Complexity Explorer Santa Fe Institute

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1/f noise

A signal or process in which the relationship between energy and frequency is an inverted power law relationship: energy is proportional to 1/frequency. Also called 'pink noise'. 1/f noise is often seen in nature with earthquakes, avalanches, etc.


In relation to a complex adaptive system, adaption is a process by which “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, J. (1995). Hidden Order: How Adaptation Builds Complexity. Reading, MA: Addison-Wesley.

adaptation (biology)

In a biological context, an adaptation is a phenotypic trait that increases an individual's or group's fitness in a particular environment. The process of adaptation occurs via modifications of the genotype or behavior of an individual or group.


Often used in 'adaptive systems', the term 'adaptive' refers to interacting entities that individually or together are able to respond to environmental changes or changes between the interacting parts. The term can refer to a temporary modification to meet a changing context, or a long-term, permanent modification.

adaptive system

A natural or artificial system which takes on new states and configurations in response to its environment, generally improving on some explicit or implicit measure of utility.

adaptive walk

A term used to describe a trajectory of changes undergone by a system to better operate in a given environment. In the context of a fitness landscape, each change can be seen as a step that either improves the performance of the system (higher elevation within landscape) or degrades the performance of the system (lower elevation within the landscape). See also fitness landscape.

affine transformation

Roughly, for two sets of elements X and Y, if f is an affine transformation that takes elements of X to elements of Y, then looks like y =  f(x) = m x + b.


An individual that exhibits the capacity to achieve a goal or to effect an outcome. Often the term "agent" implies that an agent operates within a population and interacts with other agents as well as with a more passive environment.

agent-based model

A computational simulation in which the individual components ("agents") of a system are represented and interact explicitly. An agent-based model is typically iterated over time steps, with aspects of the agents updated at each time step. Agent-based models can be contrasted with models in which the behavior of the system is based on equations and individuals are not represented explicitly.







agent-based simulation

See agent-based model.


See artificial life.


Allometry describes a comparison of physiological or morphological ratios in living systems, (e.g. brain size to body size). Comparisons may be made over one individual's lifetime, among individuals of the same species or among different species.  See also metabolic scaling.


An educated guess or an assumption, used as a starting point, possibly later verified by experimental results or mathematical justification.

anthropic principle

The anthropic principle seeks to explain why the constants of the universe are consistently in the very fine band of possibilities in which life – specifically conscious life – could occur. The 'strong' anthropic principle places a purpose to the universe: to host conscious life. The 'weak' anthropic principle instead says that the observations being made by conscious beings are necessarily occurring in the very limited set of universes which are hospitable to life.

area-preserving map

The phenomena of "area-preserving" occurs within the abstract framework of the state space of a system. It dictates that for some dynamical systems with a relatively smooth state space (in topographical terms), when the system is allowed to run for a very long time, the average state of the system over that amount of time is approximately equivalent to the average state of that system over its state space. A physical example of this might be an atom of a gas in an enclosed chamber of finite volume. In the absence of friction, gravity, or any other force that might introduce asymmetry, one could let the atom bounce from surface to surface within the chamber. If this were allowed to continue for a sufficiently long amount of time, and one averaged the location of the atom over this entire time, the average location over this time would very nearly resemble the average location over the volume of the chamber (i.e. near the center of the chamber).

In the context of cartography, area preserving describes any method whereby a representation of a three-dimensional object (i.e. the Earth) is projected on a two-dimensional surface using techniques that preserve the relative sizes of different regions on the surface.


An ancient theorist (384-322 B.C.) who developed influential, though flawed, theories of motion. His systematic approach, focus on underlying principles and wide-ranging speculation provided a starting point for later scientific thinkers.

arms race

A term coined from the era before the first World War, representing a reinforcing (i.e. positive) feedback loop between two nations seeking to have the larger armed forces. The most well known arms race is the nuclear arms race of the Cold War.

In the modern era, the term has begun to describe competitions in which there is no definite goal beyond staying ahead of one's competitor. The term is often applied in biology, as in the description of the co-evolution of hosts and parasites or pathogen response to antibiotics.

artificial intelligence

Artificial intelligence (AI) is a branch of computer science concerned with the study and design of intelligent agents, often made to emulate human thought or decision making. Agents may be designed to perform tasks such as game playing, optimization, problem solving, modeling social behavior, and many others.

artificial life

Artificial life is a field of study that examines life and life-like systems through the use of computer, mechanical, and chemical models. Some topics of interest in the field include but are not limited to the origin of life, what life is and could be, self-organization, self-replication, and evolution.


Asymmetry refers to items which display no bilateral or radial symmetry. In economics or game theory, it refers to scenarios in which one or more players have more or less information than one or more others.


A term describing the situation in which events in a system occur at times independent of other events in that system. The term is to be contrasted with synchrony, which refers to a set of events that occur at the same time, or at correlated times.


In dynamical systems, an attractor is a value or set of values for the variables of a system to which they will tend towards over enough time, or enough iterations. Examples include fixed-point attractors, periodic attractors (also called limit cycles), and chaotic (also called "strange") attractors.


The process of self-generated change. Most frequently it is a system of chemical reactions such that each reaction is aided (catalysed) by the product of another in a closed and self-perpetuating sequence.


Refers to a system or agent that is able to behave independently of any external control. This does not necessarily place it outside the effect of external systems, but simply means that it is able to initiate actions on itself or its environment. Also used to refer to teleological, or active, agents – which are those with their own goals that often serve to cause different outcomes in situations that otherwise have identical circumstances.


Literally, 'self-creation'. This describes a process by which a system creates and maintains itself. The term was coined by Humberto Maturana and Francisco Varela in the context of self-maintaining chemical systems.

