Complex systems are all around us. They are seen in the ways that migrating birds organize themselves into flocking formations and that ants communicate to successfully forage. They are seen in the ways in which humans form social networks, and in the patterns of communication, social capital, and reputation that emerge from these networks. They are seen in the emergent power-law or fractal structures of plants, snowflakes, landslides, and galaxies, as well as in similar structural patterns of wealth and income distribution, stock market fluctuations, population distributions between cities, and patterns of urban development. Complex systems are often referred to as “wholes that are more than the sum of their parts,” wholes whose behaviour cannot be understood without looking at the individual components and how they interact.
Complex behaviour arises from the interplay, in densely interconnected systems, between multiplicative causation and positive and negative feedbacks. A signature of such systems is radically disproportional causation (i.e., small causes do not always produce small effects) or what is often called “nonlinearity.” Nonlinear systems can undergo sudden flips between stable states or equilibria. A second signature is the “emergence” of structured macroscopic patterns that are the outcome of the independent microscopic interactions of the entities in the system. These macroscopic patterns — be they hurricanes in Earth’s atmosphere or boom-bust cycles in global financial markets — often have enormous causal power.
Complex adaptive systems — predominantly living systems, including human social systems — exhibit all these features; but, in addition, they survive and reproduce within dynamic selection environments. To do so, they have sets of embedded rules that guide their action in response to their external environments. These rules evolve under selection pressure.
The formal study of complex systems began in the mid-20th century in mathematics, physics, computer science, systems engineering (including cybernetics) and meteorology; more recently, ecology, social science and cognitive science have made important contributions. Researchers now apply the insights of complexity theory to the behaviour of systems as diverse as fresh-water lakes, mammalian immune systems, financial markets, social networks, the Internet, and the power grid.
Mathematically, complex adaptive systems are multi-state variable dynamical systems characterized by a moderate degree of structured interactions and interconnections. State variables in these systems are often characterized by heterogeneous parameter sets and updating rules. Spatial and network relationships are often non-uniform and violate mean field theory assumptions. As a result, mathematical representations of these systems often do not have analytical solutions. Further, system behaviour is characterized by path dependence, nonlinearities, bifurcations, and threshold behaviour. Higher-scale or aggregate output patterns are often characterized by power-law statistical distributions.
