Adaptation: A property of a complex adaptive system (cas) such that “experience guides change in the system’s structure so that as time passes the system makes better use of its environment for its own ends” (Holland, 1995) .

Adaptive Plan: One of the three key characteristics of complex adaptive systems (cas) identified by Holland (1975) is the existence of an adaptive process or plan. “The adaptive plan, progressively modifying structure by means of suitable operators, determines what structures are produced in response to the environment” (p.18). The other two key characteristics being the environment and “a measure of performance of different structures in the environment” (p 20).

Adaptive System (complex adaptive system): Specified as a set of objects: attainable structures, a set of adaptive operators for modifying structures, a set of inputs from the environment to the system, and an adaptive plan. See complex adaptive system.

Adaptive Process: See adaptive plan.

Adaptive Tension: Energy or resource differentials within the system and between the system and the environment that may lead to changes in the system which enhance fitness or performance according to some measure. Beyond certain critical values, this adaptive tension may be dissipated through dissipative structures that emerge in the process of self-organization at the edge of chaos.

Adaptive Walk: An adaptive plan that involves changing a single choice, interaction or activity, that moves the complex adaptive system (cas) to a point immediately adjacent on the fitness or performance landscape. These single-step adaptations may move the cas to greater relative performance but this process runs the risk of positioning the cas on a local peak that is not the global peak. These activities are generally associated with exploitation capabilities where individual strategic choices, other choices and processes are continuously modified to improve performance (S. Kauffman, Lobo, & Macready, 2000; S. A. Kauffman, 1993; Rivkin, 1997).

Agent (in computational modeling systems): In computational modeling, an agent is a sophisticated cellular automata that includes a self-contained program that can control its own actions based upon its perception of its environment. In addition to their own copy, or instance, of a computer program, agents may contain a genetic algorithm that varies randomly over time, thus creating continual diversity and heterogeneity in the capabilities of the agents.

Agent (in social systems): Consistent with research in computational organizational theory (Carley & Prietula, 1994), individuals or autonomous decision-making groups are the smallest entity studied. Either can be called an agent.

Agent-Based-Modeling (ABM): A sophisticated variation of cellular automaton computer simulation that tracks collections of heterogeneous agents as they interact through time. In cases where agents contain a genetic algorithm or learn based upon experience, some measure of performance or fitness, and an exogenously determined selection algorithm, are often included in the model. The evolution of the agents’ capabilities and fitness thus becomes a subject of interest.

Attractor: Although there are many definitions of Attractor, we adopt that of Hirsch, Smale and Devaney (2004): “Roughly speaking, an attractor for the flow is an invariant set that ‘attracts’ all nearby solution” (p310). See also Chaotic Attractor,Strange Attractor, Periodic Attractor and Point Attractor.

AttractorBasin: See Basinof Attraction.

Autocatalytic Sets (generality): “The generalization of autocatalytic sets of polymers is based upon the realization that that polymers can be regarded as strings of symbols.… The catalytic and other chemical rules governing the ways enzymes catalyze ligation and cleavage among proteins can be thought of as a kind of grammar. In this grammar, strings of symbols act upon strings of symbols to yield strings of symbols. An autocatalytic set is a type of collective identity operator in this space of symbol strings, or an operator which reproduces at least itself (p369).”

Autocatalytic Sets (of polymers): For any fixed probability of catalysis P, autocatalytic sets must become possible at some fixed complexity level of numbers of kinds of polymers. The achievement of catalytic closure required for self reproduction is an emergent collective property in any sufficiently complex set of catalytic polymers (Kauffman, 1993) p 310).

Artificial Life: “Also known as alife or a-life, is the study of life through the use of human-made analogs of living systems. Computer scientistChristopher Langton coined the term in the late 1980s when he held the first "International Conference on the Synthesis and Simulation of Living Systems" (otherwise known as Artificial Life I) at the Los Alamos National Laboratory in 1987.” Also see Cellular Automata and Genetic Algorithms. Above copied on October 1, 2005 from: http://en.wikipedia.org/wiki/Artificial_life

Autopoiesis: The property that “makes living beings autonomous systems…. What is distinctive about them, however, is that their organization is such that their only product is themselves, with no separation between the producer and the product” (Maturana & Varela, 1998) p 48). “The most striking feature of an autopoietc system is that it pulls itself up by its own bootstraps and becomes distinct from its environment through its own dynamics, in such a way that both things are inseparable” (pp.46-47). Also, see Stacey (Stacey, 2001), pp 236-243) for a critique of the use of the notion of autopoiesis in the context of human action.

Basin of Attraction: “…an entire volume of states which lie on trajectories flowing to that attractor… (Kauffman, 1993, p 176).”

Benard Process: See Critical Values.

Bifurcation: The characteristic of dynamical system such that even the slightest change in the value of a parameter, say μ, leads to a radical change in the dynamic behavior of the solution (Hirsch, Smale & Devaney, 2004, p 4).

Boundary (of an Organization): A delimiter on the state of potential members within a social system, such that a member or agent is said to be inside the boundary when it performs actions for the benefit of the system and outside the boundary when it does not (Hazy, Tivnan, & Schwandt, 2004).

Butterfly Effect: A term used to illustrate the property of sensitivity to initial conditions that is encountered in chaos theory. This property has been explained as follows: “One flap of a butterfly’s wings may alter the course of weather patterns forever”. Also, the butterfly-like shape of the Lorenz attractor may have inspired the name of the butterfly effect.

Cartesian Product: The Cartesian Product of two Cartesian spaces, Rn and Rm, that is, of the sets of n-tuples and m-tuples of real numbers respectively, is the Cartesian space Rn + m.

Catalytic Closure: A characteristic of a set for which ”it must be the case that every member of the autocatalytic set has at least one of the possible last steps in its formation catalyzed by some member of the set, and that connected sequences of catalyzed reactions lead from the maintained food set to all members of the autocatalytic set (Kauffman, 1993 p 299).

Cellular Automata: “Cellular Automata consist of a large grid of cells in a regular arrangement. Each cell can be in one of a small number of states and changes between these states occur according to rules which depend upon the states of the cell’s immediate neighbors. Cellular Automata form a useful framework for some models of social interaction, for example the spread of gossip between people and the formation of ethnically segregated neighborhoods” (Gilbert & Troitzsch, 2005, p8).

Chaos Theory: A term poplar in the 1980’s after the publication of James Gleick’s book Chaos in 1987. The term was used to describe in general the advances in what is now called “complexity science”. Complexity Science is the preferred term for the general area.

Chaotic Attractor: See Chaotic Behavior.

Chaotic Behavior:See chaotic dynamics. Also Wiki encyclopedia “ chaotically”

Chaotic Dynamics: Although there are many definitions of chaotic, we adopt that of Hirsch, Smale and Devaney (2004): In dynamical systems, three ingredients imply chaotic dynamics. These are: Sensitive dependence on initial conditions, 2) topological transitivity, and 3) dense periodic points. A mapping has topologic transitivity when it has dense orbits, that is, for each sequence of points in its sequence space Σ, as t increases, the sequence converges to a periodic point in a set that is dense in Σ. Periodic points form an open set in a sequence space Σ, meaning not all points close to the points in the set are included in the set. The set of periodic points is dense when its mathematical closure is the full set.

Complexity Science: The preferred generic term for the overall field under discussion that should be used to describe dynamical systems, complex systems, complex adaptive systems, chaotic behavior, a New Kind of Science (NKS) and the various methods used to explore the above. This term should replace Chaos Theory as a catch-all term.

Complex Adaptive System (cas): A complex system that has the additional property of adaptation. All complex adaptive systems are complex systems, but not all complex systems are complex adaptive systems.

Complex Analysis: A branch of mathematics that studies complex numbers. A complex number z takes the form z x + iy, where isquare root of -1. Whereas the Real numbers can be thought of as a “line” or x axis, the complex numbers form the complex plane with the x-axis representing the Real number component, and the y-axis being the “imaginary” component.

Complex Analytic Dynamics: see complex dynamics.

Complex Dynamics: This is a different usage of the term “complex” in that it refers to dynamical systems on the Complex numbers C rather than on the Real numbers, R. A complex number z takes the form z x + iy, where i square root of -1. Whereas the Real numbers can be thought of as a “line” or x axis, the complex numbers form the complex plane with the x-axis representing the Real number component, and the y-axis being the “imaginary” component.

Complex Number: A Complex Number z takes the form z x + iy, where i square root of -1. Whereas the Real numbers can be thought of as a “line” or x axis, the complex numbers form the complex plane with the x-axis representing the Real number component, and the y-axis being the “imaginary” component. See complex analysis and complex dynamics.

Complex Systems: An open system that is “made up of a large number of active elements that … are diverse in both form and capability” (Holland, 1995) .

Computational Cognitive Neuroscience: A branch of neuroscience wherein biologically realistic computer-based feedforward neural networks are used to replicate known results from psychology and neuroscience (O'Reilly & Munakata, 2000).