Baldwin effect

A process, proposed by American psychologist James Baldwin, by which acquired or learned traits or behaviors in organisms could eventually become genetically encoded.

basin of attraction

Within the phase space of a system, a basin of attraction (with respect to a given attractor) describes all the possible values for the system variables that will cause the system to tend toward a given attractor. This is similar to the way that all the surface water in a drainage basin will flow toward the point of lowest elevation. See also attractor.

bénard system

A simple physical system, consisting of a liquid in a container being heated from the bottom and which has been extensively studied by the Prigogine School because of its demonstration of self-organization and emergence. As the liquid in a container is heated from the bottom, at a critical temperature level (a far-from-equilibrium condition), there is the sudden emergence of striking hexagonally-shaped convection cells. Prigogine has termed these hexagonal cells "dissipative structures" since they maintain their structure while dissipating energy through the system and from the system to the environment. These "dissipative structures" are a good example of unpredictable emergent patterns since the direction of rotation of the convection cells is the result of the amplification of random currents in the liquid.

betweenness centrality

In graph theory or network theory, this term denotes the number of shortest paths between all pairs of nodes that pass through a given node in a graph or network.


A change or branching of the qualitative properties of a dynamical system. For example, in the logistic map, the behavior of the system repeatedly bifurcates as the growth parameter ("R") increases. The bifurcations correspond to period-doublings in periodic attractors of the system.

bifurcation diagram

A bifurcation diagram is a visualization of a dynamical system's bifurcations as a control parameter is systematically varied.

bipartite graph

Graphs that can be broken down into two disjoint sets of nodes, where edges may link nodes between two sets but not within a set. For example, imagine a graph of red and blue nodes, in which a red node can be connected to one or more blue nodes (and vice versa) but not nodes of its own color. Often useful for visualizing attributes of a population (e.g., the distribution of ethnic group memberships for individual people in a social network).


Processes of self-organization and emergence occur within bounded regions (e.g., the container holding the Bénard System so that the liquid is intact as it undergoes far from equilibrium conditions). In cellular automata the container is the electronic network itself, which is "wrapped around" in that cells at the outskirts of the field are hooked back into the field. These boundaries or containers act to demarcate a system from its environment, and, thereby, maintain the identity of a system as it changes. Boundaries channel the nonlinear processes at work during self-organization. Boundaries need to be both permeable in the sense that they allow exchange between a system and its environments as well as impermeable in so far as they circumscribe the identity of a system in contrast with its environments.

butterfly effect

The "butterfly effect" is a common way to describe the sensitivity to initial conditions in a chaotic system. It is a reference to the frequently used metaphor of a butterfly flapping its wings somewhere in the world and then much later a typhoon developing at another location as a distant, but direct, result of this.


In evolutionary or developmental biology, the term "canalization" refers to the ability of a population or species to produce the same phenotype in successive generations in spite of significant change in its environment or its genotype. The term "canalization" was coined by the British biologist C. H. Waddington.  See also robustness.

cascading failure

A process in which the failure of any component of a system increases pressure on other components, leading to their failure. A common example is the process by which a large-scale electrical blackout occurs.  Cascading failure is a common failure mode for many types of systems in which components are highly interconnected. 

catastrophe theory

In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of general singularity theory in geometry. Small changes in certain parameters of a nonlinear system can cause equilibria to appear or disappear, or to change from attracting to repelling and vice versa, leading to large and sudden changes of the behaviour of the system. However, examined in a larger parameter space, catastrophe theory reveals that such bifurcation points tend to occur as part of well-defined qualitative geometrical structures.

cellular automaton

A mathematical or computational system in which simple elements ("cells") are arrayed in a regular lattice. At a given time step, each cell is in some discrete "state" (e.g., 0 or 1), and at each time step, each cell updates its state using a function of its current state and the states of its neighboring cells. To define a particular cellular automaton, one must specify the dimensionality of the lattice, the neighborhood of a cell, the set of possible states, and the state update function used by each cell. Plural is "cellular automata".


A phenomenon seen in dynamical systems, in which the system's future behavior is highly sensitive to the initial conditions of the system. See also butterfly effect.

Church-Turing thesis

The Church-Turing thesis states that any function that is "computable" --- that is, that can be computed by an algorithm --- can be computed by a universal Turing machine (invented by Turing) or, equivalently, by the lambda calculus (invented by Church).

clustering coefficient

In graph theory and network analysis, the clustering coefficient measures the degree to which nodes exist in tightly connected groups. More specifically, the "clustering" of a node measures how many of its neighbors are also linked to one another, and the clustering coefficient of a network is the average of the clustering of each of its nodes.


The cohesiveness, coordination, and correlation characterizing emergent structures in self-organizing systems. For example, laser light is coherent compared to the light emanating from a regular light bulb, and flocks of birds exhibit qualitative and quantitatively different structures from isolated birds in flight. Oftentimes, these emergent structures are difficult to infer from only observing lower-level phenomena.



community stability

In an ecological context, the ability of an ecosystem ('a community') to withstand changes in the environment.

community structure

Community structure is one of many network descriptions, specifically the presence of groups with higher connections to themselves than to the rest of the network. Community structure can be defined several different ways, such as trait similarity and connections, or cliques defined by interconnections.

complex adaptive system

A complex, nonlinear, interactive system which has the ability to adapt to a changing environment. Such systems are characterized by the potential for the emergence of new structure with new properties. Complex adaptive systems (CASs) can evolve by random mutation, self-organization, the transformation of their internal models of the environment, and natural selection. Examples include living organisms, the nervous system, the immune system, the economy, corporations, and societies. In a CAS, semi-autonomous agents interact according to certain rules, evolving to maximize some measure like fitness to their environment.



complex network

A network with a combination of properties, often seen in real-world networks, which are not found in regular lattices or random networks. Examples include scale-free or otherwise heavy-tailed degree distributions, small-world connectivity, and high clustering.

complex system

A system composed of a large number of interacting components, without central control, whose emergent "global'' behavior—described in terms of dynamics, information processing, and/or adaptation—is more complex than can be explained or predicted
from understanding the sum of the behavior of the individual components.


