Technically, any system in which the change of values of its variables can be represented as a series of points suggesting a straight line on a coordinate plane, hence, the term “linear” for line. More generally, a linear system is one in which small changes result in small effects, and large changes in large effects. In a linear system, the components are isolated and noninteractive. Real linear systems are rare in nature since living organisms and their components are not isolated and are made-up of rich interactions.

See: Nonlinear System

Bibliography: Abraham (1982)

Algorithmic Complexity is often contrasted with another measure of complexity also relying on a measure of the algorithms needed to generate the data from a system, i.e., that of logical depth defined by the computer scientist and mathematician, Charles Bennett.

See: Complexity (Algorithmic)
Bibliography: Bennett (1982)

An equation that has been applied to various natural systems such as the changes in a system’s population over time which show a convergence to some fixed value or values over time (i.e., attractors). The logistic equation (also called a “map”) is technically a nonlinear difference equation which has been used to model the population changes in a system characterized by a predator-prey relationship between two species. The fascinating behavior of this particular equation was studied by the physicist-turned-epidemiologist Robert May and the physicist Mitchell Feigenbaum. May discovered the fascinating change of attractors as the control parameter is increased leading up to chaos, while Feigenbaum discovered his famous constant of the ratio of succeeding parameter values following the path of the period-doubling bifurcation. The logistic map and its changes of attractors has become somewhat emblematic of chaos theory since it shows how a relatively simple equation can generate such fascinating complexity.

See: Attractors
Bibliography: Peak and Frame (1994)

Images, representations, or thought schemes of how we perceive and cognize the world around us. We follow our mental models in getting about in the world, but can become trapped in limiting behaviors by being overly attached to certain mental models. That is why we need occasionally to be jogged out of the ruts of our dominant mental models by investigating new ways of looking at things. Complexity science has the promise of being a powerful tool to get us to look at our work and organizations in a new way, thereby changing our mental models of how to go about our business in the most effective manner.

See: Complex, Adaptive Systems; Internal Model
Bibliography: Senge (1990), Stacey (1996)

The management theorist Gareth Morgan’s term for processes encouraging self-organization by avoiding an overly top-down, imposed design on an organization or work group. These processes can include such elements as mission statements, guiding principles, boundaries, creative challenges, and so on. The key is for leadership to provide the minimum specifications so a work group itself has creative space to accomplish its work. Minimum specifications are analogous to the simple rules governing cell interactions in studies of cellular automata.

See: Cellular Automata; N/K Model

Bibliography: Morgan (1997); Zimmerman, Lindberg, Plsek (2001).

Electronic automatons, similar in some ways to cellular automata, that offer a simplified model of a brain. As such, neural networks are devices of machine learning that are based on associative theories of human cognition. Using various algorithms and weightings of different connections between “neurons,” they are set up to learn how to recognize a pattern such as learning a voice, recognizing a visual pattern, learning some form of robotic control, manipulating symbols, or making decisions, and so on. Generally, neural nets are composed of three layers: input neurons; output neurons; and a layer in-between where information from input to output is processed. Initially the network is loaded with a random program, then the output is measured against a desired output which prompts an adjustment in the “weights” assigned to the connectivities in response to the “error” between the actual and desired output, and this is repeated many times. In this way, the neural network learns. In a sense, a neural net has to be able to discover its own rules. Changing the rules of interaction between the “neurons” in the network can lead to interesting emergent behavior. Hence, neural nets are another tool for investigating self-organization and emergence.

See: Adaptation

Bibliography: Allman (1989).

Stuart Kauffman’s conception of understanding the evolution of complex, adaptive systems based on the fitness traits of an organism (N) and the inputs of one trait (or gene) to another. Then one can observe the fitness landscapes obtained by manipulating the N’s and K’s. Emergent patterns are thus understood in terms of what rules led to them and what implications they have for fitness.

See: Adaptation; Fitness Landscape; Self-organization
Bibliography: Kauffman (1995); Kauffman and Macready (1995)

Technically, any system where the data points representing values of its variables can be represented as a curvilinear pattern on a coordinate plane, hence, “nonlinear” for not-a-line. That is, the system’s dynamics are more appropriately represented by nonlinear and not linear functions. More generally, a system in which small changes can result in large effects, and large changes in small effects. Thus, sensitive dependence on initial conditions (the butterfly effect) in chaotic systems illustrates the extreme nonlinearity of these systems. In a nonlinear system the components are interactive, interdependent, and exhibit feedback effects. Complex adaptive systems are nonlinear systems.

Before the advent of chaos and complexity theories during the past thirty-five years, nonlinear functions were mostly relegated to appendices in textbook because of their refractoriness to analytic solutions. But now, because of both advances in computational approaches to nonlinear functions and the recognition of the crucial role played by interactions in most systems of interest, it is increasingly recognized that examples of nonlinear systems are presumably endless. Thus, human physiology is replete with nonlinear systems, such as the cardiac, circulatory, and immune systems. In addition, in social systems nonlinearity seems to be the norm since they are constituted by mutually reciprocal interactions among the members of social groupings and such interactions are appropriately modeled by nonlinear rather than linear equations.

See: Linear; Sensitive Dependence on Initial Conditions

Bibliography: Eoyang and Olson (2001); Goldstein (1994); Guastello (1995); Scott (2005); West (1990); West (2006).