Complexity

→ Understanding the processes that preserve set/category structure! → Models are used to frame problems and answer questions. They are explicit theories of why something behaves the way it does. They should help to clarify what is being considered and what is being excluded and present opportunities to suggest corrections and additions and improvements

Characteristics of complexity

 * Complexity is a subjective dimension/dependent on the agent
 * Complex phenomena in contrast to simple ones are characterized by
 * complexity: number of variables; to solve the problem a reduction of information is necessary
 * interdependence: variables are strongly interdependent; however, the degree of interdependence/closeness between the variables can vary; structuring of information is necessary
 * internal dynamic (Eigendynamik): variables can change over time even without any external influence of the problem-solver; changes are hardly predictable which creates time-pressure and makes swift decisions necessary; dependency on intervention (Eingriffsabhängigkeit); dynamic requires the problem-solver to recognize developmental trajectories (Entwicklungstendenzen) of the system
 * non-transparency (Intransparenz): not all the information (on connections and causal effects between variables) is available, either because they are unexistent or not available at hand; information has to be gathered; even if all information were availabe, the agent still doesn't know which situation he faces at any given time; non-transparency leads to indeterminancy (Unbestimmtheit)
 * polytelie: contradictory relation among the sub-goals; containing several, possibly contradictory goals/pursuit of contradictory goals are necessary in orer to solve the problem; the problem-solver has to prioritize and make compromises
 * The exact properties of the given state, goal state, and barriers are unknown to the solver at the outset
 * Complex phenomena often include difficult-to-gauge factors like exponential, time-lagged, sensitive, indirect causal dimensions/developments
 * Complexity as superposition of different possible states (quantum physics): namely the strange nature of quantum superpositions, in which a quantum system such as an atom or photon can exist as a combination of multiple states corresponding to different possible outcomes. A quantum system remains in this superposition until it interacts with, or is observed by, the external world, at which time the superposition collapses into one or another of the possible definite states (see also the Uncertainty principle)

Connectionist vs. analytical-reductionist

 * Analytical-reductionist/classical mechanics
 * The epistemological and analytical foundation of classical mechanics originating in the 17th century is considered to be Cartesian rationalism, which treats nature as a perfect machine that operates according to exact and universally valid laws.
 * Identify key input and output variables and seek to establish clear (linear) links between them
 * identify and generalise from elements (reductionist methodology) - all properties of a total system can be deduced from the properties of the system components, these natural laws can be mathematically specified
 * The path and the result of change can be mathematically calculated by means of constrained optimisation. The basic equations are linear and the equilibrium states achieved are invariably unique and stable. Linearity of relationships between the systems components is accountable for the behaviour of the total system being deducible from the behaviour of the system components and, thus, for the applicability of classical deductive and reductionist methodology.


 * Complexity
 * Draws attention to the importance of the interconnectedness of variables within systems and the qualities that emerge as a result of these interconnections
 * Characteristics of systems derive as much from the internal relationship between variables, as from intrinsic qualities of the variables themselves
 * Information is distributed across variables and only makes sense in the context of connections between them
 * Connections between (sets of) elements defy explanation in terms of regularities or linear relationships: Interconnections at any one point may be temporary, their strength may vary and sequential orders may be multidirectional dependent on particular circumstances; non-linear and dynamic ways, i.e. where factors are seen to interact in a causal relationship the effects do not necessarily relate proportionally to the cause, and few factors may interact with many and many may interact with few

Handling complexity

 * Tolerate ambiguity: if ambiguity is unbearable, this often leads to concerning oneself merely with those problems that can be solved instead of those that should be solved
 * Knowledge of how to handle complex problems can be both theoretical-explicit or implicit know-how (Handlungswissen)
 * Goals determine intentions/plans, and plans lead to decisions (whereby the plan with the highest urgency of implementation is chosen)
 * Handling goals: goal-setting, goal-elaboration, goal-pursuit
 * Ability to analyze the situation: recognize relationships, identify the system structure, information-gathering, information integration and modelling of the situation, forming hypotheses
 * Correct choice of action: prioritizing/focusing, planning and deciding, implementing the measures, prognosis and extrapolation, control of actions and modification of the strategy
 * Characteristics of the successful problem-solver:
 * Memory contents: Knowledge (domain-general, domain-specific)
 * Information processing: Strategies, monitoring, evaluation
 * Non-cognitive variables: Motivation, self-confidence, perseverance, etc.