This term means different things in different disciplines, and is not rigorously defined outside of a specific context. In general, the complexity of a system emerges from the interactions of its interrelated elements as opposed to the characteristics of those elements in and of themselves. Complexity science is the study of such emergent system behavior, and seeks to understand how the complex behavior of a whole system arises from its interacting parts. Complex behavior generally cannot be reduced to, or derived from, the sum of the behavior of the system's components.

conditional entropy

Conditional entropy measures the amount of information needed to describe one variable, given the value of another variable.


The connectionist approach presumes that high-level behaviors can be understood through interactions of networks of nodes, each of which are either uniform or relatively simple. This is a guiding philosophy behind the neural-network approach to artificial intelligence.

continuous time

The assumption used in some models (either mathematical or computational) that time is a continuous and infinitely divisible flow. Continuous time is approximated in computer simulations by floating point numbers. See also discrete time.


Convergence (in evolutionary computing) is a means of modeling the tendency for genetic characteristics of a population to stabilize over time. Convergence (in logic) refers to the property in which different sequences of transformations of the same state terminate at the same end state, independently of the path taken (they are confluent). Basically, it's when variables come together producing an emergent phenomenon.



critical phenomena

Critical phenomena refers to study of critical points in phase transitions. The melting, sublimation, and boiling curves between the phases of a typical substance represent the critical points for the phase transitions.


Criticality is a concept taken from nuclear engineering in which fissionable material is brought together in enough quantity to start a chain reaction. See also critical phenomena; self-organized criticality.


The study of control and regulatory mechanisms in both natural and artificial systems. The term was coined by mathematician Norbert Wiener in the 1940s.  The more modern fields of complex systems and artificial intelligence owe some of their concepts to the study of cybernetics in the 1940s and 1950s.


In graph theory, a cycle is a 'loop' in the graph. The term cycle is slightly more general than 'loop', as the term loop is often used to refer to a directed graph where the forward flow eventually brings one to a previous state. A cycle in a graph could also mean a source which broadcasts, and those broadcasts connect on the other side of the network


Darwinism refers to the general belief system about the natural world promoted in Darwin's "Origin of the Species" and other books, in particular, the central concepts of natural selection.


The number of links coming into or going out of the given node in a network.

degree distribution

The distribution (or histogram) giving the number or frequency of nodes in a network that have a particular degree (number of links coming into or out of that node).

deterministic dynamics

Determinism refers to mathematical models where the state at t+1 is absolutely determined by the state at time t. Deterministic dynamics refers to a simplified fluid model in which the dynamics are set by a fixed number of fluid equations. There is no probabilistic element in deterministic dynamics.

difference equation (temporal)

An equation in which the value of a variable at a given time is expressed as a function of the value of that variable at earlier times.

differential equation

A continuous differential equation relates the derivative of one or more variables with the current state and derivatives of itself or other variables. These could be used to describe how the system is changing based on how it has been changing and the state it is in. Equations like these have been used to describe fluid dynamics, populations, or any system in which change is itself dependent upon the state and previous flow of the system.


The general notion of dimension can be thought of as the number of different coordinates needed to specify a point distinctly from any other point. In the framework of a fractal dimension, dimension relates the logs of the rate of change of a measurable aspect of a fractal pattern as it is iterated.

directed graph

A graph where relationships between nodes, represented by edges, has the added constraint of how nodes may interact. This is often indicated on a graph by adding arrows to edges. 

discrete time

The assumption used in some models (either mathematical or computational) that time is divisible into discrete "steps"; a system's state is updated at each discrete time step in the model. Working with mathematical functions (maps) in discrete time potentially can yield markedly different behavior when compared to the equivalent function in continuous time.

dissipative structure

A dissipative structure is an organized structure in open systems which are operating far-from-equilibrium exchanging
energy and matter with outside environment. A dissipative system is characterized by the spontaneous appearance of symmetry breaking and the formation of complex, sometimes chaotic, structures.

A whirlpool is a dissipative structure requiring a continuous flow of matter and energy to maintain the form.



double auction

A double auction market is one in which the sellers and buyers can both give their bids simultaneously. The offers go up, and the sellers' price comes down, until they meet in some middle.

dynamical system

A system that is described by temporal change of a point in a state space (or phase space). At any given time, the system is in a particular state in its space, and it follows an evolution rule that describes how the system changes states over time. Generally, a continuous dynamical system will be described by differential equations, while a discrete dynamical system will be described by difference equations. Stochastic dynamical systems will have solutions related to a probability distribution, while deterministic dynamical systems will have exact solutions.



economic equilibria

Economic equilibria refer to points at which economic forces are stable, such as when price curve meets the demand curve. Economic equilibrium theory is a model of global economic activity that links dynamical systems with economic theory by postulating that there exists a set of prices or values which will result in an overall equilibrium.


The use of methods and tools from physics, such as those used statistical physics, critical phenomena, nonlinear dynamics, and stochastic processes, to approach problems in economics.

effective complexity

A measure, proposed by Murray Gell-Mann and Seth Lloyd, of the complexity of a system, based  on the description length of the system's regularities (and not including the random aspects of the system).

embodied cognition

Embodied cognition, such as the work of Barsalou on perceptual symbol theory, is a view of cognition that grounds all symbolic activity in the experiences of the body. For example, to understand the concept 'chair', we partially activate all recorded experiences of 'chair', building an abstraction of chair from recent and distant experiences.


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.

emergent behavior

Emergent behaviors are global-level attributes of system that arise from the interactions of the components of the system, and that are not explainable by the behavior of individual components of the system or the sum of the components acting as individuals.


The process whereby a world is brought forth by the interaction or structural coupling between an embodied agent and its medium or environment; also the study of the manner in which a subject of perception creatively matches its actions to the requirements of its situation. "Organisms do not passively receive information from their environments, which they then translate into internal representations. Natural cognitive systems...participate in the generation of meaning ...engaging in transformational and not merely informational interactions: they enact a world. " The term was coined by Francisco Varela, Evan Thompson, Eleanor Rosch, in The Embodied Mind (1991).


