The Complex World

Nonlinear Dynamical Systems as a Paradigm for International Relations Theory

Copyright © 1999 Garret Wilson

University of London, School of Oriental and African Studies

MA International Studies and Diplomacy 1998/9

International Relations, Essay 1

This dissertation is submitted in partial fulfilment of the requirements for the degree of MA International Studies and Diplomacy of the School of Oriental and African Studies (University of London).

September 27, 1999

I undertake that all material presented for examination is my own work and has not been written for me, in whole or in part, by any other person(s). I also undertake that any quotation or paraphrase from the published or unpublished work of another person has been duly acknowledged in the work which I present for examination.

Table of Contents


Viewing the international states system as a complex nonlinear dynamical systems can leverage advances in the new cross-discipline field of complexity theory, bringing insights into both the stabilities and surprises found in the world. This study demonstrates how complexity theory can bring clarifications, point out limitations, and provide extensions to the structuralist theory of Kenneth Waltz, and uncovers directions in which the paradigm of complexity theory may provide guidance and clarify expectations in the field of international relations, especially in the process of theory creation.


Sincere thanks to Dr. John Letcher and Jonathan Harju for the assistance and support in my attending SOAS. Thanks to Dr. Edward Keene who proved an excellent teacher of International Relations. As always, thanks to my family for being there. Thanks to Andreas for a random (or rather, determined but unpredicted) conversation on chaos theory and IR that neither of us knew would turn into a dissertation topic.

The image of the Mandelbrot Set (Figure 5) was created using Chaos Pro 2.0 by Martin Pfingstl, who can be reached via e-mail at .

Bifurcation diagrams were generated using Winfeed by Richard Parris, who can be reached at The latest version of Winfeed can be found on the Internet World Wide Web at

All products herein, unless otherwise specified, are trademarks of their respective holders.

About the Companion Web Site

All computer program experiments performed in this study can be found on the Internet World Wide Web at the following location:

The experiments are interactive, allowing the exact simulation conditions from this study to be duplicated, as well as permitting custom parameters to be entered so that further study can be conducted.

The following experiments are available:

These experiments are written in the Javascript 1.1 language, and therefore require a web browser with Javascript 1.1 support enabled. These programs have been confirmed to work on Microsoft® Internet Explorer 5.0 and Netscape® Navigator 4.05, but should work with any 4.x or higher generation browser.

Chapter 1: Complexity and the World

Adam Smith had a meta-narrative1. Leave a market alone, his grand story explained, and an "invisible hand" will guide agents towards profit and prosperity (Heilbroner, 71). Karl Marx was incredulous. If Smith is to be taken at his word, he explained, if the ideas of capitalism are to be taken to their logical ends, if a market is allowed to function completely unfettered by human controls, then the system is its own worst enemy. The same system forces that Smith had seen leading to prosperity would instead lead to its own destruction (Heilbroner, 162). There was no "maybe" about it. Capitalism's death was its own determined destiny, and its successor, communism, was the inevitable end of the system's dialectical development. You could count on that2.

Over 150 years later, it's easy for one to find fault with Marx, the most obvious fault being that his inevitable outcome is a bit late in arriving. It could be said that Marx's ideas only apply to a pure capitalist system (which does not and has not existed); his predictions then do not hold or instead gain another timetable. Or it could be that Marx was flatly wrong — so wrong, in fact, that instead of capitalism dying, it will adapt to its environment and, along with its cousin, liberal democracy, become the Hegelian end result of the international ecosystem's evolution (Fukuyama, 48).

If one were to ask Marx to describe the international states system during his life, the picture, while not perfectly peaceful, would nevertheless appear somewhat more pleasant than that of a century later. The year 1815 (just three years before Marx was born) (Sullivan) marked the beginning of "the longest interval without world war in the modern state system" (Nye, 57). What caused, almost 100 years after, a world war killing some 15 million people? The most readily available (and perhaps the ultimate) answer might be similar to that of the German chancellor Bethmann Hollweg shortly after war broke out: "Oh, if I only knew!" (Hollweg, quoted by Nye, 60).

It is undeniable that the assassination of Franz Ferdinand at Sarajevo by a Serbian terrorist had something to do with the beginning of World War I. But could actions of one person cause such large effects, without an environment conducive to such events? "Looking back, things always look inevitable," Joseph Nye admits, but in this case the outcome might better be described as "highly probable," due to the "deep changes in the structure of the balance of power and certain aspects of the domestic political system" (Nye, 65-66). In the setting of the early twentieth century, the situation surely appeared quite different — with 100 years of stability, regardless of structural changes and shifting alliances, is the prediction really expected that one person's actions can set off a total war in which the countries involved mobilized almost all their citizens (Carruthers, 49)?

Predictions, like opinions, seem to be had by everyone, so it's likely that at least someone at any time will have one that comes true. It is striking, however, that in 1989, after decades of a seemingly stable bipolar system3, hardly anyone predicted that within six months, "from free elections in Poland in June to the fall of Ceausescu in Romania in December, the established order" was to fall "apart at the seams" (Lundestad, 132). The Communist party was suspended in Russia and the Soviet Union was dissolved (Lundestad, 265). Again, as hindsight improves vision, a number of causes immediately spring forth, from Gorbachev's policies to the spread of communication technology. The fact remains that just a decade earlier attempts at predicting the coming international arrangement gave no better results than if one were to claim to know hurricane patterns the same amount of time in advance.

In the study of physics, the work done by Isaac Newton allows one to write equations which describe the motions of planets and other physical objects. Predicting the future position of a single planet using these equations is simple, as is the solution for two planets. In 1890 King Oscar II of Sweden offered a prize for the first person who could solve Newton's equation for more than two bodies. The person who came the closes to completing this task, referred to as the "three-body problem" or more generally, the "n-body problem," was a French mathematician named Henri Poincaré. After years of work, Poincaré finally realized that only three bodies can produce such complicated interactions that these linked equations become extremely complex. Poincaré finally had to admit defeat, and there are still some areas of the n-body problem where even contemporary computers have difficulty approximating answers (Devaney, 6).

A common feature of all these narratives mentioned, attempting to examine systems both economical and international, or trying to predict both worldwide wars and worldwide weather, is that they all initially provided models that assumed the underlying systems exhibit linear behavior. Small changes are assumed to produce small outcomes. Initial conditions, if system processes are known, are thought to matter little with ultimate inevitabilities. Similar to graphing a line using the formula y=mx+b, if the input variable x is not completely accurate (due to some measurement limitations, for example), the calculated output y will not be far removed from the real result.

During the past two decades, researchers in various fields have come to the conclusion that many systems exhibit nonlinear behavior. Chemical reactions, biological configurations, physical structures, economical cycles, and even traffic patterns have been shown to have certain similarities: the underlying systems can be shown in a mathematical sense to have complexity, and this complexity can result in behavior which in some cases is chaotic (again, in a mathematical sense) and in some cases exhibits patterns of emergent order. Complexity theory and chaos theory, the discipline from which the former has descended, have shown that diverse systems can share similar properties, regardless of the agents involved.

Kenneth Waltz notes that, "Among the depressing features of international-political studies is the small gain in explanatory power that has come from the large amount of work done in recent decades. Nothing seems to accumulate, not even criticism." (Waltz 1979, 18). Is it possible that earlier theories have overlooked crucial aspects of the international system that doomed them, if not to failure, at least to a early deaths or irrelevance? Do certain systems hold things in common that effect the accuracy of predictions? What if a variety of systems, in various disciplines, have some fundamental similarities that determine behavior patterns?