Computational Complexity Theory: Part of the theory of computation dealing with the resources required during computation to solve a given problem. The most common resources are time (how many steps it takes to solve a problem) and space (how much memory it takes). Other resources can also be considered, such as how many parallel processors are needed to solve a problem in parallel. Complexity theory differs from computability theory, which deals with whether a problem can be solved at all, regardless of the resources required.” Computational Complexity Theory gives rise to classifications about decision problem solvability such as NP-complete problems. The above quote was copied on October 1, 2005 from: http://en.wikipedia.org/wiki/Computational_complexity_theory

Continuous Dynamical System: See Dynamical Systems.

Critical Value (in the context of Dissipative Structures): Within a Benard Process an energy differential is established between warmer and cooler surfaces such that energy flows between the surfaces reducing the differential. As this differential increases beyond a certain critical value, the system undergoes a dramatic physical change in which convection begins to dissipate energy along with conduction. During this process, visible structures emerge spontaneously within the flow. Prigogine called these “dissipative structures” because they act as an accelerant in the dissipation of the energy differential (Prigogine & Stengers, 1984).

Cycle: See Cycle Graph.

Cycle Graph (or Cycle): “In graph theory, a cycle graph or cycle is a graph that consists of a single cycle, or in other words, some number of vertices connected in a closed chain. The cycle graph with n vertices is called Cn. A cycle with an even number of vertices is called “even cycle”, a cycle with an odd number of vertices is called “odd cycle”. Some sources allow cycles to repeat vertices. The cycle graphs can be “simple”, meaning that they don't repeat vertices, which is the usual definition in graph theory.” Copied on October 2, 2005 from: http://en.wikipedia.org/wiki/Directed_cycle .

Deterministic Models: Olnick P 11, INDEX

Digraph: See Directed Graph.

Directed Cycle: A directed cycle graph is a directed version of a cycle graph, with all the edges being oriented in same direction. In a directed graph, a set of edges which contains at least one edge (or arc) from each directed cycle is called a feedback arc set. Similarly, a set of vertices containing at least one vertex from each directed cycle is called a feedback vertex set. There is a directed cycle through any two vertices in a strongly connected component. Copied on October 2, 2005 from: http://en.wikipedia.org/wiki/Directed_cycle.

Directed Graph: A directed graph or digraphG is an ordered pair G:=(V, A) with V, a set of vertices or nodes, and A, a set of ordered pairs of vertices, called directed edges, arcs, or arrows. An edge e = (x, y) is considered to be directed fromxtoy; y is called the head and x is called the tail of the edge. A variation on this definition is the oriented graph, which is a graph (or multi-graph; see below) with an orientation or direction assigned to each of its edges. A distinction between a directed graph and an oriented simple graph is that if x and y are vertices, a directed graph allows both (x, y) and (y, x) as edges, while only one is permitted in an oriented graph. A more fundamental difference is that, in a directed graph (or multi-graph), the directions are fixed, but in an oriented graph (or multi-graph), only the underlying graph is fixed, while the orientation may vary. A directed graph may or may not be allowed to have loops, that is, edges where the start and end vertices are the same. By definition, this is forbidden in an oriented simple graph. Copied on October 2, 2005 from: http://en.wikipedia.org/wiki/Directed_graph

Directed Network: A Social Network or other network that can be represented by a Directed Graph.

Discrete Dynamical Systems:See Dynamical Systems.

Discrete Event Simulation: Discrete event simulation (DE) is used to model the logical outcome of a series of related events over time, in the manner of a process flow chart. In this type of simulation, the simulator maintains a queue of events and these are sorted according to the simulated time they are modeled to occur. The simulator reads the queue and triggers new events as each event is processed so that outcomes can be evaluated. It is important to be able to access the data produced by the simulation, to discover logic defects in the design, or the sequence of events that are modeled. And example of commercial software for DE is Vite’.

Dissipative Structures: Within a Benard Process an energy differential is established between warmer and cooler surfaces such that energy flows between the surfaces reducing the differential. As this differential increases beyond a certain critical value, the system undergoes a dramatic physical change in which structures emerge spontaneously within the flow. Prigogine called these “dissipative structures” because they accelerate the dissipation of the energy differential (Prigogine & Stengers, 1984). There is broad speculation that the generalized phenomenon of “dissipative structures” is closely associated with self-organization and emergence in complex adaptive systems.

Dynamical Systems: “A dynamical system is a way of describing the passage in time of all points in a given space S. The space S could be thought of, for example, as the space of states of some physical system…. When we consider dynamical systems that arise in mechanics, the space S will be the set of possible positions and velocities of the system. For the sake of simplicity, we will assume throughout that the space S is Euclidean space Rn although in certain cases, the important dynamical behavior will be confined to a particular subset of Rn. “Given an initial position X є Rn, a dynamical system on Rn tells us where X is located 1 unit of time later, 2 units of time later, and so on. At time zero, X is located at X0. One unit before time zero, X was at X-1. In general, the ‘trajectory’ of X is given by Xt. If we measure the positions of Xt using only integer time values, we have an example of a discrete dynamical system…. If the time is measured continuously with tє R, we have a continuous dynamical system. If the system depends upon time in a continuously differentiable manner, we have a smooth dynamical system” (Hirsch, Smale, & Devaney, 2004, pp 140-141). Alternatively: “A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish each spring in a lake are examples of dynamical systems. A dynamical system has a state determined by a collection of real numbers. Small changes in the state of the system correspond to small changes in the numbers. The numbers are also the coordinates of a geometrical space—a manifold. The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The rule is deterministic for a given time interval --only one future state follows from the current state.” Copied on October 5, 2005 from: http://en.wikipedia.org/wiki/Dynamical_system . See also Continuous Dynamical Systems and Discrete Dynamical System, System Dynamics and Deterministic Models.

Dynamical Systems Theory: A branch of the mathematical sciences that relies heavily on differential equations and other multi-variable analysis techniques to study dynamical systems. The field includes both linear and non-linear dynamics. See also Continuous Dynamical Systems and Discrete Dynamical System, System Dynamics and Deterministic Models.

Dynamic Capabilities: The capacity to renew competencies so as to achieve congruence with the changing business environment. This is done by appropriately adapting, integrating, and reconfiguring internal and external organizational skills, resources, and functional competencies to match the requirements of the changing environment (Teece, Pisano, & Shuen, 1997, p. 515).

Emergence: “A process by which a system of interacting subunits acquires qualitatively new properties that cannot be understood as the simple addition of their individual contributions” (Camazine et al., 2001), p. 31). Also, “a systemic process of order creation through which properties and or structures come into being that are unexpected and unpredictable, given the known attributes of the component agents and environmental forces” Lichtenstein & McKelvey” DO NOT QUOTE .

Entanglement: There are two distinct uses of this term, one in Complexity Leadership Theory, the other in quantum theory. In Complexity Leadership Theory, entanglement is the notion that different leader roles and leadership activities of various types can and do occur in the same territory, both physical and intangible (Uhl-Bien, Marion & McKelvey). For the quantum theory definition, see: Quantum Entanglement

Environment: See Objective Environment and Enacted Environment.

Epistasis or Epistatic Interaction: Kauffman

Equation-Based-Modeling: See Deterministic Models.

Evolutionary Models: See Artificial Life, Cellular Automata, Evolutionary Economics,Genetic Algorithm, and Population Ecology.

Firm or Organization: The entity upon which population-level selection acts and capability selections are aggregated.

Firm Performance: Defined in terms of the relative positioning of the focal firm against other firms with respect to the criteria driving selection for populations of organizational forms. Higher probability of survival in the population selection process is defined to be superior firm performance. In this sense, firm performance represents the firm’s current position on its idealized performance landscape. Because the specific criteria for selection may change over time, the specific firm performance variables described below may or may not reflect increased firm performance in the idealized sense as defined here. In sum, firm performance is conceptualized in this analysis as a firm-level aggregate measure of the firm’s potential for survival in a stochastic environment (Hannan & Freeman, 1989)(Hazy, 2004).

Fitness: A measure of the survival value of a particular complex adaptive system (cas) in a given selection process and a given environment.

Fitness Landscape: An organization’s fitness landscape is the topography of the payoff surface in the state space of an adaptive system interacting with its objective environment (S. A. Kauffman, 1993; Levinthal & Warglien, 1999). In later writings for the social sciences, this has been called the organization’s performance landscape (Levinthal & Siggelkow, 2001). See also Performance Landscape.

Flow: “To emphasize the dependence of a solution [to a differential equation] on the initial value xo, we will denote the corresponding solution φ (t, xo). This function φ: R X Rà R is called the flow associated to the differential equation…. Sometimes we write this function as φt (xo)” Hirsch, Smale & Devaney, 2004, p 12 .

GlobalPeak: A set of choices P = (a1, a2, a3,…, aN) is a global peak on the performance landscape if there are no other local peaks with a higher performance value.

Grammar Diversity: Grammars refers to laws of complementarity & substitutability internal to the system (Kauffman, 1993) that make up the rules for organizing and reconfiguring the capabilities within the system. “Diversity of grammars” refers to the variations in the possibilities for configuring and reconfiguring capabilities to take advantage of resources.

