Introduction to Complexity

The world is a mechanical masterpiece. Many parts work independently to author a shared existence. In our attempt to understand that existence, we have alternated between forms of narrative and experimentation, trusting both our senses and our intuitions to expand our control of the universe. In both forms, the detail of a system is viewed as the key to understanding. In a complex system, however, the detail becomes non-specific and generalized for the entire system. It is a special circumstance, one that relies on environmental factors and is characterized by universal properties.

I had to write a paper in my Complexity seminar with Alex Vespignani to explore my understanding of complexity, as seen through the two previous sessions discussions and our assigned readings. I opted to structure it around a personal use for the information, starting by listing some of the component concepts — the primitives of complexity — that comprise and define the domain. The study of complexity has origins in several domains and relies on distinct areas of knowledge. These areas manifest as universal properties, but individually do not define a complex system.

Emergence
In a deterministic view of the world, parts are made of smaller parts that, if fully known, can be manipulated to construct an intentioned reality. The world of a determinist falls into place like dominoes and can be reconstructed from any point in time simply by knowing entirely the value of a single state.

The problem with this with world-view is two-fold. First, the number of elements contained within a given system is too large to apply any equation intending to be precise. This is impractical both through data collection and from a computational viewpoint. Second, the mechanisms of the universe are littered with probable outcomes, not discrete ones. That renders any calculation irrelevant. We can use such equations only for few-bodied, real-world systems that do not rely on exactitude.

The components of a complex system interact with each other at a local level in simple, probabilistic ways. The result of many such components managing their own interactions can be a new behavior … not intended by the local activity but present at the system level. This emergent behavior is not intuitively predicted by knowing about the components. (NOTE: It is common to say that the system’s properties are beyond the design of its parts. However, design implies intention, and it is conceivable to imagine the rules and environment of a system being purposely tuned to change emergent behavior. It might be more accurate to say that the system’s emergent properties are beyond the awareness of its component interaction.)

Emergent behaviors become new properties of the larger system. While this can take any number of forms, two areas are of particular interest to complexity: Fractals and Power Laws. Fractals describe the geometry of a system that displays self-similarity at every level. These shapes are an optimal use of space that make large structures lightweight and maximize surface area. Power law is a special frequency distribution where in a graph a long tail arises due to a larger-than-normal number of larger components in the system. When viewed as a log-log graph, the curve straightens into a signature downward-sloping line. Power laws are evident in many natural and computational systems spanning a wide range of domains, from social science to transportation to health to biology.

Phase Transitions
Criticality is a point in between stable states — more ordered and less ordered systems — where complexity can arise. It is the place where phase transitions occur, such as water to ice or magnetism to non-magnetic properties. Traditional science has not explained this messy area very well. Study focused on understanding the equilibrium states, and largely ignoring the non-equilibrium dynamics.

It is the non-equilibrium state that exists in the real world, oscillating back and forth across the point of criticality. Like a spring or a pendulum, this oscillation is the result of opposing forces in the system taking turns being dominant. Through these forces, the system can respond to changes in the environment to adapt and self-organize. In this case, the point of criticality — which defines the boundary between states — is a value that comes out of the system dynamics.

Phase transition is not a property of complex systems, but by establishing a value of criticality, the transition provides a place where complexity can emerge.

Network Theory
Statistical analysis has a long history of finding correlations between one variable and another, but the data available to a statistician is flat. Statistics are a snapshot of a moving entity and will therefore miss much of the important information that defines the system. Static networks give a system depth, like a sculpture. They are much better at describing a system by including in analysis the relative orientation between variables. This, too, is incomplete since these relationships change over time. What is more helpful is a model that accounts for both depth and motion of these relationships.

Dynamic networks are a way of viewing a system in terms of relationships that change over time. By analyzing a system in this context, new insights can be mined from the information embedded in those relationships. They are important in the study of complexity not because every complex system can be viewed as a dynamic network — if not true, it is possibly a useful exercise — but rather because dynamics emphasize time as a factor and rejects the notion that objects act in isolation.

Simulation
Determinism requires great precision. Since precision is something unattainable in many-bodied systems, a new approach was needed. Simulation is a modeling technique used to explore with computers the essence of a phenomenon. The details of the interaction rules, environment and interacting agents are greatly simplified but are nonetheless able to describe the mechanics of the system.

This powerful technique is useful for describing both complex systems found in nature and those manufactured through technology. Simulation is a tool for investigation and prediction. Building a simulation forces the scientist to understand the core parameters without raising expectations of exact calculation or control. For complex systems, where self-organization strengthens the forces to overcome intervention, exact measurements would fail to describe a reality that moves in space and time.

What these components — emergence, criticality, dynamic networks and simulations — provide are the tools to understand and describe complexity.