Presented here is not a new theory of international relations, or even a theory as such. Rather, it is argued that advances in the study of complex nonlinear systems can provide insights into outcomes in the international arena. By viewing the international system as a complex system (in its technical definition), one can gain a better understanding of what to expect from theories of international relations. Certain features of world politics and seemingly contradictory descriptions can be resolved when reformed in the language of complexity theory. The concept of nonlinear dynamical systems therefore becomes not a theory of international relations, but a paradigm for theory creation, analysis, and discussion. Complexity may assist in discovering, if not what to predict as the future configuration and behavior of international actors, at least what can and cannot be accurately predicted under what circumstances.

Getting Complex

Fluid dynamics is important to many branches of the natural sciences: it is used to model the flow of air over the wing of an airplane, it explains the flow of water around a submarine, it models the flow of liquid sodium in a nuclear reactor (Coveney and Highfield, 67), and it even describes the flow of blood through the vascular system (Casti, 78). The problem with fluid dynamics is not that the behavior of individual atoms in a fluid act in a mysterious manner — in fact, they are well understood. What makes analyzing fluid turbulence so difficult is that there are so many atoms in the system interacting with each other simultaneously that calculating the minute swirls and eddies at any particular time become an enormous task, however keen a scientist may be on finding a solution. A tiny miscalculation at any particular point has the potential to affect the calculated behavior over large distances.

What makes something a fluid, after all, is not determined by any property of any of the separate atoms and molecules of which it is made. A molecule of H20 has exactly the same properties in a cube of ice as it does in a wave in the ocean. What gives a fluid its unique properties has something to do with how the molecules interact — their position and velocity; their arrangement over time. One of the most popular ways of solving fluid flow problems is the set of Navier-Stokes equations which assume that the entire fluid is a continuous entity and apply Newtonian laws of conservation of mass, momentum, and energy (Coveney and Highfield, 57).

The Navier-Stokes equations, however, are difficult to solve under high-turbulence scenarios, even with a computer: since computers are digital, they must approximate these analog equations by dividing up space and time into a grid and only analyzing the behavior of the fluid at certain points on the grid. "Thus, the computational fluid dynamicist faces a dilemma: if she subdivides space too far, then the time taken to obtain a solution to the equations will be prohibitively long because she has a very great number of points to consider; but if she settles for a cut-off that is too coarse, then she will omit important details that affect fluid behavior such as eddy structures" (Coveney and Highfield, 67).

This genre of problem has become apparent in many areas of science. In 1963, Edward Lorenz published an article in the Journal of Atmospheric Science entitled, "Deterministic Non-Periodic Flow," and in doing so sparked the study of chaos4. In attempting to replicate weather conditions by computer models of atmospheric convection, Lorenz ran identical experiments using identical inputs except for the number of digits of precision used. The two experiments soon exhibited divergent weather patterns, indicating that small changes in atmospheric conditions can cause changes weather conditions (Peak and Frame, 146). The term "butterfly effect," in which the visualization of a butterfly's flapping wings changing weather conditions around the world, has become one of the popular catchphrases of chaos theory.

Further investigations and simulations revealed that many systems exhibit similar behaviors, from traffic patterns to the stock market, making chaos theory a hot topic in mathematics. In the 1980s, chaotic theories were extended to show that complex systems may not only exhibit chaos but may also show spontaneous emergence of order by adapting to their environments. Researches such as Murray Gell-Mann and Kenneth Arrow (Nobel laureates in physics and economics, respectively) have drawn on recent work in neural networks, artificial intelligence, and chaos theory to investigate what complex systems have in common, whether their agents be atoms in a chemical reaction or organisms in a population (Waldrop, 12).

What is presented here is not meant to be a history of complexity theory; this can be found in a number of works, including those cited. Likewise, there is no attempt to demonstrate extensive mathematical proofs of the concepts of complexity theory, and an in-depth examination of the principles of complexity is left for other works. What is of concern here is that there is a body of complexity research which has begun and is still under way, and that there are ways in which it might apply to international relations.

Dynamical Systems

With that in mind, there are several aspects of dynamical systems which should be understood on their own before being applied to any specific discipline. A dynamical system is one that changes over time and is deterministic: "the future of the system is completely determined by its past." This means that a dynamical system is not random; it has a "definition of the state of the system," as well as "a rule for change called the dynamic" (Peak and Frame, 122).

One of the easiest dynamical system to visualize is that of a banking account. The state of the system is defined as the amount of money present in the account at any particular time. The dynamic is the process of compounding interest. If Bn represents the balance of the account after year n, and r represents 1 + (interest rate) - (inflation rate), the dynamic can be expressed as follows:

Bn+1 = r * Bn

This system is deterministic: The balance of any year can be unambiguously determined from previous year totals. Starting with $100, for example, and an interest rate of 5% (with no inflation), the balance in the account after 50 years would be $1146.84.5


The dynamic that governs the bank account dynamical system, Bn+1 = r * Bn, is a linear equation: plotting Bn+1 against Bn reveals a constant linear relationship. It should be noted that if a time series of bank balances is plotted for the bank account dynamical system, the outcome is not linear but exponential (Peak and Frame, 128). Compounding interest, as long as the interest rate is higher than the rate of inflation, will result each year in larger gains than the year before. A linear dynamic has produced a nonlinear outcome (Figure 1).

Figure 1: Plot of Bn+1 against Bn with corresponding time series for Bn+1 = r * Bn.
Figure 1: Plot of Bn+1 against Bn with corresponding time series for Bn+1 = r * Bn.

In the natural world, there are often limits imposed on dynamical systems growth; money is withdrawn from a bank account, populations run out of food, and (more relevant to this discussion) states declare, win, and lose wars. In these systems, the dynamics themselves are nonlinear; plotting each value in the series of such a system against its following value will not produce a straight line, or in some cases, produce no semblance of linearity.

A good example of a nonlinear dynamic is what ecologists call the logistic model of population growth (Devaney, 12), or logistic map6, which can be used to model population dynamics, e.g. as in a colony of single cells. A population with unlimited food, no predators, and no diseases (not to mention wars) would have a linear dynamic and display exponential growth. The logistic map takes into consideration that there are limits to growth that are in large part derived from the size of the current population. Taking a certain maximum population — the "carrying capacity" — one can construct an equation that allows for a certain death rate along with a birth rate that depends on the amount of free space available. In the logistic map, s can be taken to mean the intrinsic birth rate of the population measured in rescaled time units (Peak and Frame, 162):

xn+1 = s * xn * (1 - xn)

It should be obvious that the logistic map is the same equation used for an exponentially growing bank account except for the presence of (1 - xn) which compensates for a maximum value or in this case, a carrying capacity of a population. Plotting xn+1 against xn reveals that the logistic map is a nonlinear dynamic: each value in the series does not have a set relationship with the one before it, but its relationship rather depends on the actual size of the value before it. Put another way, each year (or minute or second) the population is not only determined by how large the population used to be, but by how large the population physically can be. Populations cannot grow infinitely, the temperature outside cannot get infinitely hot, and various interactions between states mean that countries cannot grow with no checks whatsoever. Dynamical systems in the real world, including the international system, are most accurately described as being nonlinear.

