Emergent Behavior of Composite Systems

Complex behaviors of systems are one of the most fascinating things in science. By understanding how existing systems work, we aim to predict behavior and hopefully design an useful mechanisms, leveraging the modeled behaviors.

What I find even more fascinating is when the properties of the aggregate do not stem, at least in an obvious manner, from the properties of the individual components. These situations illustrate how our intuition is not foolproof and how we often misjudge system behavior even when the individual behavior is know, and vice-versa.

Conway's Game of Life

The Game of Life, as described by the mathematician John Conway, is a classic example of how complex behavior can emerge from seemly simple rules at an individual level.

It goes as follows: Let there be an infinite 2-dimensional grid of cells, where each cell might be either alive or dead. The game proceeds in turns, advancing the state of the grid according to the last grid state and a small set of rules. Note the 8 closest cells are called the neighbors of a cell. The rules go as follows:

• Any live cell with one or no neighbors “dies”, becoming dead. (“underpopulation”)
• Any live cell with two or three neighbors continues to be alive on the next state.
• Any live cell with four or more neighbors “dies” and becomes dead. (“overpopulation”)
• Any dead cell with exactly three neighbors becomes a live cell. (“reproduction”)

The Wikipedia page has a number of animations showing behavior of some initial states: https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life#Examples_of_patterns. It it not obvious at all that this set of rules would lead to these observed behavior.

Other examples

• Gambler's Fallacy - Even when each individual bet is independent of the previous and next ones, in aggregate there might be a most likely outcome, considering a large enough number of bets. Any perceived patterns in the aggregate system, which can be modeled as probabilistic models, do NOT imply one can predict any individual outcomes of a single bet. The fallacy stems from our intuition blurring the lines of individual and system, leading us to misattribute properties and behaviors between the two.

• Biological Systems - Atoms are not alive and are incapable of reproducing, but certain arrangements of atoms self-replicating behavior, making up cells, tissue, organs and organisms. These complex systems spend energy to perceive and react to their their environment, capture energy sources and self-replicate. It is fascinating how live beings can emerge from plain “dead” matter and this has puzzled humans since ancient times.

• Artificial Neural Networks - ANNs are mathematical models inspired by the biological brain structure. Each “artificial neuron” might be influenced by a few “neighboring” neurons in a weighted sum, in turn influencing downstream neurons leading to the network's output. Recent research has show how such models can be trained for classifying objects in images, transcribing speech to text, translating between images. It is not obvious how this structure allows for such general behavior and why each the individual pieces of the trained model ultimately lead to the aggregate outcomes that they do.

• Markets and the Economy - Post-hoc explanations aside, there is no complete predictive model of how the market works. The market is generally modeled as a system of a large number of adversarial individuals which take self-interested decisions, but this fails to account for psychology, flock behavior and irrational decisions. It is very hard to predict the aggregate movement of the Economy, since it emerges from complex interactions between individuals, which even by themselves are very complicated.