Hamming Distance: In information theory, the Hamming distance, named after Richard Hamming, is the number of positions in two strings of equal length for which the corresponding elements are different. Put another way, it measures the number of substitutions required to change one into the other.
For example:

The Hamming distance between 1011101 and 1001001 is 2.

The Hamming distance between 2143896 and 2233796 is 3.

The Hamming distance between "toned" and "roses" is 3.

Hierarchy (of system complexity): “In application to the architecture of complex systems, ‘hierarchy’ simply means a set of Chinese boxes of a particular kind.… Opening any given box in a hierarchy discloses not just one new box within, but a whole small set of boxes; and opening any one of these component boxes discloses a new set in turn. While an ordinary set of Chinese boxes is a sequence, or complete ordering, a hierarchy is a partial ordering—specifically a tree (Simon, 1973) p 5).

Hierarchy (of agent status and influence with a system): For a system layer, or as Herbert Simon calls them, a single Chinese box (Simon, 1973), that exists within a hierarchy (of system complexity), individual components or “boxes” disclosed within the Chinese box may themselves be ordered relative to one another according to some metric, for example, status, influence, reward or power. This hierarchy of rank within a level of Hierarchy (of system complexity), that is, relative rank among the component boxes within a Chinese box, is logically distinct from the Hierarchy (of system complexity) itself in that such a relative ranking effects the interaction dynamics among the component boxes within the Chinese Box, and thus the properties of the original Chinese box, but the ranking remains logically distinct from the higher level system itself, the original Chinese box.

Individual Leadership: “A process wherein an individual member of a group or organization influences the interpretation of events, the choice of objectives or strategies, the organization of work activities, the motivation of people to achieve the objectives, the maintenance of cooperative relationships, the development of skills or confidence by members, and the enlistment of support or cooperation from people outside the group or organization” (Yukl, 1998), p. 5).

Interaction dynamics: see Epistasis.

Kinetics: The mechanism by which a physical or chemical change is effected. (Merriam-Webster’s Collegiate Dictionary 10th Edition.)

Leader Role: An abstraction associated with the need for coordination in social systems, particularly when existing structures are inadequate (Barnard, 1938; Katz & Kahn, 1966).

Leadership Activities: Specific actions and tasks that reflect the implementation of leadership methods in organizations. Examples include goal-setting practices, communication forums, and artifacts such as newsletters, formal or informal performance reviews, project management practices, etc. and other leadership routines. Practices are assumed to vary by method leading to variation in routines.

Leadership Routines: Leadership routines are defined as the organizational routines that serve organizational leadership functions. They include both leadership methods and practices. Examples include strategic reviews, internal strategy road shows, skip-level meetings, town hall meetings, budget reviews, management by objectives, etc.

Local Neighborhood (on a Performance Landscape): Positions on the performance landscape from which a local peak can be achieved through single-step micro-adaptations.

Local Peak: A set of choices P = (a1, a2, a3,…, aN) represents a local peak in the performance landscape if there is no other point on the surface that differs from P in only one dimension and that has a higher value of firm performance (Levinthal & Siggelkow, 2001).

Long-Jump Adaptation: An adaptive process, ormicro-adaptation, wherein several choices, connections or activities are changed simultaneously in an attempt to jump to a different position on the performance landscape which may lead, eventually after an adaptive walk, to a peak with the potential for higher performance than the local landscape (prior to the long-jump adaptation) would imply. Long Jump Adaptations are needed to avoid being trapped on a local performance peak. These micro-adaptations are associated with exploratory capabilities (Frenken, Marengo, & Valente, 1999; Gavetti & Levinthal, 2000; S. Kauffman et al., 2000; S. A. Kauffman, 1993; Rivkin, 1997).

Lotka-Volterra Model: See Predator-Prey Model.

Lorenz Attractor: The Lorenz attractor, introduced by Edward Lorenz in 1963, is a non-linear three-dimensional deterministic dynamical system derived from the simplified equations of convection rolls arising in the dynamical equations of the atmosphere. For a certain set of parameters the system exhibits chaotic behavior and displays what is today called a strange attractor; this was proven by W. Tucker in 2001. The strange attractor in this case is a fractal of Hausdorff dimension between 2 and 3. Grassberger (1983) has estimated the Hausdorff dimension to be 2.06 ± 0.01 and the correlation dimension to be 2.05 ± 0.01. The butterfly-like shape of the Lorenz attractor may have inspired the name of the butterfly effect in chaos theory . Copied on October 5 2005 from: http://en.wikipedia.org/wiki/Lorenz_attractor .

Lyapunov Function: In the theory of dynamical systems, and control theory, Lyapunov functions, named after Aleksandr Mikhailovich Lyapunov, are a family of functions that can be used to demonstrate the stability or instability of some state points of a system. The demonstration of stability or instability requires finding a Lyapunov function for that system. There is no direct method to obtain a Lyapunov function but there may be tricks to simplify the task. The inability to find a Lyapunov function is inconclusive with respect to stability or instability. Copied on October 5, 2005 from: http://en.wikipedia.org/wiki/Lyapunov_function .

Lyapunov Stability: Lyapunov stability is applicable to only unforced (no control input) dynamical systems. It is used to study the behavior of dynamical systems under initial perturbations around equilibrium points.

Let us consider that the origin is the equilibrium point (EP) of the system. A system is said to be stable "in the sense of Lyapunov" if for every ε, there is a δ such that:

The system is said to be asymptotically stable if as

Markovian Process: A stochastic process , but one that can be represented mathematically such that all relevant information is contained in the current state of the system. G&T p86 Olnick p 274. Also see Non-Markovian Process and Absorbing Markovian Process.

Measure of Performance: See Performance.

Meta-Capability: An organizational capability that acts upon the system itself to modify or extrapolate other capabilities of the system (Hazy, 2004).

Meta-Matrix Representation: This approach assumes that any organization can be represented as a set of matrices linking personnel, resources, and tasks. Resources include knowledge resources which can also be explicitly represented. The meta-matrix representation, originally called the PCANSS formalism, defines the interconnections between these various types of organizational elements or nodes at a point in time in an idealized sense. The matrix linking tasks to tasks is called the Precedence matrix because in represents the contingencies inherent in the task structure. The matrix linking individual personnel to individual resources (including knowledge resources) is called the Capabilities matrix. Personnel and tasks are linked in the Assignments matrix; resources to tasks in the Needs matrix; personnel to personnel in the Social Network matrix; and resources to resources in the Substitutions matrix. In theory, when all personnel, resources, and tasks are listed and their interactions are defined, a complete representation of the organization’s network configuration can be defined at a point in time (Krackhardt & Carley, 1998).

Meta-Process: Used to distinguish these processes that operate on the system from those directly supporting the organization’s tasks (So & Durfee, 1998). Leadership processes are a special case of meta-processes (Hazy, 2004).

Micro-Adaptation: A single adaptation event resulting from the action of adaptive capabilities. A micro-adaptation changes the position of the firm along its performance or fitness landscape.

Minority Game: A game commonly used in research in gametheory because it simulates the financial markets. In each time step, agents choose one of two sides, 0 or 1, and those on the minority side win a point. To see an agent-based-model of the minority game see: http://ccl.northwestern.edu/netlogo/models/MinorityGame .

Multi-Agent Modeling: See Agent Based Modeling (ABM).

New Kind of Science (NKS): A term coined and popularized by Stephen Wolfram to encompass the discoveries that have come from cellular automata modeling and analysis and their implications to information theory, computer science, science in general, mathematics, and natural philosophy.

NKS: see New Kind of Science.

New Science: Sometime used in the popular press to describe complexity science. Also, see New Kind of Science

Objective Environment: The objective environment of a firm directly affects that firm’s performance landscape and its survival independent of the perspective from within the organization. The objective environment is not independent of the organization, however, but the two are structurally coupled (Maturana & Varela, 1998), in the sense that they co-evolve with one another.

Order (of a differential equation): The order of the highest derivative that explicitly appears in the equation (Hirsch & Smale, 1974), p22).

Organization or Firm: See Firm or organization.

Organizational Capability: The presence within a firm of a dynamic process that acts to acquire and/or deploy assets, tangible or intangible, in order to perform a task or activity to produce or improve results (Maritan, 2001; Nelson & Winter, 1982; Teece, Pisano, & Shuen, 1997). Organizational capabilities are in general “knowledge-based, firm specific, socially complex, and they generally cannot be simply acquired in factor markets” (Maritan, 2001, p. 514).

Organizational Forms: Variations in structure across organizations that have different performance characteristics and thus different survival potential (Hannan & Freeman, 1989).

Organizational Leadership: The “influential increment over and above mechanical compliance with the routine directives of the organization” (Katz & Kahn, 1966, p. 528). Organizational leadership is an organization-level construct necessary due to “imperfections and incompleteness” of organizational design, the problem of managing the boundary of the organization, “changing external conditions,” “the changing internal state of the organization produced by separate dynamics of several substructures of the organization,” and the challenges of human membership in organizations (Katz & Kahn, 1966, p. 530). It is proposed organizational leadership can also considered to be an organizational meta-capability that exists with varying degrees of proficiency in all organizations (Hazy, 2004, 2006, 2007, 2008, 2009).