Plot of xn+1 against xn for the logistic map with a particular s.
Figure 2: Plot of xn+1 against xn for the logistic map with a particular s.


Although the logistic map is not meant to accurately model a real population, it does provide a insights into the principles behind population growth. The logistic map can also illustrate other aspects of nonlinear dynamical systems in general. Using a growth rate (s) of 2 and an initial population of 0.001, for example, outputs the values in Listing 1. After a few years (or days or seconds) of changes in the number of organisms, the population settles down to its equilibrium of 0.5, or half its carrying capacity.

x0: 0.001
x1: 0.001998
x2: 0.0039880159920000005
x3: 0.007944223440895105
x4: 0.015762225509632476

x14: 0.4999999999999971
x15: 0.5
x16: 0.5
x17: 0.5
x18: 0.5
Listing 1: Logistic map, s = 2.

In the language of dynamical systems, the value 0.5 is called an attractor for s = 2. Other initial populations with a growth rate of s = 2 will eventually settle down to the same equilibrium of 0.5 after several iterations7. This term can be applied to other dynamical systems as well; the nonlinear bank account dynamical system, above, will keep increasing (in theory, of course) forever. A bank account therefore has an attractor at infinity.

Limit Cycle Attractors

Although the logistic map appears straightforward enough at this point, at other growth rates the population behaves differently. Consider a growth rate of 3.1. The first few iterations reveal populations that are of different size (which is surely expected with a different growth rate), but after several iterations the values begin changing between two specific values: by iteration 150, the population is perpetually locked between two sizes, 0.5580141252026961 and 0.7645665199585943.8 The initial population size is not the governing factor; substituting other values for x0 reveals the same behavior.

x0: 0.001
x1: 0.0030969000000000005
x2: 0.009570658552209001

x150: 0.5580141252026961
x151: 0.7645665199585943
x152: 0.5580141252026961
x153: 0.7645665199585943
x154: 0.5580141252026961
x155: 0.7645665199585943
x156: 0.5580141252026961
Listing 2: Logistic map, s = 3.1.

In dynamical system parlance, the system has arrived at a limit cycle attractor (Casti, 28), its population going through a constant cycle of changes. Specifically, the behavior is a 2-cycle attractor, because two values are involved. Nonlinear dynamical systems can have a number of cycles.

Chaos — Strange Attractors

Other growth values for the logistic map display even stranger properties. Changing to a growth rate (s) of 4 with the same initial population size reveals the values shown in Listing 3. The population does not increase forever, settle at one value, or go into a fixed cycle. It displays a strange attractor (Casti, 29): sometimes the system is attracted to one point, sometimes to another. Sometimes a strange attractor may appear to have limit cycles, then abruptly be attracted to another value for a time.

x0: 0.001
x1: 0.003996
x2: 0.015920127936000002
x3: 0.06266670985000558
x4: 0.2349583733063232
x5: 0.7190117444782786
x6: 0.8081354231223248
x7: 0.6202102440689035
x8: 0.9421979888835786
x9: 0.21784375450927384
x10: 0.6815514125223083
Listing 3: Logistic map, s = 4.

Chaos has appeared — not in its common usage, which can simply mean random, but in its mathematical sense indicating unpredictability. Unpredictable here does not indicate randomness, as it has been shown that the system is entirely determined by its initial conditions and its dynamic, making the sequence deterministic. This type of behavior is more precisely referred to as deterministic chaos, although just "chaos" will be used here with that understood meaning.

High Dependence on Initial Conditions

One common trait of chaotic systems is that their strange attractors exhibit a high dependence on initial conditions. Using the same growth rate of s = 4, begin with an initial population of 0.0011, a value almost exactly the same as before. Note that at the fifth iteration, the values are similar: 0.71... versus 0.76.... As Listing 4 shows, by the ninth iterations the values are completely different: 0.21... versus 0.91.... The so-called "butterfly effect" that Lorenz discovered has arrived in the logistic map.9

x0: 0.0011
x1: 0.00439516
x2: 0.017503370274297602
x3: 0.06878800921335375
x4: 0.25622487600726923
x5: 0.762294755689315
x6: 0.7248058445515302
x7: 0.797849329021893
x8: 0.6451431088048325
x9: 0.9157339118658743
x10: 0.3086612580987899
Listing 4: Logistic map, s = 4.

Bifurcations — Transitions into Chaos

It has been shown that different growth rates for the logistic map yield radically different growth patterns for the population. While a growth rate of s = 2 exhibited a population equilibrium, s = 3.1 exhibited a limit cycle and s = 4 showed a chaotic population change. Somewhere between growth rates of s = 2 and s = 4 the population underwent a transition, or bifurcation, from stability to chaos (Peak and Frame, 132).

In the study of chaos it is often useful to examine a bifurcation diagram of a system, with inputs (in this case, the growth rate s) on the horizontal axis and the outputs (here, the population size xn) on the vertical axis. It should be noted that an initial population may not reach its attractor(s) until several iterations (see Listing 1, for example). For this reason, the first few iterations (the first 200-300, for example) are not plotted on the diagram so that the system's tendencies are evident.

The bifurcation diagram of the logistic map immediately shows some startling features. The first bifurcation happens at s = 1; at this point, the population shows positive growth for the first time. At s = 3 there is another bifurcation; populations with growth rates over s = 3 exhibit 2-cycle attractors. Near s = 3.45, the 2-cycle bifurcates into a 4-cycle, and at around s = 3.55 the 4-cycle changes into an 8-cycle. Further bifurcations quickly interact and plunge the system into chaotic cycles.

Figure 3: Bifurcation diagram, s = 0..4 (horizontally) for x values 0..1.
Figure 3: Bifurcation diagram, s = 0..4 (horizontally) for x values 0..1.

Self-Similarity — Fractals

It should also be apparent that there are similar features within the bifurcation diagram of the logistic map. At the point where the 2-cycle changes to a 4-cycle, each set of the 4-cycle resembles the 2-cycle before it. There is a similar occurrence in the 8-cycle section.

Further magnifications reveal other curiosities. At around s = 3.83 (the far right vertical blank strip seen in Figure 3), the system seems to bifurcate from chaos into a 3-cycle attractor. Furthermore (magnified in Figure 4), each "prong" of the 3-cycle itself undergoes a bifurcation into a cycle with twice the previous period (a period-doubling bifurcation). In fact, "each prong of the original attractor completely recapitulates the overall story of the whole bifurcation diagram; each prong, suitably enlarged, is a shrunken copy of the whole diagram" (Peak and Frame, 173). The self-similarity of the logistic map bifurcation diagram mean that the diagram is a fractal, with each small part carrying an image of the whole. Recent study has shown that most chaotic regions of dynamical systems are fractals (Devaney, 176).

Figure 4: Enlargement of s = 3.822..3.856.
Figure 4: Enlargement of s = 3.822..3.856.

Probably the most famous fractal image is the Mandelbrot Set, derived from a two-dimensional dynamical system closely related to the logistic map.10 For certain values of a and b, the result of this set of equations will quickly approach infinity after only a few iterations. For other values, the system will show simple attractors, limit cycle attractors, and strange attractors. A two-dimensional image can be mapped by associating a color value with how quickly a certain a, b value will cause the system to approach infinity.

Figure 5: The Mandelbrot Set.
Figure 5: The Mandelbrot Set.

