Complexity Explorer Santa Fe Institute

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In relation to a complex adaptive system, adaptation 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”.

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.

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.

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.

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.

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.

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.

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.


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.

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.


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.


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.

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.

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).

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.

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.


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.


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.


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.


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.

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.



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.


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.


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).


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. 

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.

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.  


A evolutionary algorithm (or genetic algorithm) employs a fitness function to evaluate the 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.

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. 

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.


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.

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. 


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

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.


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. 

natural selection

The disproportionate propagation of heritable characteristics that are advantageous for the system



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.


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


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. 


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.

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.

phase transition

These are state changes in the underlying system - when you change the macroscopic variables of a system sometimes its properties will abruptly change, often in a dramatic way.  These transitions may be reversible (such as the transitions between bear and bull markets or water freezing and thawing) or unidirectional (such as the development of Type II diabetes or combustion). 


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 formf(x)=ax^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.


Transmission of phenomena, such as an idea, light, sound, disease or current through a system.


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.



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. 

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

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.

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.


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. 

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".

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 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. 


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.