Entropy, in the thermodynamic sense, is the tendency of a system to move from a more ordered state to a less ordered state. In Boltzmann's statistical mechanics, the notion of "order" and "disorder", and thus the definition of entropy, corresponded to the number of possible microstates corresponding to a given macrostate. In information theory, Shannon entropy and Hartley entropy measure the distribution of discrete states in a system. A uniform distribution would have maximum entropy. Shannon entropy measures frequencies of states, while Hartley entropy ignores frequency and only examines the presence of states (out of all possible states).


The developmental processes of an organism from spores or egg to cell division and differentiation and the formation of organs. Epigenesis is contrary to the historical notion of 'preformationism'.


Equilibrium, meaning balance, occurs when all forces or influences on a system are balanced. In physics an example is when the sum of forces on an object result in no change in motion. In dynamical systems, an equilibrium is a solution or behavior of the system that does not change over time (a "fixed point" or "steady state").

ergodic process

A stochastic process such as a random walk can take on different "realizations" (where each realization has a particular initial condition and a particular sequence of steps).  A stochastic process is said to be ergodic if its average behavior over one (sufficiently long) realization is equivalent to its average behavior over all possible realizations.  A key reason to consider whether a process is ergodic is to determine if, given enough time, the process passes through all possible states with equal probability.   If it so, the ultimate behavior of the process is not dependent on the initial conditions. 

ergodic theory

A branch of mathematics that studies the long-term behavior of dynamical systems. See also ergodic process.

evolutionary arms race

An evolutionary arms race refers to the competition between two evolving populations to outperform each other. This is an analogy to a literal arms race such as the nuclear arms race between the Soviet Union and United States during the cold war. An example of this would be the rough-skinned newt (Taricha granulosa) evolving an increasingly strong toxin to protect itself from predation while the common garter snake (Thamnophis sirtalis) continues to evolve a resistance to the toxin.

evolutionary developmental biology

Evolutionary developmental biology (evo-devo) is a branch of biology that studies the evolution of developmental processes by examining differences in morphogenesis, growth, and cell differentiation across species.

evolutionary game theory

Evolutionary game theory differs from game theory by examining population effects as actors play different strategies upon each other, and use those fitness scores to reproduce. Thus, it examines both evolution of strategies (such as having a population of strategies change via selection, mutation, and recombination) and also the dynamics of the frequency of strategies in the population (such as stable centers of cooperation in a field of defectors in the evolutionary prisoner's dilemma).



Evolvability describes the capacity of a system to undergo adaptive evolution. Evolvability is the ability of a population of organisms to not merely generate genetic diversity, but to generate adaptive genetic diversity, and thereby adapt via natural selection.


An evolutionary feature, such as a structure or behavior, that is adaptive, but has non-adaptive origins.  While the feature evolved through the process of natural selection,  it may originate non-adaptively as a by-product of other adaptive trait.

Evolution as Opportunist- Quanta Magazine

A latent capacity for evolutionary innovation through exaptation in metabolic systems - Nature


The term used by the Prigogine School for those conditions leading to self-organization and the emergence of dissipative structures. Far-from-equilibrium conditions move the system away from its equilibrium state, activating the nonlinearity inherent in the system. Far-from equilibrium conditions are another way of talking about a criticalization in the values of parameters leading-up to a bifurcation and the emergence of new attractor(s) in a dynamical system.

fat tail

See long-tailed distribution.

Feigenbaum’s constants

Universal ratios discovered from bifurcation patterns occurring in one-dimensional function maps (such as the logistic map) with a single quadratic maximum, or 'hump'. The bifurcations relate to phenomena with oscillatory (cyclic) behavior, such as swinging pendulums or heart rhythms. The most well-known one, Feigenbaum's Delta, refers to the spacing between parameter values required to double the cycle's length, which decreases exponentially by a factor approaching approximately 4.669. The slightly less well-known Feigenbaum's Alpha refers to the scaling factor by which the x-values decrease for each periodic doubling on said function map, which approaches approximately 2.503.


In evolutionary biology, fitness is the relative success in reproduction, or occasionally survival, among members of a group of organisms.  In evolutionary computation (or genetic algorithms), fitness is typically an externally imposed measure of "goodness" of an individual in the population, where individuals represent candidate solutions to a problem.  See also fitness function.

fitness function

A evolutionary algorithm (or genetic algorithm) employs a fitness function to evaluate the performance or 'fitness' of various solutions to a given problem. The score that each solution receives has consequences on the probability that any particular solution will be used to generate future solutions.

fitness landscape

An n+1-dimensional function, or surface, used to visualize fitness over an n-dimensional trait space. The height of the surface at each point gives a fitness measure for a particular set of values for the traits of interest. Peaks and valleys in the surface represent local maxima and minima in the landscape.

food web

Network of organisms from selected ecosystems, typically representing predator-prey (or more generally, trophic) relationships between species of interest. Wolves, for example, might be linked to deer (since wolves prey on deer), who are in turn linked to the local vegetation.

Fourier transform

The Fourier transform decomposes a function of time (a signal) into the frequencies that make it up. The term refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time.


A geometric object or shape which exhibits self-similarity across scales.   See also fractal dimension.

fractal dimension

The term fractal dimension was originally used synonymously with Hausdorff dimension, but later grew to represent a general measure of how quickly length, area, or volume change with decreasing scale. The fractal dimension is not necessarily an integer quantity, meaning it can be a fractional value.

fractal landscape

A fractal landscape can be used to model the statistical self-similarity that exists in many natural phenomena. Fractal surfaces have been generated for use in video-games, film, and other CGI applications. For tools to generate fractal landscapes, Terragin and Grome are both accessible software packages.