The image produced from the Mandelbrot equations turns out to an extremely intricate picture with self-similar regions that display subtle differences. Each region can be "magnified" infinitely (using very small ranges of a, b values), producing endless patterns of self-similar patterns and colors that never exactly repeat. Due to their close relationship, the Mandelbrot set yields a bifurcation map identical to the logistic map (Peak and Frame, 273). Although the equations in the Mandelbrot map are not new, creating such a corresponding image as the Mandelbrot Set is virtually impossible to do by hand and therefore must be generated by electronic computers available only recently.11

Modeling Complexity with Cellular Automata

The logistic map is certainly a nonlinear dynamical system which exhibits chaos under certain conditions. It is, however, only a simple mathematical construct. While it can conceptually represent a population as a single entity, it provides no insight into the interaction of the actual components involved. While useful as a tool to explain some features of nonlinear dynamic systems, the logistic map does not reflect complexity as is found in the world, from fluid dynamics to international relations.

The use of cellular automata has become increasingly useful in the study of complex systems because it allows study of an entire system without ignoring the effects of individual components of the system. A cellular automaton is based upon the idea of space and the locality of influence: a system is distributed in space, and nearby regions have more influence than those far apart (Bar-Yam, 113). A grid of cells is used to represent the components of a system, and each cell is given a set of rules based upon its surrounding neighbors. The system evolves over several iterations by allowing each cell to interact using the given rules.

A particular class of cellular automata called lattice gases has been successfully used to model fluid flow. Instead of using a microscopic view of individual cells or taking a macroscopic picture of the system by using the Navier-Stokes equations, lattice gases allow a mesoscopic model which allows the system to be viewed as a whole while taking into account the actions of the individual components (Coveney and Highfield, 97). Rich and varied results have been seen to emerge from cellular automata, or "CA"; perhaps the most popular CA experiments has been Conway's Game of Life, which models the reproduction and death of organisms in a population based upon the population density as a representation of competition for resources (Bar-Yam, 123).

What makes CA so useful is that after several iterations they reveal complex structures and arrangements that form across great distances even though each cell only takes into account local information. A simple one-dimensional example of Conway's Game of Life illustrates this well. Figure 6 shows the results of a single-organism population in a 127-cell "environment" after a total of 64 iterations (including the first). Each cell (representing an organism) is on or off (alive or dead) at each iteration based upon a simple set of rules corresponding to the assumption that organisms have a finite life-span and population expansion does not occur under extreme population densities. The particular rules used here state that if a cell is alive in one generation, it will be dead in the next (a finite life-span), and if a cell is dead in one generation, but has one and only one live neighbor cell, it will be alive in the next generation (positive birth rate under preferred population size).

Figure 6: Sierpinski Gasket formed by a One-Dimensional Cellular Automaton.
Figure 6: Sierpinski Gasket formed by a One-Dimensional Cellular Automaton.

What is formed by this CA simulation is referred to as a Sierpinksi gasket (Casti, 234). It's notable that this image, like the bifurcation diagrams examined earlier, is self-similar or fractal — each small section of the gasket is a reduced image of the whole. What's even more remarkable is that this complex structure was formed across a large area when each "agent" in the system only had "knowledge" of the state of its immediate neighbors.12

Modeling Complexity with Neural Networks

Another automaton that is useful in studying complex systems is modeled after one of the most complex systems that exists: the human brain. The concept behind an artificial neural network is of various interconnected nodes (representing neural cells in the brain), each of which has a certain mathematical weight as a property. By the interconnection of nodes, an input can propagate through the system across various nodes and be represented as an output in another part.

The more a particular node is used, its weight increases; likewise, the less a node is used, its weight is decreased. In this way a neural network can be "trained" to "learn" concepts by applying an input and determining if the output is the correct answer to the input. The network then uses several methods to readjust the strengths of its neural connections in response to the training feedback. Through repeated training (having been given handwriting samples as input, for example), a neural network will take on a set of weights which generalizes the input. The network has then learned — is able to make correct assumptions about new inputs (to recognize letters written in previously unseen handwriting, for example) (Coveney and Highfield, 134). With a connected network of interacting nodes, neural networks have been able to learn a multitude of tasks, from voice recognition to decision-making (Peak and Frame, 337).

Figure 7: A Simple Neural Network.
Figure 7: A Simple Neural Network.

The difference between a trained and an untrained network, then, is simply the difference in weight values held by the individual nodes. The fact that a particular neural network can recognize shapes, fit a curve, or predict numerical sequences has to do with the distribution of weight values in network. Cilliers makes it clear that there is no meaning in the value of any particular weight. "The significance of a node in a network is not a result of some characteristic of the node itself; it is a result of the pattern of weighted inputs and outputs that connects the node to other nodes. The weight... does not stand for anything specific" (Cilliers, 81).

Complexity as Holism

"Meaning" or "representation" in a neural network therefore has no localized presence — these are holistic concepts that cannot be derivable from an examination of particular components of the system, yet is dependent on the combination of the components. A Sierpinski gasket is made from the interactions of the agents in the cellular automaton, yet no single agent was given any rules or instructions for creating such a figure. Liquids and gases gain their properties not from an alteration in the properties of individual molecules but through the interaction and distribution of components over space and time.

These features are what have in the past few years driven research into complex nonlinear dynamic systems. Complexity theory bridges the gap between a microscopic reductionism which ignores emergent properties of the system and a macroscopic systems view which ignores the influences each component can wield. Complex systems in general can display chaotic behavior where small changes can propagate and create proportionally large effects, yet also display configurations of stability and emergent order. These properties and tendencies from the viewpoint of complexity theory may prove useful in the analysis of the international system as well.

Chapter 2: The World of Waltz

International relations theorists are no strangers to the reductionism/holism debate. Kenneth Waltz has perhaps "made the most durable impact" regarding levels of analysis in international relations theory (Buzan, 200), especially with his classic work, Theory of International Politics (1979). In it, Waltz examines the reductionist method of analysis as applied to international politics, in which "the whole is understood by knowing the attributes and the interactions of its parts" (Waltz 1979, 18). Noting that "the same causes sometimes lead to different effects, and the same effects sometimes follow from different causes," he concludes that "reductionist explanations of international politics are insufficient..." (Waltz 1979, 37).

Waltz's solution to this predicament, borrowing from the sociological ideas of Emile Durkheim (Ruggie, 138), is that "in international politics, systems-level forces seem to be at work," meaning that "outcomes are affected not only by the properties and interconnections of variables but also by the way in which they are organized" (Waltz 1979, 39). To Waltz, "A system is composed of a structure and of interacting units. The structure is the system-wide component that makes it possible to think of the system as a whole" (Waltz 1979, 79). This structure "is defined by the arrangement of its parts... A system is composed of a structure and of interacting parts" (Waltz 1979, 80). A system, therefore, is unique because of the specific units it contains and their arrangement.

Durkheim, whose ideas influenced Waltz, further explains the idea of a system, as Ruggie notes: "Whenever certain elements combine and thereby produce, by the fact of their combination, new phenomena, it is plain that these new phenomena reside not in the original elements, but in the totality formed by their union," so that the system takes on a "specific reality which has its own characteristics" (Durkheim, quoted by Ruggie, 139). Waltz stresses that one of the fundamental properties of the international states system is that it is "decentralized and anarchic" (Waltz 1979, 88). However, "patterns of behavior nevertheless emerge," which Waltz takes to"derive from the structural constraints of the system" (Waltz 1979, 92). Indeed, "the texture of international politics remains highly constant, patterns recur, and events repeat themselves endlessly" (Waltz 1979, 66).

