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Complex Systems: A Survey
M. E. J. Newman
TL;DR
Complex systems science addresses how many interacting agents produce collective behavior that is not trivially reducible to individual behavior, across examples from physics, biology, economics, and society. This survey synthesizes the field’s theoretical tools, modeling and simulation approaches, and annotated resources. It presents complex systems theory as a collection of general theories rather than a monolithic framework, within a broad field whose unresolved questions exceed current expertise.
Problem
Complex systems are difficult to study because their many interacting parts produce collective behavior, motivating theoretical approaches to understand them.
Method
The survey organizes complex-systems methods and theoretical tools, covering simplified mathematical models, computer simulations, agent-based modeling, and annotated references.
Results
The survey synthesizes a broad interdisciplinary field spanning diverse systems, methods, applications, and general theories.
Takeaways & Limitations
Complex systems offers a broad set of methods and many open questions, with areas of ignorance exceeding current expertise.
Abstract
from arXiv · showhide
A complex system is a system composed of many interacting parts, often called agents, which displays collective behavior that does not follow trivially from the behaviors of the individual parts. Examples include condensed matter systems, ecosystems, stock markets and economies, biological evolution, and indeed the whole of human society. Substantial progress has been made in the quantitative understanding of complex systems, particularly since the 1980s, using a combination of basic theory, much of it derived from physics, and computer simulation. The subject is a broad one, drawing on techniques and ideas from a wide range of areas. Here I give a survey of the main themes and methods of complex systems science and an annotated bibliography of resources, ranging from classic papers to recent books and reviews.
I. INTRODUCTION
Complex systems science studies interacting agents whose collective behavior can exceed the sum of individual behaviors. This interdisciplinary field advances through simplified theory, computational simulation, and reviews connecting general methods with concrete systems.
- The field is relatively new, broadly interdisciplinary, and gained substantial momentum in the 1980s alongside growing academic and industrial interest.
- Complex systems comprise many interacting parts whose collective behavior can be more than the sum of individual behaviors.
- Modeling and simulation: Researchers use simplified mathematical models to abstract important qualitative elements into solvable frameworks for scientific insight.
- Modeling and simulation: More comprehensive computer simulations represent interacting parts in detail and measure the emergent behaviors that appear.
- Modeling and simulation: Agent-based simulation is a particularly important computational approach, supported by software tools developed for complex-systems research.
- The review covers modeling and simulation methods, theoretical tools, and references on individual systems such as economies and ecosystems.
II. GENERAL REFERENCES
Complex systems is a young, rapidly evolving subject with general books and reviews that organize its topics for readers.
- General books and reviews bring together relevant complex-systems topics in useful ways.
A. Books
The reviewed books range from elementary introductions to more advanced treatments, while differing in breadth, disciplinary perspective, age, and acceptance of their ideas.
- The first two books are elementary, requiring little mathematics; Mitchell targets a popular audience, while Flake offers broader and more technical coverage.
- The review labels resources as elementary, intermediate, or advanced according to their expected mathematical background.
- Three more advanced books cover important topics but none comprehensively; one has a stronger social-science flavor because its authors are economists.
- Mandelbrot’s book predates complex systems as a recognized field and is considered a readable classic, although some of its ideas remain unaccepted.
III. EXAMPLES OF COMPLEX SYSTEMS
Individual complex systems are studied within specialized disciplines, while complex-systems approaches examine common patterns across examples including ecosystems, markets, and physical systems.
- Ecosystems and stock markets are primarily studied in ecology and finance, while this review focuses on complex-systems approaches to individual systems.
- Physical systems: Physical examples include crystals, magnets, glasses, superconductors, fluids, and granular flows within condensed matter and statistical physics.
IV. COMPLEX SYSTEMS THEORY
Complex systems theory comprises multiple general theories rather than a single monolithic framework. Whether these theories will eventually form one coherent theory remains debated.
- Complex systems theory is better understood as a series of short stories than as one monolithic body of knowledge.
- The possible future integration of these theories into a single coherent theory is an open matter of debate.
A. Lattices and networks
Complex systems models specify both interaction structure and agent behavior. They represent topology with lattices or networks, with networks needed for most non-regular systems.
- Models quantify complex systems by specifying topology—who interacts with whom—and dynamics—how agents behave and interact.
- Lattices and networks are the main frameworks for representing system topology.
- Regular lattices are simple to represent, whereas most complex systems require networks for their more complicated non-regular topologies.
- Network scholarship ranges from popular introductions to lengthy technical treatments and brief or encyclopedic reviews.
B. Dynamical systems
Dynamical systems theory models agent behavior over time with coupled mathematical representations. Its continuous branch uses differential equations and exhibits emergent phenomena such as chaos and bifurcations.
- Dynamical systems theory represents agents’ behaviors over time with simple mathematical models coupled to capture interactions.
- Continuous dynamical systems are typically modeled with differential equations.
- Chaos and bifurcations are characteristic emergent behaviors of continuous dynamical systems.
C. Discrete dynamics and cellular automata
This section connects discrete-time dynamics with broader complex-systems themes, including edge-of-chaos transitions, scaling, adaptation, fitness, and information-theoretic analysis.
- C. Discrete dynamics and cellular automata: Discrete dynamical systems evolve through successive discrete time steps.
- C. Discrete dynamics and cellular automata: The logistic map transitions from ordered to chaotic regimes and motivated research on the complex-systems “edge of chaos.”
- D. Scaling and criticality: Scaling, phase transitions, and critical phenomena are fundamental physical ideas used in complex-systems theory.
- D. Scaling and criticality: Power-law distributions retain their shape when measured quantities are rescaled by a constant.
- E. Adaptation and fitness: Adaptation occurs when collective agent behavior optimizes a feature or quantity, with natural selection as a classic example.
- E. Adaptation and fitness: In complex adaptive systems, fitness measures how well an individual, group, species, or strategy performs relative to competition.
- F. Information theory: Information theory quantifies information and is frequently used to analyze and understand complex systems.
G. Computational complexity
Computational complexity studies how difficult tasks are and connects those limits to the behavior of physical, biological, and social systems. Complex-systems modeling also uses agent-based simulations to generate emergent behavior from individual interactions.
- Computational complexity: Computational complexity measures task difficulty by time or arithmetic operations, with applications beyond computer science.Its applications include evolutionary biology, molecular biology, statistical physics, game theory, and engineering.
- Computational complexity: Ground-state search can be difficult when systems have many possible states and no principle predicts which has the lowest energy.In some cases, exhaustive search may be the only reliable approach.
- Computational complexity: Computational impossibility results imply that nature may also require very long times to reach a physical system’s ground state.If the required search takes years or centuries, the system may not reach its ground state quickly or at all.
- Computational complexity: The P-versus-NP question asks whether problems whose solutions are quickly checkable can also be solved rapidly.The text states that most researchers believe P and NP differ, but this remains unproved.
- Agent-based modeling: Agent-based modeling separately simulates individual agents and their interactions so that emergent system behavior appears naturally.This approach is also called individual-based modeling.
V. CONCLUSION
Complex systems is a broad field spanning many methods and applications, while its questions remain largely unresolved. The review presents abundant resources and opportunities for further study.
- V. CONCLUSION: Complex systems spans a wide range of methods and applications, but the reviewed resources cover only a fraction of the field.The field remains active, with abundant additional resources available.
- V. CONCLUSION: Areas of ignorance in complex systems currently outnumber areas of expertise, leaving many profound questions open for investigation.The review characterizes the science as only beginning to address these questions.