unit 11 slides
TRANSCRIPT
Large networks of simple interacting elements,
which, following simple rules, produce emergent,
collective, complex behavior.
What are Complex Systems?
Core Disciplines of the Sciences of Complexity
Dynamics: The study of continually changing structure and behavior of
systems
Information: The study of representation, symbols, and communication
Computation: The study of how systems process information and act on the
results
Evolution / Learning: The study of how systems adapt to constantly
changing environments
Goals of this course:
• To give you a sense of how these topics are integrated in the study of complex systems
• To give you a sense of how idealized models can be used to study these topics
What did we cover?
Let’s review...
Dynamics and Chaos
• Provides a “vocabulary” for describing how complex systems change over time – Fixed points, periodic attractors, chaos, sensitive dependence on initial
conditions
• Shows how complex behavior can arise from iteration of simple rules
• Characterizes complexity in terms of dynamics
• Shows contrast between intrinsic unpredictability and “universal” properties
Fractals
• Provides geometry of real-world patterns
• Shows how complex patterns can arise from iteration of simple rules
• Characterizes complexity in terms of fractal dimension
Information Theory
• Makes analogy between information and physical entropy
• Characterizes complexity in terms of information content
Genetic Algorithms
• Provides idealized models of evolution and adaptation
• Demonstrates how complex behavior/shape can emerge from simple rules (of evolution)
Cellular Automata
• Idealized models of complex systems
• Shows how complex patterns can emerge from iterating simple rules
• Characterizes complexity in terms of “class” of patterns
Models of Self-Organization
• Idealized models of self-organizing behavior
• Attempt to find common principles in terms of dynamics, information, computation, and adaptation
Firefly synchronization Flocking / Schooling Ant Foraging
Ant Task Allocation Immune System Cellular Metabolism, …
Models of Cooperation
• Idealized model of how self-organized cooperation can emerge in social systems
• Demonstrates how idealized models can be used to study complex phenomena
Prisoner’s dilemma El Farol Problem
Networks
• Vocabulary for describing structure and dynamics of real-world networks – small-world, scale-free, degree distribution, clustering,
path-length
• Shows how real-world network structure can be captured by simple models (e.g., preferential attachment)
Scaling
• Gives clues to underlying structure and dynamics of complex systems (e.g., fractal distribution networks)
Goals of the Science of Complexity
• Cross-disciplinary insights into complex systems
• General theory?
√
?
Can we develop a general theory of complex systems?
That is, a mathematical language that unifies dynamics,
information processing, and evolution in complex systems ?
I.e., a “calculus of complexity” ?
Isaac Newton, 1643–1727
infinitesimal
limit
derivative
integral
“He was hampered by the chaos of language
—words still vaguely defined and words not
quite existing. . . . Newton believed he could
marshal a complete science of motion, if only
he could find the appropriate lexicon. . . .”
― James Gleick, Isaac Newton
emergence
self-organization
network
adaptation
Complex Systems, c. 2013
attractor criticality
information computation
bifurcation
nonlinearity
equilibrium
entropy fractal chaos
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.
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.
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. renormalization randomness
scaling
power law
“I do not give a fig for the simplicity on this side of
complexity, but I would give my life for the simplicity on
the other side of complexity.”
― O. W. Holmes (attr.)