Order Appears

Waltz's structure becomes much like the market entity of classical economics: "it is individualistic in origin, and more or less spontaneously generated as a byproduct of the actions of its constitutive units," (Ruggie, 140), "whose aims and efforts are directed not toward creating an order but rather toward fulfilling their own internally defined interests by whatever means they can muster... Each unit seeks its own good; the result of a number of units simultaneously doing so transcends the motives and the aims of the separate units" (Waltz 1979, 90). The observed "continuities and repetitions" (Waltz 1979, 67) are thus explained by the formation of a conceptual system entity, which none of the units alone explicitly intended to create.

Waltz takes pride in having "broken sharply away from common approaches" to modeling the international states system:

Some scholars who attempt systems approaches to international politics conceive of a system as being the product of its interacting parts, but they fail to consider whether anything at the systems level affects those parts. Other systems theorists, like students of international politics in general, mention at times that the effects of the international environment must be allowed for; but they pass over the question of how this is to be done and quickly return their attention to the level of interacting units (Waltz 1979, 99).

In explaining this break from past traditions, Waltz asserts that "most students, whether or not they claim to follow a systems approach, think of international politics" as a setting where states internally generate external effects which in turn effect the attributes of other states, such as in Figure 8. This view has states N1...3 exhibiting external effects and states X1...3 are interacting with one another. Better, says Waltz, to think of the international states system as shown in Figure 9. In the latter case, there is a structure of the international states system which not only is formed by effects of states, it in turn can assert forces that affect the attributes of the states themselves (Waltz 1979, 100).

Figure 8: System view ignoring system effects.
Figure 8: System view ignoring system effects.
Figure 9: Waltz's view of the system.
Figure 9: Waltz's view of the system.

Structural Implications

Having clarified the organization of the international states system, Waltz then explores its implications. Waltz's philosophy is that of neorealism, and Waltz sees states as finding relative gains more important than absolute gains (Waltz 1979, 105). Believing that "political structures account for some recurrent aspects of the behavior of states and for certain repeated and enduring patterns" (Waltz 1979, 117), Waltz finds a systems approach a convenient vehicle for his neorealist views:

Wherever agents and agencies are coupled by force and competition rather than by authority and law, we expect to find such behaviors and outcomes... closely identified with... Realpolitik. The elements of Realpolitik, exhaustively listed, are these: The ruler's, and later the states's, interest provides the spring of action; the necessities of policy arise from the unregulated competition of states; calculation based on these necessities can discover the policies that will best serve a state's interests; success is the ultimate test of policy, and success is defined as preserving and strengthening the state" (Waltz 1979, 117).

A state's actions can moreover be predicted, as he explains in a later work, "when we can answer this question with some confidence: How would we expect any state so placed to act" (Waltz 1986, 332)? By "so placed," Waltz is referring to the place a state occupies in the international system, again showing the importance he places on the effects generated by the latter. Waltz's "new" systems approach to international relations therefore continues to advance the "old" idea of balance-of-power: in the system there exist "unitary actors who, at a minimum, seek their own preservation and, at a maximum, drive for universal domination," but the presence of structure, which is not formed by the premeditated action of any particular state, becomes a "constraint" on the actions of states, forming a balance of power (Waltz 1979, 118).

The balance of power coupled with systems theory then "predicts that states will engage in balancing behavior, whether or not balanced power is the end of their acts" (Waltz 1979, 128). Waltz does not stop there. By assuming that all states are more or less alike in functionality but defer in capabilities (Waltz 1979, 105), Waltz greatly simplifies his approach to analyzing balances of power in the international system. Statements can be made about the arrangement of the units (structure) in terms of the capabilities (or power) of each unit. To Waltz, stability is the ability of a system to maintain the number of principle parties (those with certain relative power levels) while the system remains anarchic (Waltz 1979, 162). If the arrangement of units (structure) works to create a balance of power, there must be particular arrangements of units with certain capabilities which are more effective at achieving this stable balance than others, he surmises.

Waltz indeed finds this to be the case. A world with two great powers, he declares, is the most advantageous (Waltz 1979, 161), because a bipolar world is "dynamically stable" (Waltz 1979, 177). He brings forth a number of reasons for this. "In multipolar systems," he says, "there are too many powers to permit any of them to draw clear and fixed lines between allies and adversaries and too few to keep the effects of defection low. With three or more powers flexibility of alliance keeps relations of friendship and enmity fluid and makes everyone's estimate of the present and future relations of forces uncertain" (Waltz 1979, 168). If several units exist with large relative capabilities, the wide choice of partners for alignment (which naturally occurs in the power-balancing process) can "make for rigidity of strategy or the limitation of freedom of decision." While a system with only two powers reduces the options for alignment (there are no other units to align with), this gives greater freedom of decision to each of the powers (Waltz 1979, 169-170).

A bipolar world has no peripheries — anything that happens is of concern to both of them (Waltz 1979, 171). Furthermore, if the system in which only two powers "united in their mutual antagonism far overshadow any other," the prudence of a tempered response in any crisis will be most evident, and the powers will be "wary, alert, cautious, flexible, and forbearing" (Waltz 1979, 173). "Pressure to moderate behavior is heavy" in this situation, and bargaining is made easier. "The simplicity of relations in a bipolar world and the strong pressures that are generated make the two powers conservative" (Waltz 1979, 174). In short, Waltz finds a bipolar international system to be the most "dynamically stable" configuration available.

Out of Step with Waltz (or, Dancing to a Different Tune)

However important Waltz's ideas are to the study of international relations, he is not without critics. While acknowledging that Theory of International Politics is "one of the most important contributions" to this field (Ruggie, 138), many still feel that its formulation of the international system "is as yet only a very partial construction on which much work still needs to be done" (Buzan, 215). Seeing that Waltz's ideas hinge on the notion of dividing up the international system for purposes of analysis, many of his critics, including Barry Buzan, John Ruggie, and Richard Ashley, find fault with the exact method of this division, and seek if not to replace Waltz's theory then to modify it to correct its perceived faults.

Buzan Reconsiders

As Buzan notes, the two issues which immediately arose concerning levels of analysis are:

  1. How many, and what, levels of analysis should there be for international relations?
  2. By what criteria can these levels be defined and differentiated from one another?
  3. ...Once a levels scheme is established, one puts the pieces back together again to achieve a holistic understanding (Buzan, 201).

In examining the criteria for a definition of levels, Buzan observes that these levels can actually serve two purposes: ontological, defining levels for purposes of analysis; and epistemological, representing sources of explanation for outcomes (Buzan, 203). Buzan finds fault with Waltz most of all because his presentation makes no distinction between these two purposes of level separation. "This not only confined debate to two different types of levels (structure and unit), but also saw them as constituting the whole universe of levels" (Buzan, 207). Waltz "defined everything that is not structure as belonging to the unit... level," which "rammed the whole debate about levels into an inappropriate dyad, confining it to only two levels..." This "created a bloated and incoherent 'unit' level, to which Waltz paid relatively little attention" (Buzan, 208).