Organizational Routine: “All regular and predictable behavior patterns” (Nelson & Winter, 1982), p. 14) in organizations. The term includes “well-specified technical routines for producing things, through procedures for hiring and firing, ordering new inventory, or stepping up production of items in high demand, to policies regarding investment, research and development (R&D), or advertising, and business strategy about product diversification and overseas investment” (Nelson & Winter, 1982), p. 14).

Organizational Slack: “The pool of resources in an organization that is in excess of the minimum necessary to produce a given level of organizational output” (Nohria & Gulati, 1996), p. 1246).

Pattern (in Self-Organization): “A particular, organized arrangement of objects in space or time” (Camazine et al., 2001), p. 8).

Payoff Function: The mathematical formula for calculating the reward to agents in a game or after a defined event in an agent-based-model (ABM).

PCANSS Formalism: See Meta-Matrix Representation.

Performance: The organizational process involving transactional leadership activities in interaction with organizational capabilities that exploits the organization’s resource and capabilities position in the environment to appropriate rents from existing markets.

Performance Landscape: “A mapping of all possible sets of firm choices onto performance values (such as a profitability measures). If a firm’s set of choices is described by a vector of N choices, the performance landscape consists of N dimensions depicting the firm’s alternatives along each dimension and an N+1th dimension depicting the performance associated with each vector of N choices” (Levinthal & Siggelkow, 2001), p. 5).

Performance Landscape Dynamics: The performance landscape dynamics of the organization is the variable that is used in this analysis to represent the interplay between the strategic choices of the firm (that locate that firm on its performance landscape) and the dynamic state of the environment (that may “morph” the performance landscape), and that therefore implies that the firm may need to respond to the environment to adapt to new performance conditions (Levinthal & Siggelkow, 2001).

Period Doubling: A feature of complex systems wherein the periodic attractor of a dynamical system suddenly doubles in period as a factor in the differential equations increases, a special case of bifurcation.

Periodic Attractor: Q X Eà Q X E*

Phase Space: For an open set Q in the vector space E, and a state space, Q X E, a phase space of the system is Q X E* where E* is the dual vector space of Q. The dual vector space of Q is the vector space of all linear maps E à R. It has been shown that E is isomorphic with E* and that the relationship between the state space and the phase space of a system is given by the Legendre Transformation λ: Q X E à Q X E*. (Hirsch & Smale, 1974), p 204, 292.

Piecewise Continuous Function: A function with a finite number of bounded discontinuities in the sense that if x = c is a point of discontinuity of such a function then and both exit but are not equal (Lass, 1957).

Point Attractor: A limit point of a dynamical system in phase space.

Power Law Relationship: A power law relationship between two scalar quantities x and y means that the relationship can be written as:

where a (the constant of proportionality) and k (the exponent of the power law) are constants. Power laws can be observed when a relationship is shown on a log-log graph a straight line is observed. This is because, taking the log of both sides, the above equation is equal to:

which exhibits the same form as the equation for a line:

Power laws are observed in many fields, including physics, biology, geography, sociology, economics, war and terrorism. Power laws are among the most frequent scaling laws that describe the scaling invariance found in many natural phenomena. Scale Free Networks exhibit a power-law distribution.

Predator-Prey Model: Olnick P 105.

Prisoner’s Dilemma: A game commonly used in research in gametheory. Critical elements of the game include players, or the prisoners, each of who acts independently and in which the incentive to defect, that is, turn against the other player, rather than to cooperate, (refuse to turn against the other player) is greater for the player regardless of what he assumes the other player do. “The "dilemma" faced by the prisoners here is that, whatever the other does, each is better off defecting, that is, confessing, than remaining silent. But the outcome obtained when both confess is worse for each than the outcome they would have obtained had both remained silent”. The above quote is from: http://plato.stanford.edu/entries/prisoner-dilemma/#Sym on October 1, 2005.

Quantum Entanglement: Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. This leads to correlations between observable physical properties of the systems. For example, it is possible to prepare two particles in a single quantum state such that when one is observed to be spin-up, the other one will always be observed to be spin-down and vice versa, this despite the fact that it is impossible to predict, according to quantum mechanics, which set of measurements will be observed. As a result, measurements performed on one system seem to be instantaneously influencing other systems entangled with it. However, at this time classical information cannot be transmitted through entanglement faster than the speed of light. Copied on October 4, 2005 from: http://en.wikipedia.org/wiki/Quantum_entanglement .

Reward Attribution: An important problem is complex adaptive system (cas) research involves the distribution of the resources appropriated by the cas to the agents that comprise the system. This process is called reward attribution and can be encoded in a pay-off function or may use some other algorithm (J. H. Holland, 1975/1992).

Routines: See Organizational Routines.

Scale Free Network Topology: Scale Free Networks are networks that have the characteristic that a) the frequency of nodes with a given number of connections and b) the number of those connections, exhibit a power-law relationship. One can understand how scale free networks arise in social networks when one considers the following observation: when nodes can choose their next link, for example, for status or access to resources, there is greater marginal benefit to the node by establishing a link to the most popular node available. This leads to a scale free relationship in many types of networks. For example, knowledge networks such as collaboration networks in academia, and the Worldwide Web, have been shown to be scale-free. These are sometimes referred to as “winner-takes-allnetworks”.

Scenario: One of the strengths of the simulation method is the ability to run the model many times with identical or, in some cases, different inputs for each of the various parameters. Each of these runs is called a scenario.

Self-Organization: “Self organization is a process in which pattern at the global level of a system emerges solely from numerous interactions among the lower-level components of the system. Moreover, the rules specifying interactions among the system’s components are executed using local information, without reference to the global pattern” (Camazine et al., 2001), p. 8).

Semi-Markovian Process: Olnick p311.

Sensitive Dependence on Initial Conditions: See sensitivity to initial conditions.

Sensitivity to Initial Conditions: aaa Also see the butterfly effect.

Single Step Adaptation: see Adaptive Walk

Smooth Dynamical Systems:See Dynamical Systems.

Social Structures: Rules, resources, or transformation relationships that “bind” individuals and organize social systems (Giddens, 1984, p. 25).

State: A state of a system is information characterizing the system at a given time. In particular, it is information about the position of a point in a vector space and the first derivative of that point (Hirsch & Smale, 1974, p. 22).

State Space: a space of states is the Cartesian productRN X RN of pairs (x, v), x, v in RN; x is the position, v is the first derivative (velocity) at a point in time.

Stochastic Process: Olnick P 260. See also: Markov Process, Semi Markov Process, & Non-Markovian Process

Strange Attractor: See Chaotic Attractor and Chaotic Behavior.

Strategic Leadership: The organizational process aimed at appropriately adapting, integrating, and reconfiguring internal and external organizational skills, resources, and functional competencies to match the requirements of the changing environment (Teece, Pisano, & Shuen, 1997, p. 515).

Structural Coupling: “When there exists a history of recurrent interactions leading to the structural congruence between two (or more) systems” (Maturana & Varela, 1998, p. 75).

Sugarscape: A path breaking Cellular Automata model (Epstein & Axtell, 1996).

System’s Dynamic State: Whether the system is stable or unstable, or at the chaotic boundary between the two. Stable systems tend to dampen signals that perturb the system. Unstable systems amplify such disturbances and can drive the system out of a dynamic equilibrium associated with positions within a basis of attraction. Perturbations of a system in a position near the edge of chaos can cause the system to change state and shift to another attractor. One example of this is in the context of periodic attractors is bifurcation (Gleick, 1987; Lichtenstein, 1995; McKelvey, 1999; Prigogine & Stengers, 1984; Thietart & Forgues, 1995; Waldrop, 1992).

Tag: Identifying information within an agent that may be made available to other agents during an interaction. The information from the tag may be used by other agents to inform the specific nature of the interaction dynamics. Tags can identify agents with particular skills, certifications, resources, knowledge, histories, pedigrees, status, membership or rights and/or responsibilities of membership, for example. Tags can improve interaction efficiency by relaying information about traits, e.g., trustworthiness, competence, etc.

Tension: See adaptive tension.

Unity: “An entity brought forth by an act of distinction” (Maturana & Varela, 1998), p. 40). An act of distinction “distinguishes what has been indicated from its background” (p. 40)

Winner-Takes-All Networks: See Scale-Free-Network.