Game of Life

Conway's Game of Life (by British Mathematician John Conway) is a two-dimensional cellular automaton with a simple set of rules. At a given time step, a cell is in one of two statesalive or dead.  The state of a cell at the next time step is a function of its current states and the states of cells in its Moore neighborhood (the surrounding 8 cells). The rules of Life are: 1) live cells with fewer than two live neighbors die (underpopulation); 2) live cells with more than three live neighbors die (overpopulation); 3) dead cells with exactly three live neighbors become alive (reproduction); 4) all other cells do not change state.   Despite the apparent simplicity of the individual cells' behavior, the cellular automaton can exhibit elegant, complex patterns over time. A wide variety of patterns can be achieved, with the user varying only the initial configurations of the cells.

game theory

The mathematical study of decision making in situations of cooperation or conflict with multiple actors, each trying to maximize their own gain or utility. 

genetic algorithm

Genetic algorithms are a family of computational search and learning methods inspired by biological evolution.  Evolution takes place on a population of individuals, each of which represents a candidate solution to a given problem.  At a given generation, each individual's fitness is calculated according to a user-defined fitness function.  A selection process probabilistically chooses the fittest individuals to reproduce (with variation resulting from crossover and mutation); their offspring make up the next generation.   The algorithm runs for either a fixed number of generations, or until an individual is found whose fitness is above a user-defined threshold. 

genetic drift

Genetic drift is the change in allele frequencies for a gene from one generation to the next due to random sampling events in a population. This is different from selection, where allele frequencies change because of a variation in reproductive fitness associated with each allele.


The genotype of an organism is that organism's full hereditary information, even if not expressed.

genotype-phenotype map

The often complex linkage between particular genes in the genotypes and their expression is known as a genotype-phenotype map. In some cases, the modification of one gene will effect several changes in the phenotype (pleiotropy). Conversely, in some cases change to in the genotype will cause no phenotypic change. 

Hamming distance

In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In another way, it measures the minimum number of substitutions required to change one string into the other, or the minimum number of errors that could have transformed one string into the other.

The Hamming distance between:

"complex" and "simplex" is 2.
11011011 and 01101001 is 3.
112358 and 336915 is 6.

Hausdorff dimension

The Hausdorff (or Hausdorff-Besicovitch) Dimension is a metric that can be used to calculate a fractal dimension of an object and is generalized as N = S^D. In this formulation, N is the number of pieces an object can be divided up into equally that have the same appearance as the original object (i.e. a square being divided up into smaller squares.) S is the scale of N in relation to the larger object (i.e. when the parts of a 4x4 square are made into 16 1x1 smaller squares, the smaller pieces in relation to the whole object is said to have the scale factor of 4, or S of 4). Finally, D is the dimension of the object, which operates with the scaling factor, in the manner of S^D, to describe how the length, area, and volume change as the object shrinks or gets larger. When calculating fractal shapes D is solved for by taking the natural logarithm of N and dividing it by the natural logarithm of S: D = ln N / ln S.

heavy tail

See long-tailed distribution.

heterogeneous agent models

Heterogeneous agent models allow one to have different types of agents with distinct behavior. Sometimes, these agent types are mathematically modeled as a continuum of agents. Unlike agent-based models, which are algorithmic and typically solved by simulation, heterogeneous agent models are formulated mathematically and typically solved analytically or numerically.

highly optimized tolerance

Highly optimized tolerance (HOT) is a proposed mechanism, developed by J. Carlson and J. Doyle, for explaining why complex systems often give rise to power-law scaling behavior. HOT involves optimizing systems to be robust in the face of uncertain environments, by making tradeoffs among yield, the cost of resources, and the tolerance to risks. The optimization mechanisms can be either natural (e.g., natural selection) or human-engineered. Thus HOT applies to both natural and human-designed complex systems.


Homeostasis refers to the ability of an organism or other system to maintain a stable internal environment, even as its external environment changes. This is often important for regulating the continued function of the system, and usually employs balancing feedback mechanisms (for example, burning available calories to maintain body temperature in cold weather).


A category of autocatalytic or self-replicative biochemical reaction networks where the constituents in the network act as catalysts for the production of other constituents (themselves catalysts) in the network. In addition, each constituent is able to instruct its own production, and is therefore autocatalytic. Hypercycles, which are a novel class of cyclic networks with nonlinear reaction kinetics, provide a principle and a mechanism for self-organization because they allow the evolution of a set of functionally coupled self-replicative entities.


A property of dynamical systems in which different stable states can co-exist depending on both the system's history and current trajectory.

information theory

Information theory is theory concerned with mathematical analysis of unpredictability in a system (quantified as "Shannon entropy") and its applications to communication and other models of information storage and transmission.


Isomorphism refers to the similarity between two systems or two processes such that the defined patterns of relations/operations among each system's elements are perfectly analogous and map neatly one-to-one between the systems, at some level of coarse graining. For example, two dynamical systems may be considered isomorphic if they can be represented with the same set of differential equations. Recognizing isomorphisms between analogous phenomena (such as bird flocking and fish schooling) allows prediction of the dynamics of both systems with the same general model.

keystone species

A keystone species is one that plays an important role in the composition or structure of its ecosystem for some reason other than abundance. If this species were removed, the ecosystem would become unstable and possibly collapse. For example a predator may be necessary to control the population size of its prey.


A formal grammar that recursively outputs symbols and strings of symbols based on a set of production rules in order to represent geometric shapes and patterns. An initial symbol or string is used as a seed to create a new string; this new string is then used as the seed for the next group of symbols. Symbols representing line length, line angle, and scale can be used to easily represent geometric patterns that are fractal. L-systems are named after the botanist Aristid Lindenmayer who pioneered their development and used them to categorize the algorithmic structure of plants.