By pushing everything into two levels, unit and structure, Waltz therefore either ignores certain features of the system or relegates them to the unit level. Buzan worries that this loses several sources of explanation. One of these, interaction capacity, "is about the technological capabilities, and the shared norms and organizations, on which the type and intensity of interaction between units in the system, or within a unit, depend. These things clearly fall outside the meaning of structure, and represent a different source of explanation from it" (Buzan, 210). Another, process, is responsible for explaining international society, regimes, and the "many recurrent patterns [that] have been found in the system and subsystem level in these often very complex dynamics, including war, alliance, the balance of power, arms racing and the security dilemma, and the whole range of international political economy patterns arising from protectionist and liberal policies on trade and money" (Buzan, 211).

In reality, says Buzan, "No one level of unit or source of explanation is always dominant in explaining international events." "In international relations generally, all the levels are powerfully in play." It may even be possible to argue that "structures are mutually constitutive: states make the structure, and the structure makes states" (Buzan, 213). Waltz, admittedly, does allude to this situation considering the process of socialization:

A influences B. B, made different by A's influence, influences A... Each is not just influencing the other; both are being influenced by the situation their interaction creates... The behavior of the pair cannot be apprehended by taking a unilateral view of either member. The behavior of the pair cannot, moreover, be resolved into a set of two-way relations because each element of behavior that contributes to he interaction is itself shaped by their being a pair. They have become parts of a system (Waltz 1979, 74-75).

It is soon evident, though, that in his analysis Waltz relegates the result of this interaction to the system level and equates the overall system with structure in direct opposition to the unit level, furthering what Buzan feels is an artificial division into two levels.13

Ruggie Continues14

Ruggie also feels such a strict division into two levels is incomplete. "...An entire dimension of social totalities is missing from Waltz's model," Ruggie complains, "because he drops the second analytical component of political structure, differentiation of units, when discussing international systems" (Ruggie, 145). Differentiation of units should be the second component of structure, and should not be ignored as does Waltz; instead, it should be included because "it serves as an exceedingly important source of structural variation" (Ruggie, 146).

Unit differences are not all that are missing from Waltz's model. Borrowing from Waltz's own source, Durkheim, Ruggie discovers that while Waltz attends to Durkheim's notion of volume by giving importance to the number of great powers in a system, he ignores dynamic density. This "aggregate quantity, velocity, and diversity of transactions that go on within society" goes hand in hand with the interaction capacity Buzan felt was missing from Waltz's ideas. "But Waltz... banishes such factors to the level of process" (Ruggie, 151), which Buzan and Ruggie both note is relegated from cause to effect in his international scheme. Buzan and Ruggie at least agree that Waltz "goes too far" in his division of levels, making "unit-level processes... all product and... not at all productive" (Ruggie, 153).

Ashley: "Poor Neorealism"

No friend of neorealism, Richard Ashley sets out to "challenge not individual neorealists but the neorealist movement as a whole" (Ashley, 257). Although admittedly phrasing his arguments "in deliberately exaggerated terms" (Ashley, 257), he investigates the centrality of structuralism to neorealism via Waltz's rendition, along with claims that it "transcends the confines of utilitarianism, statism, and positivism — perhaps enriching them by disclosing their deeper historical significance," and finds primarily "disappointment."

The reasons he gives for his displeasure with neorealism sound similar to arguments advanced by those more friendly towards Waltz's ideas:

...Neorealists slide all too easily between two concepts of the whole, one structuralist... and one atomist and physicalist. The structuralist posits the possibility of a structural whole — a deep social subjectivity — having an autonomous existence independent of, prior to, and constitutive of the elements. From a structuralist point of view, a structural whole cannot be described by starting with the parts as abstract, already defined entities, taking note of their external joining, and describing emergent properties among them... By contrast, the atomist conception describes the whole precisely in terms of the external joinings of the elements, including emergent properties produced by the joinings and potentially limiting further movement or relations among the elements" (Ashley, 287).

Ashley sees Waltz's formulation as an attempt to fuse these two views, but with a result that ends up back in the atomist camp:

...Waltz understands "international structure"... as an external joining of states-as-actors who have precisely the boundaries, ends, and self-understandings that theorists accord to them on the basis of unexamined common sense. In turn — and here is the coup — Waltz grants this structure a life independent of its own independent of the parts, the states-as-actors15; and he shows in countless ways how this structure limits and disposes action on the part of states such that, on balance, the structure is reproduced and actors are drown into conformity with its requisites. But how is the independence of this structural whole established?... Waltz establishes the independence of the structured whole from the idealized point of view of the lone, isolated state-as-actor, which cannot alter the whole and cannot rely on others to aid it in bringing about change in the whole's deepest structures (Ashley, 287-288).

Criticism of Waltz, from friends and foes alike, seems therefore to have a common thread. The main complaint seems to be that Waltz "goes too far" (Ruggie, 153) in his dissection of levels, pushing everything into two divisions, the levels of unit and structure. This artificial division leaves out important elements which shape the international landscape, such as methods and frequencies of interactions between units. In forcing explanations into artificial categories, some features gain exaggerated roles or are lost altogether, with miscellaneous items being relegated to the unit level (Buzan, 208) and the structural level taking on a "life independent of its own independent of the parts" (Ashley, 287), completely controlling and receiving no influence from the units themselves (Ruggie, 153).

Has Waltz went overboard in the sea of structuralism? In rejecting reductionism, has Waltz ironically analyzed and dissected the system to such an extent that he has reduced the whole to merely another part? Has Waltz's structure merely become another unit in the system, meaning that Ashley is correct in saying that really, "Waltz's is an atomist conception of the international system" (Ashley, 288)? If so, what effects does this bring about in the analysis of international relations? Furthermore, can recent advances in the study of nonlinear complex dynamic systems shed any light on the situation?

Chapter 3: Balance of Power Cellular Automaton

in order to investigate the picture of international system stability set forward by Waltz, this study has taken a cue from the field of complexity theory and designed a cellular automaton which encodes some of the basic assumptions of modern neorealist thought. This CA, as with CA in general, is not meant to accurately reflect the exact configuration of any real-world situation16. Rather, the value of CA, as discussed earlier, is that they can give a general idea of the activities of a system as a whole while taking into consideration the behavior of individual components.

This cellular automaton models the international states system as a one-dimensional row of cells, each of which represents a state.17 Following Waltz and neorealists in general, the following assumptions are made when constructing the model:

While these assumptions are not universally reflected in the real world, they more or less represent what neorealists expect to happen as a general rule. More relevant to this discussion, they are reasonably faithful to the assumptions of Waltz's theory, allowing one to investigate the predictions about configuration and stability set forth by that theory.