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## Glossary of Terms

Adaptation:A property of acomplex adaptive system (cas)such that “experience guides change in thesystem’sstructure so that as time passes thesystemmakes better use of its environment for its own ends” (Holland, 1995) .Adaptive Plan:One of the three key characteristics ofcomplex adaptive systems (cas)identified by Holland (1975) is the existence of an adaptive process or plan. “The adaptive plan, progressively modifying structure by means of suitable operators, determines what structures are produced in response to the environment” (p.18). The other two key characteristics being theenvironmentand “ameasure of performanceof different structures in the environment” (p 20).Adaptive System (complex adaptive system):Specified as a set of objects: attainable structures, a set of adaptive operators for modifying structures, a set of inputs from the environment to the system, and an adaptive plan. Seecomplex adaptive system.Adaptive Process:Seeadaptive plan.Adaptive Tension:Energy or resource differentials within the system and between the system and the environment that may lead to changes in the system which enhance fitness or performance according to some measure. Beyond certaincritical values, this adaptive tension may be dissipated throughdissipative structuresthat emerge in the process ofself-organizationatthe edge of chaos.Adaptive Walk:Anadaptive planthat involves changing a single choice, interaction or activity, that moves thecomplex adaptive system (cas)to a point immediately adjacent on the fitness orperformance landscape. These single-step adaptations may move thecasto greater relative performance but this process runs the risk of positioning thecason alocal peakthat is not theglobal peak. These activities are generally associated with exploitation capabilities where individual strategic choices, other choices and processes are continuously modified to improve performance (S. Kauffman, Lobo, & Macready, 2000; S. A. Kauffman, 1993; Rivkin, 1997).Agent (in computational modeling systems):In computational modeling, anagentis a sophisticatedcellular automatathat includes a self-contained program that can control its own actions based upon its perception of its environment. In addition to their own copy, or instance, of a computer program, agents may contain agenetic algorithmthat varies randomly over time, thus creating continual diversity and heterogeneity in the capabilities of the agents.Agent (in social systems):Consistent with research in computational organizational theory (Carley & Prietula, 1994), individuals or autonomous decision-making groups are the smallest entity studied. Either can be called an agent.Agent-Based-Modeling (ABM):A sophisticated variation of cellular automaton computer simulation that tracks collections of heterogeneous agents as they interact through time. In cases where agents contain agenetic algorithmor learn based upon experience, some measure of performance or fitness, and an exogenously determined selection algorithm, are often included in the model. The evolution of the agents’ capabilities and fitness thus becomes a subject of interest.Attractor:Although there are many definitions of Attractor, we adopt that of Hirsch, Smale and Devaney (2004): “Roughly speaking, an attractor for theflowis an invariant set that ‘attracts’ all nearby solution” (p310). See alsoChaotic Attractor,Strange Attractor,Periodic AttractorandPoint Attractor.AttractorBasin:SeeBasinof Attraction.Autocatalytic Sets (generality):“The generalization ofautocatalytic sets of polymersis based upon the realization that that polymers can be regarded as strings of symbols.… The catalytic and other chemical rules governing the ways enzymes catalyze ligation and cleavage among proteins can be thought of as a kind of grammar. In this grammar, strings of symbols act upon strings of symbols to yield strings of symbols. An autocatalytic set is a type of collective identity operator in this space of symbol strings, or an operator which reproduces at least itself (p369).”Autocatalytic Sets (of polymers):For any fixed probability of catalysis P, autocatalytic sets must become possible at some fixed complexity level of numbers of kinds of polymers. The achievement ofcatalytic closurerequired for self reproduction is an emergent collective property in any sufficiently complex set of catalytic polymers (Kauffman, 1993) p 310).Artificial Life: “Also known asalifeora-life, is the study of life through the use of human-made analogs of living systems. Computer scientist Christopher Langton coined the term in the late 1980s when he held the first "International Conference on the Synthesis and Simulation of Living Systems" (otherwise known as Artificial Life I) at the Los Alamos National Laboratory in 1987.” Also seeCellular AutomataandGenetic Algorithms. Above copied on October 1, 2005 from: http://en.wikipedia.org/wiki/Artificial_lifeAutopoiesis:The property that “makes living beings autonomous systems…. What is distinctive about them, however, is that their organization is such that their only product is themselves, with no separation between the producer and the product” (Maturana & Varela, 1998) p 48). “The most striking feature of an autopoietc system is that it pulls itself up by its own bootstraps and becomes distinct from its environment through its own dynamics, in such a way that both things are inseparable” (pp.46-47). Also, see Stacey (Stacey, 2001), pp 236-243) for a critique of the use of the notion of autopoiesis in the context of human action.Basin of Attraction:“…an entire volume of states which lie on trajectories flowing to that attractor… (Kauffman, 1993, p 176).”Benard Process:SeeCritical Values.Bifurcation:The characteristic of dynamical system such that even the slightest change in the value of a parameter, say, leads to a radical change in the dynamic behavior of the solution (Hirsch, Smale & Devaney, 2004, p 4).μBoundary (of an Organization):A delimiter on the state of potential members within a social system, such that a member or agent is said to be inside the boundary when it performs actions for the benefit of the system and outside the boundary when it does not (Hazy, Tivnan, & Schwandt, 2004).Butterfly Effect:A term used to illustrate the property ofsensitivity to initial conditionsthat is encountered inchaos theory. This property has been explained as follows: “One flap of a butterfly’s wings may alter the course of weather patterns forever”.Also, thebutterfly-like shape of theLorenz attractormay have inspired the name of the butterfly effect.Cartesian Product:The Cartesian Product of two Cartesian spaces, Rn and Rm, that is, of the sets of n-tuples and m-tuples of real numbers respectively, is the Cartesian space Rn + m.Catalytic Closure:A characteristic of a set for which ”it must be the case that every member of theautocatalytic sethas at least one of the possible last steps in its formation catalyzed by some member of the set, and that connected sequences of catalyzed reactions lead from the maintained food set to all members of theautocatalytic set(Kauffman, 1993 p 299).Cellular Automata:“Cellular Automata consist of a large grid of cells in a regular arrangement. Each cell can be in one of a small number of states and changes between these states occur according to rules which depend upon the states of the cell’s immediate neighbors. Cellular Automata form a useful framework for some models of social interaction, for example the spread of gossip between people and the formation of ethnically segregated neighborhoods” (Gilbert & Troitzsch, 2005, p8).Chaos Theory:A term poplar in the 1980’s after the publication of James Gleick’s bookChaosin 1987. The term was used to describe in general the advances in what is now called “complexity science”. Complexity Science is the preferred term for the general area.Chaotic Attractor:SeeChaotic Behavior.Chaotic Behavior:See chaotic dynamics. AlsoWiki encyclopedia “ chaotically”Chaotic Dynamics:Although there are many definitions ofchaotic, we adopt that of Hirsch, Smale and Devaney (2004): Indynamical systems, three ingredients imply chaotic dynamics. These are:Sensitive dependence on initial conditions, 2) topological transitivity, and 3) dense periodic points. A mapping hastopologic transitivitywhen it has dense orbits, that is, for each sequence of points in its sequence spaceΣ, astincreases, the sequence converges to a periodic point in a set that is dense inΣ. Periodic points form an open set in a sequence spaceΣ, meaning not all points close to the points in the set are included in the set. The set of periodic points isdensewhen its mathematical closure is the full set.Complexity Science:The preferred generic term for the overall field under discussion that should be used to describedynamical systems,complex systems,complex adaptive systems,chaotic behavior, aNew Kind of Science (NKS)and the various methods used to explore the above. This term should replace Chaos Theory as a catch-all term.Complex Adaptive System (cas):Acomplex systemthat has the additional property ofadaptation. All complex adaptive systems arecomplex systems, but not all complex systems are complex adaptive systems.Complex Analysis:A branch of mathematics that studies complex numbers. A complex numberztakes the formzx + iy, whereisquare root of-1. Whereas the Real numbers can be thought of as a “line” or x axis, the complex numbers form the complex plane with the x-axis representing the Real number component, and the y-axis being the “imaginary” component.Complex Analytic Dynamics:seecomplex dynamics.Complex Dynamics:This is a different usage of the term “complex” in that it refers to dynamical systems on the Complex numbersrather than on the Real numbers,C. A complex numberRztakes the formz x + iy, whereisquare root of-1. Whereas the Real numbers can be thought of as a “line” or x axis, the complex numbers form the complex plane with the x-axis representing the Real number component, and the y-axis being the “imaginary” component.Complexity Theory:May refer toComputational Complexity Theory, but the term is also “sometimes used as a broad term addressing the study ofcomplex systems,including subjects such aschaos theory,artificial life, andgenetic algorithms”. Quote taken on October 1, 2005 from: http://en.wikipedia.org/wiki/Complexity_theory .Complex Number:A Complex Numberztakes the formz x + iy, whereisquare root of-1. Whereas the Real numbers can be thought of as a “line” or x axis, the complex numbers form the complex plane with the x-axis representing the Real number component, and the y-axis being the “imaginary” component. Seecomplex analysisandcomplex dynamics.Complex Systems:Anopen systemthat is “made up of a large number of active elements that … are diverse in both form and capability” (Holland, 1995) .Computational Cognitive Neuroscience:A branch of neuroscience wherein biologically realistic computer-based feedforward neural networks are used to replicate known results from psychology and neuroscience (O'Reilly & Munakata, 2000).Computational Complexity Theory:Part of the theory of computation dealing with the resources required during computation to solve a given problem. The most common resources aretime(how many steps it takes to solve a problem) andspace(how much memory it takes). Other resources can also be considered, such as how many parallel processors are needed to solve a problem in parallel. Complexity