I present a definition of emergent behavior, and give a discussion why I think that the concept of emergent behavior is something you cannot express in mathematics. As a result, emergent behavior is a concept that only exists in the eye of the beholder, not in any mathematical form.
When a collection of individuals, each following specific rules are studied, unexpected behavior of the whole group may pop out of the system. Very simple examples are numerous:
All of these higher level behaviors cannot be described with the same words to describe the agents by themselves.
"Emergence is the process of deriving some new and coherent structures, patterns and properties in a complex system. Emergent phenomena occur due to the pattern of interactions between the elements of a system over time. Emergent phenomena are often unexpected, nontrivial results of relatively simple interactions of relatively simple components." (from http://en.wikipedia.org/wiki/Emergence)
In a discussion with Michael Vose and Alden Wright uponj my arrival at Schloss Dagstuhl, we were wondering whether you could find a mathematical way to distinguish emergent behavior from an individual's behavior. Or whether a mathematical definition of emergent behavior actually exists. Personally, I don't think you can define "emergent behavior" mathematically.
The emergent behavior of a group of individuals is inherently and implicitly defined by the system, as it is given by the individual rule base, and the combination of these rules when many individuals are put together. If you are able to describe the individuals' rules mathematically, and pop them together, the emergent behavior should be the result of this system. At least, mathematically.
The rules of each individual can be seen as the cause of the emergent behavior, and the emergent behavior itself is simply what follows from these rules. As the emergent behavior is implicitly defined by the behavior of the individuals, you cannot give a mathematical method to make a distinction between the global emergent behavior and the individuals' behaviors, as both systems are the same systems, except that we seem to use other words for the description of the behaviors at the different levels.
In contrast with this mathematical impossibility of not being able to define emergent behavior mathematically, a lot of scientists seem to be using this concept regularly. So, how can one detect "emergent behavior"?
Emergent behavior is obvious used as a term to denote the behavior of your system that you did not expect. When implementing a simulation of economic agents, you are not thinking about what might happen on a global scale when you put many of these agents together in your simulation. As the global behavior of the whole system is more interesting than what you would've expected, or as more comes out of the system than what you have put in, the thus discovered behavior is defined "emergent".
But what does this really mean? Mathematically, the emergent behavior is explicitly defined in the individuals' rules, but a scientist doesn't always figure out beforehand what behavior might pop out on a higher, more global level.
I think this is so because our brain is limited in many ways. More limited than a mathematical proof. The fact that we do not think of the implicit global behavior of the system when we look at the individiuals' rules, although these exist, makes us actually look more stupid than we are. We are surprised by the global behavior and thus call it emergent, although it has been programmed in the system, but we are not smart enough to anticipate it beforehand.
"Emergent behavior" seems to be a concept that only exists in the eye of the beholder. When the emergent behavior is first detected, it is often a surprising result from a simulation, mostly of a large set of individuals following their rules, put together in this one simulation. The whole seems to be bigger than the sum of its parts, but is this really true? From a mathematical point, the global emergent behavior is implcitly given by the individuals' rules. As a result, there is no way to make a mathematical distinction between what is emergent and what is not.
It seems like scientists that discover emergent behavior do not seem to be capable of anticipating this global behavior given the sets of rules of the individuals. After the emergent behavior has been discovered, a mathematical analysis of the system may indeed show the observed global behavior as a result of the individuals' rules being put together. As a result, the emergent behavior can be explained, and the surprise of discovering the global behavior fades away because of this mathematical derivation.
As such, I do believe that "emergent behavior" is only in the eye of the beholder, but is not a concept that could be taken lightly. If the concept of "emergent behavior" is used to describe an observed phenomenon that was not anticipated for, a mathematical analysis should be undertaken to show where this new global behavior originates mathematically. As such, the behavior loses its "emergent" adjective and becomes an explicit part of the system. "Emergent behavior" thus does not exist.