Lagrangian coherent structure

Lagrangian coherent structures (LCSs) are the scale dependent key material lines that organize fluid-flow transport. They are dominant coherent structures which has been found in many geophysical systems such as oceans and the atmosphere. They act as a transport barriers to fluid flow. This means that flux across the LCS is negligible for the lifetime of the structure.

leverage point

A leverage point is a place within a complex system where a small shift at one point can produce large changes elsewhere. Related to tipping points

limit cycle

A limit cycle is a trajectory in a dynamical system that repeats itself (i.e., is cyclical, or periodic) and that is an attractor for at least one neighboring trajectory. 

limit points

Limit points are points in the phase space or state space of a system. There are three kinds of limit points: attractors, repellers, and saddle points. A system will tend toward an attractor, and away from a repeller, similar to the way in which a ball rolling across a smooth landscape will roll toward a basin and away from a hill. A saddle point is so-named because it resembles an equestrian saddle. It therefore functions as an attractor to systems originating in "higher" regions, and as a repeller to systems originating from "lower" regions.


A link can be thought of as a connection, relation, or association that when represented graphically illustrates a pathway with an origin and endpoint, referred to as a 'node' or a 'vertex'. Also called 'edges' and 'arcs', links can be directed or undirected.

logistic map

The logistic map is the simple recurrence relation: x_(n + 1) = x_n * R * ( 1 - x_n ), which produces chaotic behavior under certain parameters .  

This map is a version of Verhulst's logistic model of population growth, where x_n represents the population (as a fraction of carrying capacity) at time n, and R is the growth rate.  In order to keep x_n between 0 and 1, R is restricted to be between 0 and 4. 

long-tailed distribution

A statistical distribution characterized by a skewed, or a slow-decaying, tail of  extreme values.  Such distributions are often contrasted with Normal (Gaussian) distributions, which do not have skewed tails.  The significant weight of these extreme values in the population skew the mean away from 'typical' values in both median and mode.  In some cases, long-tailed distributions are characterized by power laws.  In the context of complex networks, long-tailed distributions are commonly found in a network's degree distribution

Lyapunov exponent

The Lyapunov exponent is a value that describes the rate at which two nearly identical trajectories within a dynamic system's phase space will separate from one another per unit of time. It is indicative of a system's sensitivity to initial conditions:  a large Lyapunov will reflect a system in which a very small perturbation of the system will cause a large change in the trajectory of the system within its phase space. 

For N-dimensional systems, there will be N Lyapunov exponents, one that measures the rate of separation along each dimension.  The largest of these is called the "maxiumal Lyapunov exponent" or MLE.  A positive MLE is one indicator that a system is chaotic. 


Evolutionary changes at the level of species or higher taxonomic groups.  Macroevolution occurs on geologic time scales, to be contrasted with microevolution, which occurs on short time scales.


A macrostate is a unique, discrete, and observable characteristic or property of a defined entity or system. Macrostates are considered the “ensemble” expression that correspond to an ensemble of microstates. For example, in classical thermodynamics the macrostates of matter defined by temperature, pressure, and volume can correspond to a huge number of microstates, each defined by the positions and velocities of  large numbers of individual particles that comprise the gas, liquid, or solid.

Markov chain

A random process that undergoes transitions from one state to another, in which the probability distribution of the next state depends only on the current state and not on the sequence of events that preceded it.

maximum entropy

A technique to model a complicated system where there are many interactions or effects that are too difficult, or uninteresting, to track. Max-ent works when you want to describe a system as a probability distribution over possible configurations (i.e., descriptions of the data at a fine-grained level).


A living organism's metabolism is the set of chemical reactions occuring within a cell that are required for the survival, maintenance, and growth of that organism.


Changes in allele frequencies within a species or population, where the changes are due to selection, migration, mutation, recombination, and genetic drift.   Microevolution occurs on short time scales, to be contrasted with macroevolution, which occurs on geologic time scales.


A microstate is a detailed description of the configuration of low-level components making up a system.  For example, in statistical mechanics, a microstate might be the locations and velocities of the individual particles making up a gas.  This is contrasted with macrostates, such as temperature and pressure, which are global measures of a system.  A particular macrostate typically corresponds to many possible microstates of the system. 


Morphogenesis refers to the formation of physical structures or body shape in an organism.

mutual information

Mutual information is the amount of information shared by two variables, often input and output. This could also be described as the decrease in entropy in one variable when the other is known.


A network (or graph) is a collection of elements, called vertices or nodes, connected by edges or links.  Edges typically are either one-way or two-way connections.   A network is often represented by an adjacency matrix.

network diameter

The diameter of a network is the longest minimum-path length between any two nodes in a network.


A node (also called a vertex) is a single element in a network.


Noise is an error or undesired random disturbance of a useful information signal. The noise is a summation of unwanted or disturbing energy from natural and sometimes man-made sources.


Refers to the class of nondeterministic polynomial time problems:  those for which there is no known polynomial-time algorithm to solve the worst-case instances, but for which there is a polynomial-time algorithm to verify whether a candidate solution to a problem instance is indeed a solution.   See also P.


An NP-complete problem is one that is (1) in NP and (2) any other problem in NP can be translated, in polynomial time, into an instance of the given NP complete problem. 


Ontogeny refers to the formation and development of an organism (or other persistent identifiable living system) over its lifetime.


An orbit refers to a time-ordered list of states of a dynamical system as it moves through its phase space.  This is analogous to the concept of an orbit in astronomy and subatomic physics.


Refers to the class of polynomial time problems:  those for which every instance can be solved in polynomial time (with respect to the size of the input).    See also NP.

Pareto optimality

A situation where no indivdual stands to gain further utility, whether in terms of money, energy, or some other asset, without doing so at the expense of another individual.

path dependence

Path dependence refers to the idea that current and future states, actions, or decisions depend on the sequence of states, actions, or decisions that preceded them---namely their (typically temporal) path.  For example, the very first fold of a piece of origami paper will determine which final shapes are possible; origami is therefore a path dependent art.

path length

In a network (or graph),  path length refers to the shortest distance (i.e., minimum degrees of separation) between two nodes (or vertices).   A network's average path length can be used to consider the efficiency of information flow or dissemination through a network.


