The level at which a system is observed. For example, one can observe the coast of England from a satellite or from a jet liner or from a low flying plane, or from walking along the coast, or from peering down into the sand and rocks on a cove beach that you are standing on. Each of these perspectives is of a different scale of the actual coast of England. Fractals are geometric patterns that are self-similar on different scales.

See: Fractal; Power Law

Bibliography: Kaye (1989); Schroeder (1991)

A type of network, studied by means of graph theory, in which some of the nodes act as highly connected hubs but most other nodes have a lower degree of connectivity. They are called “scale-free” because their structure and dynamics are independent of the system’s size defined in terms of number of nodes. A scale-free network has the same features no matter what number of nodes is in the network. Scale-free networks also exhibit a power law distribution and may be more resilient in the face of loss of connectednes than hub networks which cannot withstand the loss of the hub.

See: Fractal; Scale; Graph Theory; Small Worlds

Bibliography : Barabási (2002); Watts (1999).

A process in a complex system whereby new emergent structures, patterns, and properties arise without being externally imposed on the system. Not controlled by a centralized, hierarchical “command and control” center, self-organization is usually distributed throughout a system. Self-organization requires a complex, nonlinear system under appropriate conditions, variously described as “far-from-equilibrium” or criticalization. Studied in physical systems by Ilya Prigogine and his followers, as well as the Synergetics School founded by Hermann Haken, self-organization is now studied primarily through computer simulations such as cellular automata, Boolean networks, and other phenomena of artificial life. Self-organization is recognized as a crucial way for understanding emergent, collective behavior in a large variety of systems including: the economy; the brain and nervous system; the immune system; ecosystems; and the modern large corporation or institution. The emergence of new system order via self-organization is thought to be a primary tendency of complex systems in contrast to the past emphasis on the degrading of order in association with the principle of entropy (second law of thermodynamics). In recent perspectives, rather than fighting against entropy, self-organization can be understood as a way that the total entropy of a complex system along with its environment(s) increases.

Now that we have a better handle scientifically on how self-organization takes place, it is easier to recognize instances of it in the world around us. For example, self-organization could be an appropriate way of understanding how a hospital staff may spontaneously re-organize itself to respond more effectively to a sudden influx of critically ill patients. This is what seems to have happened, for example, at Beekman Downtown Hospital in Manhattan during the tragedy of 9-11-2001 when the staff coalesced into novel treatment teams to handle the tremendous inflow of seriously wounded victims. Self-organization may also take place in innumerable other ways, for example, the change in family dynamics that results when a family member enters a hospice program, or the emergence of novel ways to provide care to a seriously ill that comes from interactions among the patient, nurses, physicians, other healthcare professionals, support staff and family members when patients have multiple chronic diseases.

See: Coherence; Dissipative Structures; Emergence; Far-from-equilibrium

Bibliography: Eoyang & Olson (2001); Goldstein (1994); Nicolis (1989); Nicolis & Prigogine (1989).

Formulated by the late physicist Per Bak, a phenomenon of sudden change in physical systems in which they evolve naturally to a critical state at which abrupt changes can occur. That is, when these systems are not in a critical state (i.e., they are characterized by instability), output follows from input in a linear fashion, but when in the critical state, systems characterized by self-organized criticality act like nonlinear amplifiers, similar to but not as extreme as the exponential increase in chaos due to sensitive dependence on initial conditions. That is, the nonlinear amplification in a self-organized, critical system follows a power law instead of an exponential law. Such systems are self-organized in the sense that they reach a critical state on their own. Examples of such systems include avalanches, plate tectonics leading to earthquakes or stock market systems leading to crashes. Because these systems follow power laws, and because fractals also show a similar mathematical pattern, it may be that many naturally occurring fractals, such as tree growth, the structure of the lungs, and so on, may be generated by some form of self-organized criticality.

See: Bifurcation; Catastrophe; Instability; Power Law; Self-organization

Bibliography: Bak (1996).

The property of chaotic systems in which a small change in initial conditions can have a hugely disproportionate effect on outcome. Sensitive dependence on initial conditions is popularly captured by the image of the butterfly effect. Sensitive dependence on initial conditions makes the behavior of chaotic systems largely unpredictable because measurements at initial conditions always will contain some amount of error. The late mathematical metereologist Edward Lorenz uncovered this concept in his work on weather forecasting. He noticed that a seemingly insignificant difference in an initial parameter in a forecasting system modeled on his computer led to very different forecasts.

See: Chaos; The Butterfly Effect

Bibliography: Lorenz (1993); Ott (2003).

The management/complexity theorist Ralph Stacey’s term for the set of informal relationships or networks among people in an organization which exists in tandem with the official and”legitimate” network or hierarchy. The shadow organization is not focused on the same stabilizing objective as the official organization, so it is a ripe ground for the instability required for self-organization and the emergence of more adaptable organizational structures and processes. Effective leaders take into consideration both the mainstream and the shadow systems, even capitalizing, according to Stacey on the potential friction between them.