theory differs from computability theory, which deals with whether a problem can be solved at all, regardless of the resources required.” Computational Complexity Theory gives rise to classifications about decision problem solvability such asNP-complete problems. The above quote was copied on October 1, 2005 from: http://en.wikipedia.org/wiki/Computational_complexity_theoryContinuous Dynamical System:See Dynamical Systems.Critical Value (in the context of Dissipative Structures):Within a Benard Process an energy differential is established between warmer and cooler surfaces such that energy flows between the surfaces reducing the differential. As this differential increases beyond a certaincritical value,the system undergoes a dramatic physical change in which convection begins to dissipate energy along with conduction. During this process, visible structures emerge spontaneously within the flow. Prigogine called these “dissipative structures” because they act as an accelerant in the dissipation of the energy differential (Prigogine & Stengers, 1984).Cycle:SeeCycle Graph.Cycle Graph (or Cycle): “Ingraph theory, a cycle graph or cycle is a graph that consists of a single cycle, or in other words, some number of vertices connected in a closed chain. The cycle graph withnvertices is calledCn. A cycle with an even number of vertices is called “even cycle”, a cycle with an odd number of vertices is called “odd cycle”. Some sources allow cycles to repeat vertices. The cycle graphs can be “simple”, meaning that they don't repeat vertices, which is the usual definition in graph theory.” Copied on October 2, 2005 from:http://en.wikipedia.org/wiki/Directed_cycle .Deterministic Models:Olnick P 11, INDEXDigraph:SeeDirected Graph.Directed Cycle:Adirected cycle graphis a directed version of acycle graph, with all the edges being oriented in same direction. In a directed graph, a set of edges which contains at least one edge (orarc) from each directed cycle is called a feedback arc set. Similarly, a set of vertices containing at least one vertex from each directed cycle is called a feedback vertex set. There is a directed cycle through any two vertices in a strongly connected component. Copied on October 2, 2005 from: http://en.wikipedia.org/wiki/Directed_cycle.Directed Graph:Adirected graphordigraphGis an ordered pairG:=(V,A) withV, a set ofverticesornodes, andA, a set of ordered pairs of vertices, calleddirected edges,arcs, orarrows. An edgee= (x,y) is considered to be directedfromxtoy;yis called theheadandxis called thetailof the edge. A variation on this definition is theoriented graph, which is a graph (or multi-graph; see below) with an orientation or direction assigned to each of its edges. A distinction between a directed graph and an orientedsimplegraph is that ifxandyare vertices, a directed graph allows both (x,y) and (y,x) as edges, while only one is permitted in an oriented graph. A more fundamental difference is that, in a directed graph (or multi-graph), the directions are fixed, but in an oriented graph (or multi-graph), only the underlying graph is fixed, while the orientation may vary. A directed graph may or may not be allowed to haveloops, that is, edges where the start and end vertices are the same. By definition, this is forbidden in an oriented simple graph. Copied on October 2, 2005 from: http://en.wikipedia.org/wiki/Directed_graphDirected Network:ASocial Networkor other network that can be represented by aDirected Graph.Discrete Dynamical Systems:See Dynamical Systems.Discrete Event Simulation:Discrete event simulation (DE) is used to model the logical outcome of a series of related events over time, in the manner of a process flow chart. In this type of simulation, the simulator maintains a queue of events and these are sorted according to the simulated time they are modeled to occur. The simulator reads the queue and triggers new events as each event is processed so that outcomes can be evaluated. It is important to be able to access the data produced by the simulation, to discover logic defects in the design, or the sequence of events that are modeled. And example of commercial software for DE is Vite’.Dissipative Structures:Within a Benard Process an energy differential is established between warmer and cooler surfaces such that energy flows between the surfaces reducing the differential. As this differential increases beyond a certaincritical value,the system undergoes a dramatic physical change in which structures emerge spontaneously within the flow. Prigogine called these “dissipative structures” because they accelerate the dissipation of the energy differential (Prigogine & Stengers, 1984). There is broad speculation that the generalized phenomenon of “dissipative structures” is closely associated withself-organizationandemergenceincomplex adaptive systems.Dynamical Systems:“A dynamical system is a way of describing the passage in time of all points in a given spaceS. The spaceScould be thought of, for example, as the space of states of some physical system…. When we consider dynamical systems that arise in mechanics, the spaceSwill be the set of possible positions and velocities of the system. For the sake of simplicity, we will assume throughout that the spaceSis Euclidean spaceRnalthough in certain cases, the important dynamical behavior will be confined to a particular subset ofRn. “Given an initial positionX є Rn, a dynamical system onRntells us whereXis located 1 unit of time later, 2 units of time later, and so on. At time zero,Xis located atX0. One unit before time zero,Xwas atX-1. In general, the ‘trajectory’ ofXis given byXt. If we measure the positions ofXtusing only integer time values, we have an example of adiscrete dynamical system…. If the time is measured continuously withtє R, we have acontinuous dynamical system. If the system depends upon time in a continuously differentiable manner, we have asmooth dynamical system” (Hirsch, Smale, & Devaney, 2004, pp 140-141). Alternatively:“A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish each spring in a lake are examples of dynamical systems. A dynamical system has astatedetermined by a collection of real numbers. Small changes in the state of the system correspond to small changes in the numbers. The numbers are also the coordinates of a geometrical space—a manifold. Theevolution ruleof the dynamical system is a fixed rule that describes what future states follow from the current state. The rule is deterministic for a given time interval --only one future state follows from the current state.” Copied on October 5, 2005 from: http://en.wikipedia.org/wiki/Dynamical_system . See alsoContinuous Dynamical SystemsandDiscrete Dynamical System, System DynamicsandDeterministic Models.Dynamical Systems Theory:A branch of the mathematical sciences that relies heavily on differential equations and other multi-variable analysis techniques to studydynamical systems.The field includes both linear and non-linear dynamics. See alsoContinuous Dynamical SystemsandDiscrete Dynamical System, System DynamicsandDeterministic Models.Dynamic Capabilities: The capacity to renew competencies so as to achieve congruence with the changing business environment. This is done by appropriately adapting, integrating, and reconfiguring internal and external organizational skills, resources, and functional competencies to match the requirements of the changing environment (Teece, Pisano, & Shuen, 1997, p. 515).Emergence:“A process by which a system of interacting subunits acquires qualitatively new properties that cannot be understood as the simple addition of their individual contributions” (Camazine et al., 2001), p. 31). Also, “a systemic process of order creation through which properties and or structures come into being that are unexpected and unpredictable, given the known attributes of the componentagentsand environmental forces” Lichtenstein & McKelvey” DO NOT QUOTE .Entanglement:There are two distinct uses of this term, one in Complexity Leadership Theory, the other in quantum theory.In Complexity Leadership Theory, entanglement is the notion that different leader roles and leadership activities of various types can and do occur in the same territory, both physical and intangible (Uhl-Bien, Marion & McKelvey). For the quantum theory definition, see:Quantum EntanglementEnvironment:SeeObjective EnvironmentandEnacted Environment.Epistasis or Epistatic Interaction:KauffmanEquation-Based-Modeling:SeeDeterministic Models.Evolutionary Models:SeeArtificial Life,Cellular Automata,Evolutionary Economics,Genetic Algorithm, andPopulation Ecology.Feedforward Neural Network: “Afeedforward neural networkis aneural networkwhere connections between the units donotform adirected cycle. This is different fromrecurrent neural networks.” The above is quoted from: http://en.wikipedia.org/wiki/Feedforward_neural_networksFirm or Organization:The entity upon which population-level selection acts and capability selections are aggregated.Firm Performance:Defined in terms of the relative positioning of the focal firm against other firms with respect to the criteria driving selection for populations of organizational forms. Higher probability of survival in the population selection process is defined to be superior firm performance. In this sense, firm performance represents the firm’s current position on its idealized performance landscape. Because the specific criteria for selection may change over time, the specific firm performance variables described below may or may not reflect increased firm performance in the idealized sense as defined here. In sum, firm performance is conceptualized in this analysis as a firm-level aggregate measure of the firm’s potential for survival in a stochastic environment (Hannan & Freeman, 1989)(Hazy, 2004).Fitness:A measure of the survival value of a particularcomplex adaptive system (cas)in a given selection process and a givenenvironment.Fitness Landscape:An organization’s fitness landscape is the topography of the payoff surface in the state space of an adaptive system interacting with its objective environment (S. A. Kauffman, 1993; Levinthal & Warglien, 1999). In later writings for the social sciences, this has been called the organization’sperformance landscape(Levinthal & Siggelkow, 2001). See alsoPerformance Landscape.Flow:“To emphasize the dependence of a solution [to a differential equation] on the initial valuexo, we will denote the corresponding solution φ (t, xo). This functionφ: R X Rà R iscalled theflowassociated to the differential equation…. Sometimes we write this function as φt (xo)” Hirsch, Smale & Devaney, 2004, p 12 .GlobalPeak:A set of choices P = (a1, a2, a3,…, aN) is a global peak on theperformance landscapeif there are no otherlocal peakswith a higher performance value.Grammar Diversity:Grammars refers to laws of complementarity & substitutability internal to the system (Kauffman, 1993) that make up the rules for organizing and reconfiguring the capabilities within the system. “Diversity of grammars” refers to the variations in the possibilities for configuring and reconfiguring capabilities to take advantage of resources.Hamming Distance:Ininformation theory,theHamming distance, named after Richard Hamming, is the number of positions in two strings of equal length for which the corresponding elements are different. Put another way, it measures the number ofsubstitutionsrequired to change one into the other.For example:

