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

by John H. Miller

The behavior of systems as we transition between the continuous and discrete is often surprising. Many systems do not smoothly move between these two realms, but instead exhibit quite different patterns of behavior, even though from the outside they seem so “close.”

If heterogeneity is a key feature of complex systems, then traditional social science tools—with their emphases on average behavior being representative of the whole—may be incomplete or even misleading.

individual bee’s temperature thresholds for huddling and fanning are tied to a genetically linked trait. Thus, genetically similar bees all feel a chill at the same temperature and begin to huddle; similarly, they also overheat at the same temperature and spread out and fan in response.

Hives with genetic diversity produce much more stable internal temperatures. As the temperature drops, only a few bees react and huddle together, slowly bringing up the temperature. If the temperature continues to fall, a few more bees join into the mass to help out. A similar effect happens when the hive begins to overheat. This moderate and escalating response prevents wild swings in temperature. Thus, the genetic diversity of the bees leads to relatively stable temperatures that ultimately improve the health of the hive.

Here, average behavior leads to wide temperature fluctuations whereas heterogeneous behavior leads to stability.

Again, notice how average behavior is misleading. The average threshold of the heterogeneous hive is identical to that of the homogeneous hive, yet the behaviors of the two hives could not be more different. It is relatively difficult to get the homogeneous hive to react, while the heterogeneous one is on a hair trigger.

The difference of response between the two systems is due to feedback. In the temperature system, heterogeneity introduces a negative feedback loop into the system: when one bee takes action, it makes the other bees less likely to act. In the defense system, we have a positive feedback loop: when one bee takes action, it makes the other bees more likely to act.

In economics, formal modeling usually proceeds by developing mathematical models derived from first principles. This approach, when well practiced, results in very clean and stark models that yield key insights. Unfortunately, while such a framework imposes a useful discipline on the modeling, it also can be quite limiting. The formal mathematical approach works best for static, homogeneous, equilibrating worlds. Even in our very simple example, we are beginning to violate these desiderata. Thus, if we want to investigate richer, more dynamic worlds, we need to pursue other modeling approaches. The trade-off, of course, is that we must weigh the potential to generate new insights against the cost of having less exacting analytics.

Consider the earlier problem of modeling the weather. Rather than worrying about the position and characteristics of every atom in the atmosphere (an impossible task to be sure), we first use equivalence classes to collapse this space down to, say, measures of pressure and humidity. We next develop notions of how patterns of pressure and humidity are transformed over time.

We would like to be able to develop a theory that helps us understand how states of the world (composed of lower-level entities and interaction rules) are transformed into higher-level entities. Some initial work on this topic has been done with cellular automaton models, where it has been shown that under some conditions a variety of seemingly different interaction rules imply only a few distinct types of high-level behavior (Wolfram, 2002).

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.

As we have discussed, emergence is a phenomenon whereby well-formulated aggregate behavior arises from localized, individual behavior. Moreover, such aggregate patterns should be immune to reasonable variations in the individual behavior.

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. The remarkable implication of this theorem is that, for an amazing variety of underlying agent behaviors, the global behavior that emerges is described by a simply specified, common form.

The Law of Large Numbers works because as we add more and more independent agents to the world, the vagaries of the stochastic elements, quite literally, average out. With only a few agents, these stochastic elements make it impossible to predict with any certainty the aggregate behavior because individual variation overwhelms any potential predictability, but as we increase the number of agents involved in the world, individual variations begin to cancel one another out, and systemwide predictions become possible.

The key feature of disorganized complexity is that the interactions of the local entities tend to smooth each other out. In the case of the Law of Large Numbers, an unusually high value for one random value is compensated for by an unusually low value of another.

the vast majority of social science theory focuses on exactly these two types of outcomes. Nonetheless, there are many canonical examples of “large events” that arise in social systems, such as stock market crashes, riots, outbreaks of war and peace, political movements, and traffic jams.

While we have ample evidence, both empirical and experimental, that under organized complexity, systems can exhibit aggregate properties that are not directly tied to agent details, a sound theoretical foothold from which to leverage this observation is only now being constructed.

As theoretical tools become more widely accepted, their foundations tend to be rarely revisited and even get forgotten by new generations of users. This is unfortunate, as a clear knowledge of the foundations often enhance the resulting theory.

The emphasis here is on understanding nature, not on the tools used to gain this understanding.

While axiomatic rigor is not required for theoretical work in physics, there is still a high premium on good theory—just not on the tools used to develop the theory.

This relationship between mathematics and theory provides an interesting contrast to the norms that have developed in other fields such as economics.

The agent-based object approach can be considered “bottom-up” in the sense that the behavior that we observe in the model is generated from the bottom of the system by the direct interactions of the entities that form the basis of the model. This contrasts with the “top-down” approach to modeling where we impose high-level rules on the system—for example, that the system will equilibrate or that all firms profit maximize—and then trace the implications of such conditions. Thus, in top-down modeling we abstract broadly over the entire behavior of the system, whereas in bottom-up modeling we focus our abstractions over the lower-level individual entities that make up the system.

Anderson’s hypothesis suggests that even if we can fully uncover the microfoundations of behavior—for example, acquire a complete specification of the psychological aspects of behavior or the probability of interaction—we may still not have a simple way to understand their macrolevel implications.

good models tend to have a number of other properties: for example, their implications tend to be robust to large classes of changes in the underlying structure, they tend to produce “surprising” results that motivate new predictions, they can be easily communicated to others, and they are fertile grounds for new applications and contexts. As the structure of a model becomes more complicated, many of these desirable features are lost, and we move away from modeling toward simulation.

Computational models are often thought to be brittle, in the sense that slight changes in one area can dramatically alter their results. This fear is perhaps due to the experience of having a computer program crash after some seemingly innocuous input or alteration. Indeed such crashes are rather dramatic, though they are not unique to computational tools. For example, we see similar collapses in a mathematical model when we alter our assumptions about, say, the form of the utility function or the compactness of the strategy space.

The lack of equilibria in complex systems models does not imply a lack of regularities.

Models that settle into equilibrium tend to include primarily negative feedbacks. When a firm makes positive profits, other firms enter and wipe out those profits. Here, actions are offset by other actions. In contrast, systems that generate complexity tend to include positive feedbacks as well. When one politician takes a new policy position, it creates incentives for other politicians to move as well, and when those politicians move, they set in motion an endless sequence of further movements.

However, in systems with positive feedback, we loose some predictability. Small differences can build upon themselves and create large differences, making precise prediction difficult.

“We repeat most emphatically that our theory is thoroughly static. A dynamic theory would unquestionably be more complete and therefore preferable. But there is ample evidence from other branches of science that it is futile to try and build one as long as the static side is not thoroughly understood.”

Unlike traditional tools, computational methods are able to incorporate heterogeneous agents easily.