A Complex system is a collection of agents (i.e., objects, actors, or quantities) and the relationships among them, which when taken together, form an entity that can be represented as a distinct category of events in the environment.
Because interactions occur among agents at a lower level of scale than the overall system, these interactions are called fine-grained. An example of this is how interactions among individual genes results in the expression of traits in the phenotype of the relevant organism.
In the case of human beings, individuals interact with others to form inter-generational families, groups of only distantly related individuals, firms, and governments and to create art, evolve scientific knowledge, develop technologies, and discover economic and political innovations.
The common theme is that in general, fine-grained interactions are too difficult or too numerous to precisely measure, model and monitor in a timely manner. Further, the many interaction effects that are being expressed locally might be too “complex” in that they make it impractical or even impossible to measure, model and monitor each individual interaction to specify the precise nature of its impacts on the overall complex system.
This modeling challenge, measured by what is called descriptive complexity, is highlighted in the realization that complex systems are dynamic and always changing. As such, to be meaningful in any practical way, this work would need to be done in a timely manner, and completed before the system played out on its own. This would be the only way that the resulting outputs could be used to intervene in the ongoing dynamics of the system before it’s too late to bother. If the work can’t be done in a timely manner, the unfolding of the complex system is itself the most efficient model of itself. In this can would be most efficient to just wait for the system itself to unfold.
How Interacting Agents Form a Complex System
A collection of agents (i.e., objects, actors, or quantities) and the relationships among them can, when taken together, form a complex system. However, such a “system” is only relevant to an observer if it can be recognized as a distinct conceptual entity, a recognizable category of events within a broader ecosystem that is relevant to an observer.
A collection of agents and the relationships among them are relevant if patterns can be recognized at the system level and if these can be identified and their outputs predicted before future events unfold. For example, a collection of young men in tight formation walking in the same direction with resonating cadence and wearing similar athletic apparel could be recognized as a “team” category of some sort. Therefore, based on generic models related to this team category and its outputs, it would not be unreasonable for an observer to predict that “the collection of agents” would cross the street together. The observer could thus preemptively adjust one’s gait with respect to crossing the street in a manner that resonate with their rhythms and patterns. This would tend to limit what might be called “interference costs.”
A Complex System Defined
Thus, a complex system is a collection of agents (i.e., objects, actors, or quantities) and the relationships among them that can be recognized as an entity. This occurs when an observer can predict some pattern or regularity in its collective behavior before events unfold. It is a category or type that is relevant to an observer as an entity in the ecosystem. Often, these regularities are identified through attractors in dynamical systems models. A swarm of honeybees is recognized as a complex system in the biological sciences, for example, because distinct collective properties of the swarm can be observed, but also because there is value for some observer in predicting the behavior of the swarm as an entity. Likewise, short term weather patterns like typhoons and longer term climate patterns like droughts are recognized and modeled as physical complex systems.
Complex Systems and Information
Put in terms of information theory, complex system must 1) encode information into their interaction structure, which they do in the patterns and regularities that emerge, and 2) that information must be decoded by an observer, which occurs when a categorical representation enables the observer to use the decoded information to predict the system’s behavior before it occurs. The first of these is called the ontological condition and implies that complex systems are models of themselves; the latter is called the epistemological condition and implies that the complex system has been modelled by an observer.
 Crutchfield J.P. & Feldman, D.P. (1997). Statistical complexity of simple one-dimensional spin systems, Physics Review E 55 (2), 1239-1242
 Hidalgo, C. (2015). Why Information Grows: The Evolution of Order, from Atoms to Economics. New York: Basic Books.