Complex Adaptive Systems: An Introduction to Computational Models of Social Life (Princeton Studies in Complexity Book 14)

by John H. Miller

Complexity is a deep property of a system, whereas complication is not. A complex system dies when an element is removed, but complicated ones continue to live on, albeit slightly compromised. Removing a seat from a car makes it less complicated; removing the timing belt makes it less complex (and useless). Complicated worlds are reducible, whereas complex ones are not.

For every complex problem, there is a solution that is simple, neat and wrong. —H. L. Mencken Things should be made as simple as possible—but no simpler. —Albert Einstein Nothing is built on stone; all is built in sand. But we must build as if the sand were stone. —Jorge Luis Borges

This ability to ignore is a crucial component of scientific progress as it allows us, just like the parent trying to stop the endless regress of a three-year-old’s “why” questions, to say “just because.” Of course, the art of good science is knowing when to say “just because,” for if we are able to invoke that incantation at the right moment, the sand underlying our model’s foundation will turn to stone.

Good modeling requires that we have just enough of the “right” transparencies in the map. Of course, the right transparencies depend on the needs of a particular user.

Here we present a more formal approach to the ideas underlying modeling—a model of modeling. Our discussion relies on the mental modeling ideas developed by Holland et al. (1986) for creating artificial learning systems. We focus on trying to model a world that varies over discrete time steps (though one could obviously apply these ideas to other types of systems). The basic outline of the underlying ideas are presented in figure 3.2. In the top half of the figure, we represent the real world that we are interested in modeling; in the lower half, we depict our model.

The requirement that the maps between the model and the real world must be commutative in this way is known as a homomorphism. Thus, the goal of modeling under this view is to find a set of equivalence classes and a transition function that results in a useful homomorphism.

One view of modeling complex systems, which (at least implicitly) is held by many scientists, is the reductionist hypothesis. This hypothesis suggests that, if we can just get the right simplifications in the model, we will understand everything—if true, then the world around us, including the social world, is “just particle physics.”

Of course, as pointed out by Anderson (1972), the fallacy here is that the reductionist hypothesis does not imply a “constructionist” one. Even if we know the fundamentals of a particular system, we may not be able to use that knowledge to reconstruct higher-level systems. It may be, as Anderson says, that “the whole becomes not only more than but very different from the sum of its parts.”

Models need to be judged by what they eliminate as much as by what they include—like stone carving, the art is in removing what you do not need. Even though a computational model may require thousands of lines of code, if done well it can still embody the simplicity and elegance that is demonstrated in a mathematical model existing in only a few equations.

1“The Cartographers Guilds struck a Map of the Empire whose size was that of the Empire, and which coincided point for point with it. The following Generations, who were not so fond of the Study of Cartography as their Forebears had been, saw that that vast Map was Useless, and not without some Pitilessness was it, that they delivered it up to the Inclemencies of Sun and Winters.” Jorge Luis Borges and Adolfo Bioy Casares, On Exactitude in Science (1946), English translation from Jorge L. Borges, A Universal History of Infamy (London: Penguin Books, 1975).

He intends only his own gain, and he is in this, as in many other cases, led by an invisible hand to promote an end which was no part of his intention. —Adam Smith, Wealth of Nations

Any sufficiently advanced technology is indistinguishable from magic. —Arthur C. Clarke, Profiles of the Future

The usual notion put forth underlying emergence is that individual, localized behavior aggregates into global behavior that is, in some sense, disconnected from its origins. Such a disconnection implies that, within limits, the details of the local behavior do not matter to the aggregate outcome.

The theorem, the Law of Large Numbers (and its various offshoots, including the Central Limit Theorem), was developed by statisticians over the past few hundred years. It is of interest because it provides some relatively general conditions under which a certain type of aggregate behavior can emerge from the stochastic, microlevel actions of individual agents.

Thus, in such systems there is a stable, aggregate property (here the expected value of the common distribution) that emerges from aggregating the activities of the agents.

The Central Limit Theorem provides another example of such a result (see figure 4.2). If we add the assumption that the variance of the common distribution is finite, then the distribution of the average (our aggregate property) will converge to a well-known “normal” form.

Thus, in cases of disorganized complexity, it should be easy to derive fairly precise emergence theorems based on fundamental concepts that are centuries old. Unfortunately, disorganized complexity accounts for only one part of our world.

Explorations of complex systems have begun to identify the emergent properties of interacting agents—for want of a better term, organized complexity. We often see unanticipated statistical regularities emerging in complex systems. These regularities go beyond the usual bounds covered by Central Limit Theorems and such.

The use of computers seems thus not merely convenient, but absolutely essential for such experiments which involve following the games or contests through a very great number of moves or stages. I believe that the experience gained as a result of following the behavior of such processes will have a fundamental influence on whatever may ultimately generalize or perhaps even replace in mathematics our present exclusive immersion in the formal axiomatic method. —Stanislaw Ulam, Adventures of a Mathematician

The ability to analyze systems of “adaptive” agents systematically is an area of great promise for social scientists, but it does face a potentially serious scientific challenge: can we create a coherent science of adaptive agents? One advantage of optimization-based models is that there is typically only one way for an agent to be optimal while, as we all have experienced at one time or another, there appears to be an infinity of ways for an agent to be “dumb.”

model classes on actual human behavior, using trained models that are essentially limited, hampered in their ability to reach optimal outcomes, in proportion to human actors...

Many of our existing analytic tools avoid an emphasis on dynamic processes and focus on equilibrium states. When transition paths are short and conditions are stable, such an approach may provide a good description of the world. In natural systems, however, equilibria are usually associated with the death of the system. The conditions that favor equilibrium analysis are likely the exception rather than the rule in many complex adaptive social systems. If so, the techniques that we traditionally use to analyze these systems may be like trying to “understand running water by catching it in a bucket.”

Whether we actually need to model heterogeneity is an important research question. It may be the case that given sufficient agent heterogeneity, the aggregate behavior of the system may no longer depend on the various details of each agent, and abstracting this behavior into a single representative agent is feasible.

A second area in which we are forced to simplify our models is in the amount of asymmetry we have in the system. Like homogeneity, symmetry assumptions dramatically simplify calculations, and so they are used even though asymmetry may be a pervasive and influential feature of social systems. We know from some mathematical models that simple asymmetries in, say, information can alter our prediction of a single, well-behaved equilibrium point to one where there are multiple equilibria linked to, say, agent expectations. Computational models that use agent-based objects can easily accommodate asymmetries.

An Eightfold Mapping of Agent-Based Object Models Path Focus View Information and connections Intention Goals Speech Communication among the agents Action Interaction Livelihood Payoffs Effort Strategies and actions Mindfulness Cognition Concentration Model focus and heterogeneity

Far better an approximate answer to the right question, which is often vague, than the exact answer to the wrong question, which can always be made precise. —John Tukey, Annals of Mathematical Statistics

Few things are harder to put up with than the annoyance of a good example —Mark Twain, The Tragedy of Pudd’nhead Wilson

It is all very beautiful and magical here—a quality which cannot be described. You have to live it and breath it, let the sun bake it into you. The skies and land are so enormous, and the detail so precise and exquisite, that wherever you are, you are isolated in a glowing world between the macro- and the micro- where everything is sideways under you, and over you, and the clocks stopped long ago. —Ansel Adams, Letter to Alfred Stieglitz

The introduction of noise into a model provides another example of the interest in between. In noiseless systems, agents quickly get stuck, often in inferior configurations. With a lot of noise in the system, chaos reigns, and little, other than frenetic movement, is possible. Adding just a bit of noise to a system, however, often induces order and leads it toward optimization.