The above assumptions were encoded into the cellular automaton in the following manner. Each cell, representing a state, begins with a particular power level and a growth factor. For each iteration:

  1. The cell's power level is multiplied by its power factor to derive the cell's new power quantity. That is, power = power × growthFactor.
  2. If a cell's power level makes its position out of balance, then its power and growth factor are divided between the neighboring cells. Specifically, an imbalance occurs when the cell's power level, divided by the sum of the power levels of each neighboring cell, exceeds the imbalance factor in both cases. If the power levels of both neighbors combined can meet or exceed the imbalance factor, the cell's power level and growth factor are reduced by half the difference between the cell's power and the largest neighbor's power, and distributed between neighboring cells based upon their relative power levels. The cell's growth factor is reduced and distributed in the same proportion. That is, if all the following conditions exist:
    1. power / powerneighbor1imbalanceFactor
    2. power / powerneighbor2imbalanceFactor
    3. power / (powerneighbor1 + powerneighbor2) < imbalanceFactor

    Then the following actions occur:18

    1. balanceDifference = (power - mostPowerfulNeighbor) / 2
    2. growthDifference = growthFactor × (balanceDifference / power)
    3. power = power - balanceDifference
    4. powerneighbor1 = powerneighbor1 + (powerneighbor1 / (powerneighbor1 + powerneighbor2) × balanceDifference)
    5. powerneighbor2 = powerneighbor2 + (powerneighbor2 / (powerneighbor1 + powerneighbor2) × balanceDifference)
    6. growthFactor = growthFactor - growthDifference
    7. growthFactorneighbor1 = growthFactorneighbor1 + (growthFactorneighbor1 / (growthFactorneighbor1 + growthFactorneighbor2) × growthDifference)
    8. growthFactorneighbor2 = growthFactorneighbor2 + (growthFactorneighbor2 / (growthFactorneighbor1 + growthFactorneighbor2) × growthDifference)

However complicated this set of equations may initially appear, they are in reality one of the simplest mathematical encapsulations of the basic tenants of Waltz's structuralism. Indeed, in relation to the real-world international states system, this model is orders of magnitude simpler than even the simplest unit.

Balance of Power Cellular Automaton Result Analysis

As is common with complex systems and their models, analysis of the results of only a few simulation configurations could fill many volumes.19 Here the emphasis will be on how results apply to Waltz's predictions, as well as what they imply about the international states system in general. Note also that all simulations performed here use a row of ten cells over 50 iterations (not including initial conditions) with initial growth factors all set to 1.25 and a critical imbalance threshold factor of two.20 (That is, states all initially grow at a rate of 125% per iteration, and there is a power imbalance if any state's power is twice as much as both neighbors combined.) A shaded cell indicates a great power.

Perhaps the most straightforward case is that of a single great power (cell #3), whose power more than exceeds the critical imbalance factor selected:

1 2 3 4 5 6 7 8 9 10
10 10 50 10 10 10 10 10 10 10

As intuition might lead one to predict, the result after 50 more iterations is that, with all cells growing at the same rate, none are able to reach a point powerful enough to balance the power of cell #3 (see Figure 10). This does not necessarily contradict Waltz; although in general he claims that a two-power system is more stable than others, he does not preclude the possibility of a single controlling power.

Figure 10: [10, 10, 50, 10, 10, 10, 10, 10, 10, 10]
Figure 10: [10, 10, 50, 10, 10, 10, 10, 10, 10, 10]

Next consider a case in which there are initially two great powers. Cell #3 still exceeds the imbalance factor of both neighbors, although not to so great an extent. Cell #9 shows a similar situation.

1 2 3 4 5 6 7 8 9 10
10 10 30 10 10 10 10 10 30 10

After 50 more iterations (Figure 11), there are, as Waltz would predict, still only two great powers in the system.21 Note that one of the two final powers is not the same as in the initial conditions, as Waltz expected.22

Figure 11: [10, 10, 30, 10, 10, 10, 10, 10, 30, 10]
Figure 11: [10, 10, 30, 10, 10, 10, 10, 10, 30, 10]

Things get more interesting with different starting configurations. Cell #3 is again made so great a power that its neighbors cannot overcome it, as is cell #5. Cells #7 and #8 have large powers, but they are not "great" powers because they do not exceed the critical imbalance factor with their immediate neighbors.

1 2 3 4 5 6 7 8 9 10
10 10 40 10 30 10 30 30 10 10
Figure 12: [10, 10, 40, 10, 30, 10, 30, 30, 10, 10]
Figure 12: [10, 10, 40, 10, 30, 10, 30, 30, 10, 10]

At the end of the simulation (Figure 12), hardly any expectations are fulfilled; four great powers have appeared. And to further the confusion, consider the results of changing the initial power of one, seemingly inconsequential, unconnected cell, cell #1. This "state" does not have great power status, and is remote from all great powers:

1 2 3 4 5 6 7 8 9 10
15 10 40 10 30 10 30 30 10 10
Figure 13: [15, 10, 40, 10, 30, 10, 30, 30, 10, 10]
Figure 13: [15, 10, 40, 10, 30, 10, 30, 30, 10, 10]

This one, seemingly inconsequential change in the initial configuration of a minor player in the system has altered the eventual configuration of great powers (Figure 13); instead of two great powers (as the system began and Waltz would predict to continue) or four great powers (as one might expect from the previous simulation), this small change has propagated through the system to bring about a stable configuration of three great powers.

Balance of Power Cellular Automaton Result Implications

After only a few CA simulations, a few implications are apparent:

  1. Instead of repeatedly showing stability only when the system contains two great powers, as Waltz predicted, the system exhibited several stable configurations.
  2. The number of great powers in each stable configuration was highly dependent on the initial configuration and arrangement of powers.
  3. At times a change in even a non-great power's initial state could influence the eventual number and configuration of great powers.

These observations imply that an analysis of the international states system as complex nonlinear dynamical system would be profitable, and brings insight into the shortcomings of Waltz's model as well.23 To begin with, Waltz's assertion that in analyzing structure, outcomes are determined "not by all of the actors that flourish within them but by the major ones" (Waltz 1979, 93) is incorrect.24 Small actors can have eventual large effects on the entire system, and ignoring their presence during analysis can radically skew expected effects.

Furthermore, a system's eventual stable configuration in the future cannot be determined simply from its present arrangement of capabilities — its past configurations must also be taken into account. In other words, the system's future configuration is highly dependent on its initial conditions. Such a concept is missing from Waltz's scheme, with drastic consequences that Waltz failed to comprehend.

"Ashley accuses me of excluding history from the study of international politics," Waltz complained. "How can one incorporate history into the type of theory I constructed" (Waltz 1986, 340)? Complexity theory reveals the answer: "...the states of a complex system are determined not only by external circumstances, but also by the history of the system" (Cilliers, 66). A complex system can be said to have several attractors, and a neural network can have several configurations which provide expected outputs, but the specific attractor or configuration reached will depend on its initial configuration. In fact, the international states system at any point in time could be said to incorporate its entire history in its configuration, in the same way a neural network encodes meaning in a holistic combination of its values.

Waltz, in attempting to create a holistic model, transformed his whole, the structure, into yet another component of the system. The entity of structure was assumed to take complete control over the units, as Ruggie explained, with little or no interaction between them. Waltz failed to realize that while unit interactions create emergent properties of the system, the units can in turn interact with emergent properties producing more emergent properties, and these emergent properties can in turn interact, and so on ad infinitum. Even though an attempt was made at creating a holist picture of the international system, Waltz's model was, as Ashley illustrated, ultimately atomist and reductionist.

The most crucial problem with Waltz's picture of the world, therefore, is that, after going to great lengths to prove the international states system complex, he draws conclusions based upon a linear view of system processes. Waltz does at times comes close to discovering his oversight:

The three-body problem has yet to be solved by physicists. Can political scientists or policy-makers hope to do better in charting the courses of three or more interacting states? Cases that lie between the simple interaction of two entities and the statistically predictable interactions of very many are the most difficult to unravel (Waltz 1979, 192).

