Complexity: A Very Short Introduction

John H. Holland, Oxford University Press 2014
  1. Complex systems
  2. Complex physical systems(CPS)
  3. Complex adaptive systems(CAS)
  4. Agents, networks, degree, and recirculation
  5. Specialization and diversity
  6. Emergence
  7. Co-evolution and the formation of niches
  8. Putting it all together
Notable People: Herbert Simon, Newton, Burk, von Neumann, Nobel Laureate Phil Anderson, Waldrop, Turing, Maxwell, Navier-Stokes, Noam Chomsky,  J. H. Brown, B. J. Enquist, G. B. West,  Denes Konig, Duncan Watts and Mark Newman,

Complexity - objects with many interconnected parts, now a scientific field.
Complex systems - large variety, interaction between objects, new ongoing adaptations and interactions, which has emergent behaviour.
Emergence - action of the whole is more than the sum of the actions of the parts.
Computational complexity - to classify and compare the practical difficulty of solving problems about finite combinatorial objects.
Levels of complexity - a difficulty level for computational complexity, of a problem.
Non-linear system - the output is non proportional to the input of the system.
Self-organization - spontaneous arrangement of order through no external agency.
Chaotic behaviour - mathematical theories for dynamical systems like, economics, biology, weather.
Fat-tailed behaviour - rare events occur more frequently than predicted by normal(bell-curve) distribution.
Adaptive interaction - interactive agents modify their strategies in diverse ways as experience accumulates.
Complex physical systems - geometric arrays of elements where interactions depend on the effects propagated from nearest neighbours, usually expressed by differential equations.
Cellular automaton - collection of cell in a grid(or arrangement) which have state, and are influenced by neighbouring cells state in a discrete time step rate.
Complex adaptive systems - elements that are not fixed, called agents, that learn and adapt in response to interactions with other agents. Agents usually don't converge on optimal strategy, as new strategies are always emerging thus increasing the interactions, entanglement, and overall complexity. Complex feedback loops increase the difficulty of analysis.
Perpetual novelty - a characteristic of complex systems is the abundance of possibilities that are produced with a limited number of rules/laws.
Law - or rules that must be obeyed, or are the foundation of a system.
State - a single configuration/representation of values produced from the laws.
Partial differential - a very small immeasurable change, variables of interest are fixed. (look into this)
Partial differential equation - two or more independent variables, an unknown function (dependent on those variables), and partial derivatives of the unknown function with respect to the independent variables.
Universal grammar -  theory of the genetic component of the language faculty.
Generators - vocabulary(primitives.)
Operators - combining generators into meaningful strings.
Corpus - a set to describe the state that occur under the grammar's rules.
Critical point - derivative is undefined or zero, inside an interval of known values.
Attractor -  set of numerical values toward which a dynamical system tends to evolve, for a wide variety of starting conditions of the system.
Self-organized criticality - a point in a dynamical system, where the critical point is the attractor.
Self-similarity - a curve constructed by a repeated process of the same construction. In the limit, the curve is everywhere discontinuous.
Fractal curve - (ambiguous) snowflake-like curves of fractals.
State trajectory - (not sure at all).
Wave function - probabilistic trajectory.
Scaling - (not quite sure)a property related to self-similarity, a new way to examine CPS and complexity.
Zipf's Law - approximately, the most frequently used word in a language will occur twice as often as the second most frequent word, three times as often as the third...
Power laws - words ordered according to usage plotted against frequency of usage result in an exponential curve. Neither a necessary nor sufficient condition f or complexity.
Network(graph theory) - tool to study complexity, used to model pairwise connections between objects.
Tree - an undirected graph in which any two vertices are connected by exactly one path.
Scale-free networks - (investigate)ex. fractal resource distribution network.
Small-world networks - arranged so most nodes in the network are connected only to nearest neighbours with few long range connections between clusters of nodes.

Hierarchical organization is closely tied to emergence. Often with each level having its own law, but constrained by not violating the laws of the earlier levels.
The combination of top down and bottom up effects is a feature in complex systems
Behaviours of complex systems: self-organization into patterns, chaotic behaviour, fat-tailed, behaviour, adaptive interaction.
Analyzing complexity looks for recurring patterns in the system.
Patterns help steer systems towards one goal, called motifs, derived rules, lemmas, etc in other disciplines. To exploit these possibilities, analysis depends on methods to discover and exploit recurring patterns in generated systems.
In CPS, laws constrain the way a given initial state can change, almost always formulated using partial differential equations, variable of the equations specify the state.
In CAS elements are adaptive agents, that change as the agents adapt. Changing interactions between adaptive agents are not additive. Non linearity rules out direct use of PDE, and most disciplines involving CAS have no standard language for describing the interaction of agents. First, we must discover mechanisms that generate data with a precise language for describing adaptive interactions of large number of agents, then to collect and organize data.
Both CPS and CAS use the time dimension to order the elements generated by the grammar.
Use formal grammar to convert common features of complex systems into stylized facts to be examined carefully within the grammar, similar in analyzing complex systems like network theory and data-mining.
Chaotic systems are deterministic, but small changes to the initial conditions provide radically different endpoints.

chapter 2 complex physical systems(CPS) - dynamics