Download - Unit 11 Slides
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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. 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.)