- The Hamming distance between
- The Hamming distance between
- The Hamming distance between "

Copied on Octomer 5, 2005 from: http://en.wikipedia.org/wiki/Hamming_distance .1011101and1001001is 2.2143896and2233796is 3.toned" and "roses" is 3.Hierarchy (of system complexity):“In application to the architecture of complex systems, ‘hierarchy’ simply means a set of Chinese boxes of a particular kind.… Opening any given box in a hierarchy discloses not just one new box within, but a whole small set of boxes; and opening any one of these component boxes discloses a new set in turn. While an ordinary set of Chinese boxes is a sequence, or complete ordering, a hierarchy is a partial ordering—specifically a tree (Simon, 1973) p 5).Hierarchy (of agent status and influence with a system):For a system layer, or as Herbert Simon calls them, a single Chinese box (Simon, 1973), that exists within ahierarchy (of system complexity),individual components or “boxes” disclosed within the Chinese box may themselves be ordered relative to one another according to some metric, for example, status, influence, reward or power. This hierarchy of rank within a level ofHierarchythat is, relative rank among the component boxes within a Chinese box(of system complexity),is logically distinct from the,Hierarchyitself in that such a relative ranking effects the interaction dynamics among the component boxes within the Chinese Box, and thus the properties of the original Chinese box, but the ranking remains logically distinct from the higher level system itself, the original Chinese box.(of system complexity)Individual Leadership:“A process wherein an individual member of a group or organization influences the interpretation of events, the choice of objectives or strategies, the organization of work activities, the motivation of people to achieve the objectives, the maintenance of cooperative relationships, the development of skills or confidence by members, and the enlistment of support or cooperation from people outside the group or organization” (Yukl, 1998), p. 5).Interaction dynamics:seeEpistasis.Kinetics:The mechanism by which a physical or chemical change is effected. (Merriam-Webster’s Collegiate Dictionary 10th Edition.)Leader Role: An abstraction associated with the need for coordination in social systems, particularly when existing structures are inadequate (Barnard, 1938; Katz & Kahn, 1966).Leadership Activities:Specific actions and tasks that reflect the implementation of leadership methods in organizations. Examples include goal-setting practices, communication forums, and artifacts such as newsletters, formal or informal performance reviews, project management practices, etc. and otherleadership routines. Practices are assumed to vary by method leading to variation inroutines.Leadership Routines:Leadership routines are defined as theorganizational routinesthat serve organizational leadership functions. They include both leadership methods and practices. Examples include strategic reviews, internal strategy road shows, skip-level meetings, town hall meetings, budget reviews, management by objectives, etc.Local Neighborhood (on a Performance Landscape):Positions on the performance landscape from which a local peak can be achieved through single-step micro-adaptations.Local Peak:A set of choices P = (a1, a2, a3,…, aN) represents a local peak in the performance landscape if there is no other point on the surface that differs from P in only one dimension and that has a higher value of firm performance (Levinthal & Siggelkow, 2001).Long-Jump Adaptation:Anadaptive process, ormicro-adaptation, wherein several choices, connections or activities are changed simultaneously in an attempt to jump to a different position on the performance landscape which may lead, eventually after anadaptive walk, to a peak with the potential for higher performance than the local landscape (prior to the long-jump adaptation) would imply. Long Jump Adaptations are needed to avoid being trapped on a local performance peak. These micro-adaptations are associated with exploratory capabilities (Frenken, Marengo, & Valente, 1999; Gavetti & Levinthal, 2000; S. Kauffman et al., 2000; S. A. Kauffman, 1993; Rivkin, 1997).Lotka-Volterra Model:SeePredator-Prey Model.Lorenz Attractor:TheLorenz attractor, introduced by Edward Lorenz in 1963, is a non-linear three-dimensional deterministicdynamical systemderived from the simplified equations of convection rolls arising in the dynamical equations of the atmosphere. For a certain set of parameters the system exhibitschaotic behaviorand displays what is today called astrange attractor; this was proven by W. Tucker in 2001. The strange attractor in this case is afractalof Hausdorff dimension between 2 and 3. Grassberger (1983) has estimated the Hausdorff dimension to be 2.06 ± 0.01 and the correlation dimension to be 2.05 ± 0.01. The butterfly-like shape of the Lorenz attractor may have inspired the name of thebutterfly effectinchaos theory. Copied on October 5 2005 from: http://en.wikipedia.org/wiki/Lorenz_attractor .Lyapunov Function:In the theory ofdynamical systems, and control theory, Lyapunov functions, named after Aleksandr Mikhailovich Lyapunov, are a family of functions that can be used to demonstrate the stability or instability of some state points of a system. The demonstration of stability or instability requires finding a Lyapunov function for that system. There is no direct method to obtain a Lyapunov function but there may be tricks to simplify the task. The inability to find a Lyapunov function is inconclusive with respect to stability or instability. Copied on October 5, 2005 from: http://en.wikipedia.org/wiki/Lyapunov_function .Lyapunov Stability:Lyapunov stability is applicable to only unforced (no control input)dynamical systems. It is used to study the behavior of dynamical systems under initial perturbations around equilibrium points.Let us consider that the origin is the equilibrium point (EP) of the system. A system is said to be stable "in the sense of Lyapunov" if for every ε, there is a δ such that:

The system is said to be asymptotically stable if as

A system is stable in the sense of Lyapunov if there exists a

Lyapunov functionfor the system. Taken on October 5, 2005 from: http://en.wikipedia.org/wiki/Lyapunov_stabilityMarkovian Process:A stochastic process , but one that can be represented mathematically such that all relevant information is contained in the current state of the system. G&T p86 Olnick p 274. Also seeNon-Markovian ProcessandAbsorbing Markovian Process.Measure of Performance:SeePerformance.Meta-Capability:An organizational capability that acts upon the system itself to modify or extrapolate other capabilities of the system (Hazy, 2004).Meta-Matrix Representation:This approach assumes that any organization can be represented as a set of matrices linking personnel, resources, and tasks. Resources include knowledge resources which can also be explicitly represented. The meta-matrix representation, originally called the PCANSS formalism, defines the interconnections between these various types of organizational elements or nodes at a point in time in an idealized sense. The matrix linking tasks to tasks is called thePrecedence matrixbecause in represents the contingencies inherent in the task structure. The matrix linking individual personnel to individual resources (including knowledge resources) is called theCapabilities matrix. Personnel and tasks are linked in theAssignments matrix; resources to tasks in theNeeds matrix; personnel to personnel in theSocial Network matrix; and resources to resources in theSubstitutions matrix. In theory, when all personnel, resources, and tasks are listed and their interactions are defined, a complete representation of the organization’s network configuration can be defined at a point in time (Krackhardt & Carley, 1998).Meta-Process:Used to distinguish these processes that operate on the system from those directly supporting the organization’s tasks (So & Durfee, 1998). Leadership processes are a special case of meta-processes (Hazy, 2004).Micro-Adaptation:A single adaptation event resulting from the action of adaptive capabilities. A micro-adaptation changes the position of the firm along its performance or fitness landscape.Minority Game:Agamecommonly used in research ingametheorybecause it simulates the financial markets. In each time step, agents choose one of two sides, 0 or 1, and those on the minority side win a point. To see an agent-based-model of the minority game see: http://ccl.northwestern.edu/netlogo/models/MinorityGame .Multi-Agent Modeling:SeeAgent Based Modeling (ABM).New Kind of Science (NKS):A term coined and popularized by Stephen Wolfram to encompass the discoveries that have come fromcellular automatamodeling and analysis and their implications to information theory, computer science, science in general, mathematics, and natural philosophy.NKS:seeNew Kind of Science.New Science:Sometime used in the popular press to describecomplexity science. Also, seeNew Kind of ScienceNP-Complete Problems:see http://en.wikipedia.org/wiki/NP-completeObjective Environment:The objective environment of a firm directly affects that firm’s performance landscape and its survival independent of the perspective from within the organization. The objective environment is not independent of the organization, however, but the two are structurally coupled (Maturana & Varela, 1998), in the sense that they co-evolve with one another.Order (of a differential equation):The order of the highest derivative that explicitly appears in the equation (Hirsch & Smale, 1974), p22).Organization or Firm:SeeFirm or organization.Organizational Capability:The presence within a firm of a dynamic process that acts to acquire and/or deploy assets, tangible or intangible, in order to perform a task or activity to produce or improve results (Maritan, 2001; Nelson & Winter, 1982; Teece, Pisano, & Shuen, 1997). Organizational capabilities are in general “knowledge-based, firm specific, socially complex, and they generally cannot be simply acquired in factor markets” (Maritan, 2001, p. 514).Organizational Forms:Variations in structure across organizations that have different performance characteristics and thus different survival potential (Hannan & Freeman, 1989).Organizational Leadership:The “influential increment over and above mechanical compliance with the routine directives of the organization” (Katz & Kahn, 1966, p. 528). Organizational leadership is an organization-level construct necessary due to “imperfections and incompleteness” of organizational design, the problem of managing the boundary of the organization, “changing external conditions,” “the changing internal state of the organization produced by separate dynamics of several substructures of the organization,” and the challenges of human membership in organizations (Katz & Kahn, 1966, p. 530). It is proposed organizational leadership can also considered to be an organizational meta-capability that exists with varying degrees of proficiency in all organizations (Hazy, 2004, 2006, 2007, 2008, 2009).Organizational Routine:“All regular and predictable behavior patterns” (Nelson & Winter, 1982), p. 14) in organizations. The term includes “well-specified technical routines for producing things, through procedures for hiring and firing, ordering new inventory, or stepping up production of items in high demand, to policies regarding investment, research and development (R&D), or advertising, and business strategy about product diversification and overseas investment” (Nelson & Winter, 1982), p. 14).Organizational Slack:“The pool of resources in an organization that is in excess of the minimum necessary to produce a given level of organizational output” (Nohria & Gulati, 1996), p. 1246).Pattern (in Self-Organization):“A particular, organized arrangement of objects in space or time” (Camazine et al., 2001), p. 8).Payoff Function:The mathematical formula for calculating the reward to agents in agameor after a defined event in anagent-based-model (ABM).PCANSS Formalism:SeeMeta-Matrix Representation.Performance:The organizational process involving transactional leadership activities in interaction with organizational capabilities that exploits the organization’s resource and capabilities position in the environment to appropriate rents from existing markets.Performance Landscape:“A mapping of all possible sets of firm choices onto performance values (such as a profitability measures). If a firm’s set of choices is described by a vector of N choices, the performance landscape consists of N dimensions depicting the firm’s alternatives along each dimension and an N+1th dimension depicting the performance associated with each vector of N choices” (Levinthal & Siggelkow, 2001), p. 5).Performance Landscape Dynamics:The performance landscape dynamics of the organization is the variable that is used in this analysis to represent the interplay between the strategic choices of the firm (that locate that firm on its performance landscape) and the dynamic state of the environment (that may “morph” the performance landscape), and that therefore implies that the firm may need to respond to the environment to adapt to new performance conditions (Levinthal & Siggelkow, 2001).Period Doubling:A feature of complex systems wherein the periodic attractor of a dynamical system suddenly doubles in period as a factor in the differential equations increases, a special case ofbifurcation.Periodic Attractor: Q X Eà Q X E*Phase Space:For an open set Q in the vector space E, and astate space, Q X E, a phase space of the system isQ X E* where E* is the dual vector space of Q. The dual vector space of Q is the vector space of all linear maps E à R. It has been shown that E is isomorphic with E* and that the relationship between the state space and the phase space of a system is given by the Legendre Transformation λ: Q X E à Q X E*. (Hirsch & Smale, 1974), p 204, 292.Piecewise Continuous Function:A function with a finite number of bounded discontinuities in the sense that if x = c is a point of discontinuity of such a function then and both exit but are not equal (Lass, 1957).Point Attractor:Alimit pointof a dynamical system inphase space.Power Law Relationship:A power law relationship between two scalar quantitiesxandymeans that the relationship can be written as:where