A pathogen is an infectious organism that can cause illness in a host organism.   Examples of possible pathogens include bacteria, viruses, and fungi.


Pertaining to systems that undergo repeated cyclic behaviors, in a continuous system, the period refers to the length of time per full cycle, such as the time it takes for a pendulum to swing back and forth to its original position. For discrete systems such as those usually expressed by recursive equations or computer simulations, the period may be measured by the length of the sequence of repeated states. For example, a system that cycles through  three  values before repeating has a period of three.

phase portrait

A phase portrait of a dynamic system refers to a summary of every trajectory from every conceivable initial condition in the phase space of that dynamic system. This is usually plotted on a graph along axes of the relevant state variables, and may illuminate stable orbits and attractors.

phase space

A phase space is a multidimensional space that represents all possible states of a dynamical system. The dimensions of the phase space reflect the variables of the system. Each point in the phase space therefore represents a possible state of the system. See also state space.


An organism's observed characteristics, such as morphology, development, or behavior.


Phylogeny describes the evolutionary history of a gene, species, genus, or other evolutionary unit.

power law

A power law is a functional relationship between two variables of the form f(x) = \alpha x^{k}.  The degree distribution among nodes in growing scale free networks and other phenomena involving preferential attachment, such as continuously self-compounding acquisition proportionate to current assets, create power-law distributions.

principle of maximum entropy

The principle of maximum entropy states is based on the premise that when estimating the probability distribution, one should select that distribution which leaves the largest remaining uncertainty (i.e., the maximum entropy) consistent with the constraints. That way, no additional assumptions or biases have been introduced into the calculations.

public-goods games

Public-goods games are generalized economic games where a finite amount of a resource is available to a community, whose members decide how much of this resource each of them utilize. Collectively, the community is richest by everyone conserving the resource for future renewal, although there is always temptation for a greedy individual to use more than their fair share. This incentive is due to the possibility of reaping all the benefits from individual enrichment without incurring most of the negative consequences, despite the total net harm to the community outweighing this private benefit. Multiple farmers' cattle grazing on a common pasture is a classic example of this type of game. This situation is often modeled as an N-player prisoner's dilemma.

punctuated equilibrium

The pattern can be described as long periods of evolutionary stability with brief periods of rapid evolutionary change found (either periodically or randomly) along the timeline. It's validity for speciation is contentious, but the pattern of fits and starts in rates of change in evolutionary processes has been identified in several (natural and artificial) evolutionary systems. Many independent factors (either endogenous or exogenous) are sufficient for producing punctuated equilibria. Originally proposed by Stephen Jay Gould to explain patterns of new species in the fossil record that did not fit Darwin's principle of gradualism.

random network

A graph where all the nodes, initially unconnected, are linked to other nodes at random with a set probability – usually denoted k. For example, a random network specified with k=0.5 gives all pairs of nodes in the network a 50% chance of being connected.  Among the many notable outcomes is that of an approximate Poisson degree distribution. With this distribution, the graph quickly forms a cohesive giant component as the average degree per node surpasses 1.    Also known as an Erdös-Rényi graph.

random walk

A type of stochastic process in which an entity (e.g., a molecule, a animal, a stock price) follows a path consisting of  a series of steps in random directions. For very small steps, a completely random walk approaches Brownian motion.





recurrence relation

In mathematics, a recurrence relation is an equation that recursively defines a sequence. Once one or more initial terms are given, each further term of the sequence is defined as a function of the preceding terms. The Fibonacci numbers are the archetype of a linear, homogeneous recurrence relation with constant coefficients. The logistic map is another common example.


A repeller is one of the three types of limit points in the phase space of a dynamical system, with the other two being attractors and saddle points. Repellers have the effect of directing the trajectory of a system away from themselves within the phase space.


Resilience describes the ability of a system to persist and maintain its core functions and/or purpose in the presence of disturbances, stresses or other changes in its environment.


Robustness describes the ability of a system to maintain a certain behavior, trait, or characteristic regardless of changing environmental conditions. Robustness is often contrasted with optimization, especially in human-constructed systems.  This is because the quality of being robust usually requires that one or more components of a system operate at sub-optimal levels in some situations in order to maintain the ability to operate acceptably at any level under most conceivable scenarios. Conversely, a system that is highly optimized typically operates very well under certain conditions but then becomes less than operational if those conditions change.

saddle point

A saddle point refers to a point in the phase space of a system that attracts the trajectory of a system when it originates from some directions, and repels the trajectory of a system when it originates from other directions. The name is derived from its resemblance to an equestrian saddle when the phase space is visualized in three-dimensional space.

scale-free network

A scale-free network is one that has a power-law degree distribution.  Under such a degree distribution, the vast majority of nodes have low degree and only a small fraction of nodes (hubs) have high degree.  Because of this property, scale-free networks tend to be robust to random node failure but vulnerable to targeted attacks upon hubs.

second law of thermodynamics

The second law of thermodynamics states formally that isolated systems always increase in entropy. This could be equivalently stated by noting that they trend towards a state of uniformity or equal mixing.


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.

self-organized criticality

Self-organized criticality refers to the notion that there can be systems which have critical points that will be reached spontaneously via the system's dynamics, without the need for careful tuning of parameters.   More specifically, the critical point is an attractor.  The term "self-organized criticality" was coined by Per Bak, Chang Tao, and Kurt Wiesenfeld in a 1987 paper in Physical Review Letters


Self-similarity is a phenomena that occurs when the structure of a sub-system resembles the structure of the system as a whole, and then the structure of a sub-system within that sub-system resembles the structure of the larger sub-system, and so on.  Self-similarity is the defining property of fractals

sensitive dependence on initial conditions

A system's sensitivity to initial conditions refers to the role that the starting configuration of that system plays in determining the subsequent states of that system. When this sensitivity is high, slight changes to starting conditions will lead to significantly different conditions in the future. Sensitive dependence on initial coditions is a defining property of chaos in dynamical systems theory. 