But in the end Waltz chooses a model that is essentially linear and makes predictions based upon assumptions of linearity, not heeding his own warnings. For this oversight Waltz should not be severely chastised, for although Poincaré first noticed complexity and chaos in the n-body problem over 100 years ago, it was not until recently that such phenomena was at all studied or understood. Indeed, no one had ever even seen the intricate image of the Mandelbrot Set until only a year after Waltz's Theory of International Politics was first published.25 Waltz's Theory is a solid first step based upon the underlying principles that were soon to start the study of complexity. Waltz was guilty of oversight, not blindness, and a realization of the shortcomings of his model can provide a view of new directions for international relations theory.

Chapter 4: New Directions

As noted earlier, the purpose of this work is not so much to reach a new destination but to point out new directions and paradigms that may be helpful in investigating the international states system. This study has shown how complexity theory can bring to light shortcomings in present models, particularly the neorealist structural model made popular by Waltz. The following is but a sample of the multitude of areas of international study to which complexity theory can bring new perspectives.

Bifurcations and Chaos Control

Waltz, while noting that "structures... may suddenly change" (Waltz 1979, 70), relegated the cause of this to the individual units of a system (Waltz 1979, 72). In recounting how complexity theory can be applied to the social sciences, David Byrne points out that it is common in a complex system for transitions between states to be rapid, not gradual, and that this reflects "small scale perturbations in controlling variables" as in the "Lorenz or butterfly attractor" (Byrne, 23). While some changes such as the collapse of Soviet communism do not endanger world peace, other sudden events such as the beginning of World War I from its small catalyst are less desired.

Peak and Frame relate how an understanding of underlying system dynamics can allow small, appropriately-placed perturbations to restrain a system from a chaotic attractor to a stable orbit, something that could prove useful in stabilizing airplane wings, for example (Peak and Frame, 233). Byrne furthers this position by postulating that by recognizing bifurcation points, there is a possibility not only to"maintain the stability of the system" but to bring about a "transformation of the system to a better of two alternatives" (Byrne, 41). Although the study of "control of chaos in dynamical systems is still in its infancy" (Peak and Frame, 239), further investigation has the potential to reveal the extent to which NGOs, the UN, and other organizations provide positive and/or negative feedback to control or further chaos in the complex international system.

The Feigenbaum Number and International Politics

In analyzing the logistic map discussed earlier, Mitchell Feigenbaum discovered that its period-doubling path to chaotic behavior had certain universal characteristics. As one goes deeper and deeper into the process of bifurcations, the spacings between bifurcations approximate the fixed ratio 4.6692..., the Feigenbaum Number. Further investigation has revealed this same ratio in electronic circuits, fluid flow turbulence, and contractions in chick heart muscle cells.

Most relevant here is the work of Alvin Saperstein, who analyzed the ratio of armaments in the Soviet Union to those in Nazi Germany leading up to World War II. Although working with limited data, he believes he has found some evidence for period doubling in the ratio of the Feigenbaum number, and even asserts that this ratio may have achieved full chaotic behavior by the time the war started (Peak and Frame, 176-186). Only further study will reveal the relevance of the Feigenbaum Number to the events on the international stage.

Complexity and Emergence

Complex systems do not only exhibit chaos — they can also, as the patterns of cellular automata and the learning of neural networks demonstrate, create emergent properties that "cannot be obtained by summing the behaviors of [their] constituent parts" (Holland, 122). The most recent studies of complex systems are examining how these systems "undergo spontaneous self-organization" (Waldrop, 11), reminiscent of Waltz's observation that, in the international states system, "Order may prevail without an orderer..." (Waltz 1979, 77). Further study of how and why emergent properties form in complex systems may provide better descriptions of international configurations such as those seen by Waltz26 .

Complexity and the Neo-Neo Debate

One of the main disputes between neorealists and neoliberals is over the ease and likelihood of international cooperation: neorealist believe that, since states are more concerned about their own interests than the interests of others, cooperation is harder to achieve and maintain than do neoliberals (Baldwin, 5). In the late 1970's Robert Axelrod27 held a computer tournament to study cooperation with an iterated Prisoner's Dilemma28, a model related to the cellular automata discussed earlier. The strategy that out of all submitted entries gave the highest payoff over 200 iterations was named "TIT FOR TAT" and was "nice" in the sense that it would never defect first and was "forgiving" by rewarding good behavior with cooperation (Waldrop, 264).

Further investigations by Axelrod involved using a genetic algorithm29 to literally evolve strategies in the iterated Prisoner's Dilemma that would yield the highest payoff for a particular agent. In a computer simulation using twenty individuals per generation over 151 iterations, the median agent (from a random start) evolved strategies that acquired better results than TIT FOR TAT, yet had many cooperative traits in common, including "don't rock the boat," "forget," and "accept an apology" (Axelrod, 20). Complexity theory may therefore bring more insight into how much of a contradiction actually exists between self-help and cooperation in the international states system.

Complexity and Postmodernism

Jean-Francois Lyotard's The Postmodern Condition defines postmodern as "an incredulity towards metanarratives," implying that "post-modernism is essentially concerned with deconstructing, and distructing any account of human life that claims to have direct access to 'the truth.'" Richard Ashley has performed a typical postmodern "double reading" of the concept of anarchy and found the "seemingly natural opposition between anarchy and sovereignty... is in fact a false opposition" (Smith, 181).

After noting the way in which, for example, a neural network has representation not in any particular value in a node but in a holistic combination of the presence of all nodes — that is, representation is not local but distributed — Paul Cilliers has argued "that the postmodern approach is inherently sensitive to complexity, that it acknowledges the importance of self-organization whilst denying a conventional theory of representation" (Cilliers, 113). Seeing the international states system as a complex nonlinear dynamical system may show more ways in which postmodern philosophy can be applied to the study of international relations.

Simply Important

Adam Smith and Karl Marx30 each spent years creating a grand scheme that would describe the world they knew. Any shortcomings in their models do not invalidate the important contributions each has made to the understanding of the economic world, the social world, and even the international political world. While those after them have not been content to accept their unaltered models as perfectly clear pictures, they have been able to extend significant portions of their respective ideas in the quest for a better understanding of society.

It should not be overlooked, however, that the concepts set forth by each of these thinkers did have their share of flaws — limitations that ultimately made their predictions fail. An investigation of the complexities of the world, a study of the properties which emerge in highly connected and interactive systems, can not only help create better theories but can provide a healthy perspective of how much a theory of such a system can be expected to predict, whether one is attempting to describe worldwide economic patterns or the configuration of international power concentrations.

The language of complexity theory can transform discussions of international relations and bring about completely new perspectives on old postulates. Did Marx's predictions ignore strange attractors in the economic world? Is Fukuyama's thesis of the liberal democracy as a stable attractor in the world reasonable or likely? Did the international states system become more complex around the turn of the 20th century, allowing a single event to illustrate connectionistic chaos, or was the problem instead a temporary reduction of complexity, and is recent stability related to the emergent stabilizing properties of NGOs and other international organizations? Although shifting paradigms can initially lead to more questions, not fewer, such transformations of viewpoint have the potential to bring to light artificial discrepancies and incorrect assumptions, as well as lead to more productive and unified outcomes in the future.

Complexity theory is in its infancy, and complex nonlinear dynamical systems are only now beginning to be understood. The chaos and order, the surprises and monotony, the instability and solidity of such systems are only now being discovered, and many applications of complexity theory to international relations have yet to be explored. Nevertheless, it is evident that the nonlinear aspects of international politics can have a great impact. In the international states system, complexity cannot simply be ignored.