a(the constant of proportionality) andk(the exponent of the power law) are constants. Power laws can be observed when a relationship is shown on a log-log graph a straight line is observed. This is because, taking the log of both sides, the above equation is equal to:which exhibits the same form as the equation for a line:

Power laws are observed in many fields, including physics, biology, geography, sociology, economics, war and terrorism. Power laws are among the most frequent scaling laws that describe the scaling invariance found in many natural phenomena.

Scale Free Networksexhibit a power-law distribution.Predator-Prey Model:Olnick P 105.Prisoner’s Dilemma:Agamecommonly used in research ingametheory. Critical elements of the game include players, or the prisoners, each of who acts independently and in which the incentive to defect, that is, turn against the other player, rather than to cooperate, (refuse to turn against the other player) is greater for the player regardless of what he assumes the other player do. “The "dilemma" faced by the prisoners here is that, whatever the other does, each is better off defecting, that is, confessing, than remaining silent. But the outcome obtained when both confess is worse for each than the outcome they would have obtained had both remained silent”. The above quote is from: http://plato.stanford.edu/entries/prisoner-dilemma/#Sym on October 1, 2005.Quantum Entanglement:Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. This leads to correlations between observable physical properties of the systems. For example, it is possible to prepare two particles in a single quantum state such that when one is observed to be spin-up, the other one will always be observed to be spin-down and vice versa, this despite the fact that it is impossible to predict, according to quantum mechanics, which set of measurements will be observed. As a result, measurements performed on one system seem to be instantaneously influencing other systems entangled with it. However, at this time classical information cannot be transmitted through entanglement faster than the speed of light. Copied on October 4, 2005 from: http://en.wikipedia.org/wiki/Quantum_entanglement .Recurrent Neural Network: “A recurrent neural network is aneural networkwhere the connections between the units form adirected cycle. Recurrent neural networks must be approached differently thanfeedforward neural networks,both when analyzing their behavior and training them. Recurrent neural networks…” often exhibitchaotic behavior.“Usually,dynamical systems theoryis used to model and analyze them.” Quoted areas are from on October 2, 2005: http://en.wikipedia.org/wiki/Recurrent_neural_networks.Reward Attribution:An important problem iscomplex adaptive system (cas)research involves the distribution of the resources appropriated by the cas to theagentsthat comprise the system. This process is calledreward attributionand can be encoded in apay-off functionor may use some other algorithm (J. H. Holland, 1975/1992).Routines:SeeOrganizational Routines.Scale Free Network Topology:Scale Free Networks are networks that have the characteristic that a) the frequency of nodes with a given number of connections and b) the number of those connections, exhibit a power-law relationship. One can understand how scale free networks arise in social networks when one considers the following observation: when nodes can choose their next link, for example, for status or access to resources, there is greater marginal benefit to the node by establishing a link to the most popular node available. This leads to a scale free relationship in many types of networks. For example, knowledge networks such as collaboration networks in academia, and the Worldwide Web, have been shown to be scale-free. These are sometimes referred to as “winner-takes-allnetworks”.Scenario:One of the strengths of the simulation method is the ability to run the model many times with identical or, in some cases, different inputs for each of the various parameters. Each of these runs is called a scenario.Self-Organization: “Self organization is a process in which pattern at the global level of a system emerges solely from numerous interactions among the lower-level components of the system. Moreover, the rules specifying interactions among the system’s components are executed using local information, without reference to the global pattern” (Camazine et al., 2001), p. 8).Semi-Markovian Process:Olnick p311.Sensitive Dependence on Initial Conditions:Seesensitivity to initial conditions.Sensitivity to Initial Conditions:aaa Also seethe butterfly effect.Single Step Adaptation:seeAdaptive WalkSmooth Dynamical Systems:See Dynamical Systems.Social Structures:Rules, resources, or transformation relationships that “bind” individuals and organize social systems (Giddens, 1984, p. 25).State:Astateof a system is information characterizing the system at a given time. In particular, it is information about the position of a point in a vector space and the first derivative of that point (Hirsch & Smale, 1974, p. 22).State Space:a space ofstatesis theCartesian productRN X RNof pairs (x, v),x, vinRN;xis the position,vis the first derivative (velocity) at a point in time.Stochastic Process:Olnick P 260. See also: Markov Process, Semi Markov Process, & Non-Markovian ProcessStrange Attractor:SeeChaotic AttractorandChaotic Behavior.Strategic Leadership:The organizational process aimed at appropriately adapting, integrating, and reconfiguring internal and external organizational skills, resources, and functional competencies to match the requirements of the changing environment (Teece, Pisano, & Shuen, 1997, p. 515).Structural Coupling:“When there exists a history of recurrent interactions leading to the structural congruence between two (or more) systems” (Maturana & Varela, 1998, p. 75).Sugarscape:A path breakingCellular Automatamodel (Epstein & Axtell, 1996).System’s Dynamic State:Whether the system is stable or unstable, or at the chaotic boundary between the two. Stable systems tend to dampen signals that perturb the system. Unstable systems amplify such disturbances and can drive the system out of a dynamic equilibrium associated with positions within a basis of attraction. Perturbations of a system in a position near the edge of chaos can cause the system to change state and shift to another attractor. One example of this is in the context of periodic attractors is bifurcation (Gleick, 1987; Lichtenstein, 1995; McKelvey, 1999; Prigogine & Stengers, 1984; Thietart & Forgues, 1995; Waldrop, 1992).Tag:Identifying information within an agent that may be made available to other agents during an interaction. The information from the tag may be used by other agents to inform the specific nature of the interaction dynamics. Tags can identify agents with particular skills, certifications, resources, knowledge, histories, pedigrees, status, membership or rights and/or responsibilities of membership, for example. Tags can improve interaction efficiency by relaying information about traits, e.g., trustworthiness, competence, etc.Tension:Seeadaptive tension.Unity:“An entity brought forth by an act of distinction” (Maturana & Varela, 1998), p. 40). 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