See: Edge of Chaos; Far-from-equilibrium
Bibliography: Stacey (1996)

A type of graph network in which the connectivity among nodes leads to the formation of pathways linking an unusually large number of nodes. The small world phenomenon was made famous in the play(movie) “Six Degrees of Separation” and the “Kevin Bacon number”, which refers to the idea that any actor can be linked through his or her film roles to the actor Kevin Bacon. In both of these, it has been shown mathematically and through experimentation that nearly everyone on the planet is remarkably linked by no more than six linkages in a network comprising all the relationships between people.

See: Graph Theory; Scale-free Network

Bibliography: Barabási, A. L. (2002); Watts (1999).

The opposite of “instability,” the property of a system which stays pretty much the same after being disturbed by internal or external forces or events. For example, the deeper the keel of a sailboat, the more stable it is regarding the wind and currents. A running gyroscope is stable with respect to changes affecting its centrifugally determined level plane. Stability is sometimes used as a synonym for equilibrium or with the state of a system circumscribed within a particular attractor regime.

See: Equilibrium; Far-from-Equilibrium; Instability

Bibliography: Nicolis (1989); Nicolis & Prigogine (1989); Ott (2003).

Two terms coined by the editor of Wired Magazine Kevin Kelly for two antithetical management processes. “Clockware” are rational, standardized, controlled, measured processes; whereas “swarmware” are processes including experimentation, trial and error, and risk-taking. Clockware processes are seen in linear systems whereas swarmware is what happens in complex systems undergoing self-organization as a result of the nonlinear interaction among components.

See: Cellular Automata; Complex Adaptive System; Self-organization

Bibliography: Kelly (1994).

A theory about the emergence of new biological forms put forward by Lynn Margulis which posits that cooperation or symbiosis among two or more distinct types of organisms can lead to the emergence of radically novel types of organisms. It is believed that primitive organisms called eukaryotes incorporated certain elements of the aerobic bacteria that had been ingested into them and that out of this symbiotic relation, the more advanced prokaryotic cells resulted with the novel features of nuclei and membranes. Symbiogenesis manifests a new interpretation of evolution whereby other mechanisms besides variation and selection may be at work. It also represents a growing recognition of the importance of cooperative relationships among species instead of the more typical emphasis on competition and predator-prey relationships.

See: Co-evolution; Emergence

Bibliography: De Duve (2005); Margulis & Sagan (2002); Reid (2007)

A phenomenon that can occur in complex systems in which system components or agents align themselves in a startling coherence. A striking example can be seen in the dramatic synchronization of lighting in certain species of fireflies (what we used to call “lightning bugs” as children). This can be seen inside the Great Smoky National Park near Elmont, Tennessee, during mid-June at about 10 PM every night when thousands of fire flies flash together according to a highly synchronized pattern: After six seconds of total darkness, thousands of lights flash in perfect synchrony six times in three short seconds; the pattern then repeats itself over and over again. A similarly synchronization of firing among fireflies can be observed in parts of Thailand. Research has shown that synchronization takes place without any “leader” firefly. Instead synchronization develops out of the interaction among the fireflies. Specifically, under the right conditions, signals from one to the other become resonated in concert. In human systems, synchronization is evident during sporting events when fans in a stadium combine movement into the famous “wave” of hands.

A destructive kind of synchronization was responsible for the collapse of the Tacoma Narrows Bridge on December 11, 1940. A confluence of high winds and too much structural coherence that was built into the bridge led to a resonance of vibrations affecting the bridge leading to the collapse of the bridge’s structure.

See: Coherence

Bibliography: Strogatz (2003)

A collection of measurements of the variable(s) of a system as it evolves over time. Traditionally, times series data were graphed with time on the x-axis and some system variable on the y-axis. For example, the time series of an oscillating (periodic) system such as a forced pendulum or a metronome would show a curve depicting the speed of the pendulum bob going up and down like hills and valleys over time. However, as the result of dynamical systems theory, time series are now usually graphed in phase or state space with either two or more variables marking each dimension, or one variable is mapped against a time lagged version of the same variable. By graphing times series data in phase space, attractors can be identified more easily. Our ability to graph such times series and to determine their attractors has been greatly accelerated by the rise of the personal computer.

See: Attractor; Phase Space

Bibliography: Guastello (1995); Ott, Sauer, Yorke 1994)

 

A hypothetical “universal” computer envisioned by the great English mathematician and founder of modern computer languages, Alan Turing (who incidentally helped the British break the famous “Enigma Code” of the Germans during World War II.) Turing used this concept of a “universal computer” to prove that there were some mathematical problems which could not be solved by a “mechanical” procedure (or algorithm) generated on a computer, that is, there are certain well-defined mathematical problems which are not computable.

See: Algorithm; Church-Turing Thesis
Bibliography: Goertzel (1993); Penrose (1989); Sulis in Robertson and Combs (1995)