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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.  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.)