This webpage, which is under construction, aims to provide a tutorial on complex systems and recent studies in this field.
What is complex systems?
There is no consensus definition about complex systems yet. According to Herbert A. Simon, winner of 1972 Turing Award, "Roughly, by a complex system I mean one made up of a number of parts that interact in a nonsimple way. In such systems, the whole is more than the sum of the parts, not in an ultimate, metaphysical sense, but in the important pragmatic sense that, given the properties of the parts and the laws of their interaction, it is not a trivial matter to infer the properties of the whole." (The Architecture of Complexity, 1962)
The key problems of complex systems are difficulties with their formal modeling and simulation. Complex systems usually invovle with non-linear cause-and-effect relationships among variables and stochastic procesess that are hard to be analytically solved.
Take the emergence of cities as an example. Human residents and locales of acitivities are very hierarchial systems, which emerge from individual firms and organizations' decision-making and interactions.
Readings:
Walter Christaller & Central Place Theory
Interesting topics in complex systems/science:
* Network analysis
Networks can be investigated from the micro-scopic and the macro-scopic persepctive. The micro-scopic method mainly focuses on the relationships/interactions between nodes (agents). This method has the advantage of examing the dynamic process of a system and may help reveal the cause-and-effect relationship between an input and an output. The macro-scpic method sheds light on the macro patterns, topology, or system structure, etc. When a variety of outputs are availabe, people are mostly interested in the patterns which are stable.
* Self-organization
Self-organization means the capability of a system that self-organizes and/or adapts to certain stable state(s). Many real-world phenomena have been found to be self-organized, such as the world airline networks, critical mass, herd behavior, etc.
* Chaotic Systems and Complexity
Chaos theory, originating from the realm of statistical physics, depicts the behavior of certain systems which are senstive to initial inputs. (wiki). One famous example is the butterfly effect.
Complexity is a general concept to describe the organization/state of a system, either from a mirco-scopic or a macro-scopic perspctive. It is often related with the concepts of emergence and self-orgnization.
Modeling techniques:
Agent-based Modeling
Agent-based modeling is a technique to simulate the interactions among agents in the system. There are two important factors: agents and rules. Agents are the units in a system which can make decisions. Rules are a set of pricinple followed by agents; rules often can be represented through mathematical formulations or logics. A system comprised of interactive multiple agents is called a Multi-Agent System (MAS).
Cellular Automata
Cellular automata (CA) is a discrete modeling technique, describing a system as a set of cell and a set of rules. The cells, which can be one-dimentional or multi-dimentional, usually have finite states. They change states by following certain rules. CA can vividly display how macro-patterns emerge from micro-behhaviors of mutilple cells.
Monte Carlo Methods
Monte Carlo methods are computational methods of using repeated random sample to generate outputs. This method is often used when there is uncertainty in inputs.
Readings:
Torrens, P.M. (2000) How cellular models of urban systems work (1. theory). Working paper. Centre for Advanced Spatial Analysis (UCL), London, UK.
Batty, M. (1997) Cellular automata and urban form: a primer. Journal of the American Planning Association, 63, 266-277.
Model validation and calibration:
Model validation and caliration is a fast-developing area in recent years. Hot topics in this realm include, but not limited to, pattern recognition, measures of match, process-related validation, and measures of complexity.
Pattern recognition
Measures of match
Process-related validation
Measures of complexity
Research Institutions:
New England Complex Systems Institute