Shannon entropy

Shannon entropy, described by mathematician Claude Shannon, is a measurement of the unpredictability of a "message source" (or, equivalently, of an ensemble of messages).   The message source can be any system that produces output, and whose output can be modeled by a random variable.  Shannon entropy is typically measured in units of "average bits per message", and was shown by Shannon to be the minimum number of bits (on average) needed to encode a message from the given message source. 

Shannon information

See Shannon entropy.

small-world network

A network in which most pairs of nodes are not connected, but there are a small number "long-distance" links that result in low average path length and a high clustering coefficient


The state of a system represents the current location of that system within its phase space, which in turn represents the set of current values for the parameters used to define and measure that system.

state space

The state space of a system is a metaphorical region representing every possible state of a given system. See also phase space.

statistical mechanics

The field of statistical mechanics seeks to explain how macroscopic behaviors of a system emerge from the statistical properties of large numbers of "microscopic" components making up the system.   Statistical mechanics was originally developed by Ludwig Boltzmann as a foundation for thermodynamics, but the ideas and techniques of statistical mechanics have since been applied in a large number of fields, ranging from physics to social sciences. 

stochastic process

A stochastic process is process whose behavior (i.e., transition from one state of the system to a successor state) has random or probabilistic components.  A classic example is a random walk.

strange attractor

A strange attractor is a type of attractor that has fractal structure. Strange attractors most often occur in dynamical systems in regimes exhibiting chaos.

sub-linear growth

Sub-linear and super-linear growth refer to departures from a linear growth curve. Super-linear growth means "growing faster than linear growth", and sub-linear growth means "growing slower than linear growth".

super-linear growth

Super-linear and sub-linear growth refer to departures from a linear growth curve. Super-linear growth means "growing faster than linear growth", and sub-linear growth means "growing slower than linear growth".


Synchrony refers to events that occur simultaneously, or at correlated times.   Contrast with asynchrony.

synthetic biology

A field that uses gene sequencing and other artificial methods of manipulating DNA to create biological constructs to fulfill certain tasks, such as inhibiting bacteria growth, having nanotechnology deliver medicine inside of white blood cell sheaths, or any artificially created biological system.

system dynamics

System dynamics is one approach to the study of complex systems over time. System dynamics focuses on stocks, flows, and feedback, often described by differential equations.


Thermodynamics is a field of the natural sciences that focuses on the relationships among heat, energy, work and entropy.

time series

A time series is a set of sequential measurements taken at regular time intervals to indicate the change in the state of a system (i.e. its behavior) over time.

tipping point

Places where a small change in the input can dramatically affect the outcome. 

The term is said to have originated in the field of epidemiology when an infectious disease reaches a point beyond any local ability to control it from spreading more widely. A tipping point is often considered to be a turning point. The term is now used in many fields to describe almost any change that is likely to lead to additional consequences. In the butterfly effect of chaos theory, for example, the small flap of the butterfly's wings that in time leads to unexpected and unpredictable results could be considered a tipping point. 


An approach to modeling, economics or computing which focuses on centralized control. The decentralized approach is therefore called "bottom-up".

tragedy of the commons

When a group of people share a resource, such as the original English commons of a town, each person has an incentive to use the resource but no incentive to restore the resource, sometimes resulting in a mentality that may ruin the shared resource for all. This is called the "tragedy of the commons".


The trajectory of a system refers to the sequential states of that system, or its path, across its phase space, originating at the point of its initial condition within the phase space. See also orbit.

Turing machine

The British mathematician Alan Turing devised a hypothetical machine to carry out mathematical "algorithms".   This construct is now termed a "Turing Machine".   A given Turing machine consists of a linear "tape" containing readable/writable "cells", a "tape head" that can either read or write a symbol on the current tape cell or move to the left or the right on the tape, a set of "states" that the machine can be in at a given time, a initial state that the machine starts in, and a set of rules that map the current state and contents of the current tape symbol into an action for the tape head to take.  The machine proceeds over a series of time steps until it reaches the "halt" state (if it is a "halting" machine).  Turing's goal in devising this construct was to formalize the notion of "definite procedure" (or "algorithm").  A given Turing machine corresponds to what we would now call a "program" or "algorithm".  A universal Turing machine corresponds to what we would now call a "programmable computer".  



universal Turing machine

A universal Turing machine is a Turing machine that can emulate the operation of any other Turing machine on a given input.   The universal Turing machine, designed by British mathematician Alan Turing, gave an early blueprint for programmable computers. 

utility (economics)

In economics, the term utility refers to the set of values an economic agent puts on various goods (or services).


Versatility, in the context of networks, is a measure of a single node in a network.  It is defined in as the standard deviation of degree differences between the node concerned and the neighbors of the node. This measure quantifies the heterogeneity of the neighbors of a node. In social networks a higher value of versatility means the node connects with a wide variety of people.

von Neumann neighborhood

In cellular automata, the von Neumann neighborhood is classically defined on a two-dimensional square lattice and is composed of a central cell and its four adjacent cells.

white noise

A signal or process that contains equal energy at every frequency within a certain range. Auditory white noise, for example, contains roughly equal energy at every audible sound frequency.

Zipf's law

Zipf's law is an empirical observation that, in certain systems,  if values (e.g., word frequencies in a text) are ranked from most to least frequent, each value is inversely proportional to its rank in the list. The term originates from the name of the American linguist George Kingsley Zipf, who noted the phenomena in reference to the frequency distribution of words in any given language, and the fact that the frequency of each word is inversely proportional to its rank in the list. This same model has been applied to various types of data, such as the relative populations of different cities, the income rankings of individuals, and the size rankings of corporations.