the following note:
- An idea that runs throughout this kind of research is that complexity lies somewhere between order and disorder, predictability and surprise. "Nobody disputes that there are some characteristics of systems that make them more complicated," said Dr. Erica Jen, vice president for academic affairs at the Santa Fe Institute. "And those characteristics are neither highly ordered nor completely random. A string of numbers with all the same digits is very uninteresting. but a number like pi that has all this structure in it is very interesting....As Dr. Lloyd continues to hammer away at a definition, he likes to ask his colleagues what they mean by complexity. After puzzling over the matter, they usually answer with something like this: "I can't define it for you, but I know it when I see it." "That," he said, "remains the tried and true definition.
It is obvious from the above that the field of complexity permits, even invites, speculation. In this book I admit there is some of that. Certain concepts such as macro-versus micro-nonlinearity, unpunctuated equilibrium, the complexity shuttle, and the edge of equilibrium are constructions. But then, I too know complexity when I see it. We are dealing with the messy process by which science assumes human dimensions and relevancy. Science applied to technology yields engineering; the application of science to human affairs produces philosophy.
The world of nonlinearity has been discovered there is no turning back or ignoring the fact. Further, it has been found to be fundamental to national security processes and warfare. While John Holland, Seth Lloyd, and other scientists search for general principles, we must make do with what we have and work with it. That is the object of this work.
| Coping with the Bounds Index | Foreword | Acknowledgments | Introduction | Part One Introduction | Chapter 1 | Chapter 2 | Chapter 3 | Chapter 4 | Part Two Introduction | Chapter 5 | Chapter 6 | Chapter 7 | Chapter 8 | Chapter 9 | Chapter 10 | Conclusion | Appendix 1 | Appendix 2 | Appendix 3 | Appendix 4 | Appendix 5 | Appendix 6 | Notes |