 * 1) Goal elaboration
 * 2) Identifying the reality structure
 * 3) Information gathering
 * 4) Prognosis
 * 5) Planning and deciding
 * 6) Control of the effects
 * 7) Modification of the constructed model of reality and devised strategies

Definitions & Terms

 * Complexity classes are concerned with the rate of growth of the requirement in resources as the input n increases
 * Any complexity classes can be characterized in terms of the mathematical logic needed to express them
 * A fractal (also: expanding/evolving symmetry) is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale. If the replication is exactly the same at every scale, it is called a self-similar pattern.
 * A self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales → or for example a community? (has similar properties at the local, communal, national, regional level)
 * In mathematics, an invariant is a property, held by a class of mathematical objects, which remains unchanged when transformations of a certain type are applied to the objects

Set & category theory

 * In classical/naive set theory, a set is described as a well-defined collection of objects, called the elements/members of the set
 * The membership of elements in a set is assessed in binary terms according to a bivalent condition → fundamental binary relation between an object x and a set A: if x belongs to A it is denoted by x ∈ A
 * Most (mathematical) objects such as numbers, relations, functions, etc. are defined in terms of sets
 * A set is a gathering together into a whole of definite, distinct objects of our perception or of our thought—which are called elements of the set
 * The intersection of A and B is the set of all objects which are both in A and in B. It is denoted by A ∩ B
 * Finally, the relative complement of B relative to A, also known as the set theoretic difference of A and B, is the set of all objects that belong to A but not to B. It is written as A \ B or A − B


 * Categories include sets, groups and topologies
 * The arrows between two sets designates functions from one set to another: an arrow represents a process connecting two objects, or in many cases a "structure-preserving" transformation connecting two objects
 * Instead of focusing merely on the individual objects (e.g., groups) possessing a given structure, category theory emphasizes the morphisms – the structure-preserving mappings – between these objects; by studying these morphisms, we are able to learn more about the structure of the objects
 * A topological spaces allows for the definition of concepts such as continuity, connectedness, and convergence

Fuzzy sets

 * Fuzzy set: fuzzy sets are sets whose elements have degrees of membership
 * The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise

Complex social systems

 * Interaction in social complex system involves exchanging information and continuous adaptation to new information and knowledge. Rules, norms, conventions, ethics and the legal system create a framework in which human beings can act. This framework itself has emerged from the interaction between human beings and continues to evolve. The government is an institution that results from this interaction. Social complex systems show evolutionary dynamics. They are constantly in a flux, without having objectives of their own and without any knowable endpoint or equilibrium.
 * Complex social systems evolve without knowable endpoint or equilibrium. They are open-ended. This follows from the human condition that the future is fundamentally uncertain.
 * In complex social systems, the critical variables can often be influenced only indirectly
 * Decisions usually have not only the intended but also side effects and future effects
 * Check Panarchy

Causal loop diagram

 * Aids in visualizing how different variables in a system are interrelated
 * Causal loop diagrams consist of a set of nodes/variables (things, actions or feelings) connected by causal links (edges/arrows) with polarities (+ and – signs) and delays (||)
 * Together, these create positive and negative feedback loops that describe the circles of cause and effect that take on a life of their own
 * In systems thinking, there are two basic types of causal loops: reinforcing and balancing. In a reinforcing loop, change in one direction is compounded by more change.

The elements of a CLD

 * "+": a positive causal link, which means the two nodes change in the same direction (i.e. if the node in which the link starts decreases, the other node also decreases); a negative causal link means that if one variable increases, the other decreases
 * "R": a closed cycle, namely a reinforcing loop, whereby the effect of a variation in any variable propagates through the loop and returns to the variable reinforcing the initial deviation → i.e. if a variable increases in a reinforcing loop the effect through the cycle will return an increase to the same variable; reinforcing loops are associated with exponential increases/decreases
 * "B": a closed cycle, namely a balancing loop or, the opposite of the reinforcing loop, i.e. if a variable increases in a balancing loop the effect through the cycle will return a decrease to the same variable; balancing loops are associated with reaching a plateau
 * "||": a delay situations where it takes time before the effect plays out; if delays are relatively long, that could lead to a lag in responsiveness or inability to adapt, whereas if delays are very short or non-existent, the system might be more sporadic


 * To determine if a causal loop is reinforcing or balancing, one can start with an assumption, e.g. "Node 1 increases" and follow the loop around. It is reinforcing if, after going around the loop, one ends up with the same result as the initial assumption
 * For example: The amount of bank balance will (positively and thus denoted with a +) affect the amount of the earned interest. The earned interest gets added to the bank balance (also a positive link). The causal effect between these nodes forms a positive reinforcing loop.

Create a CLD

 * How to determine, which variables to add to it?
 * A variable is relevant and therefore to be added if:
 * ...it has many (heavily weighted) causal links pointing towards it or, if it has many causal links originating from it - i.e. if it has a multiplying position in the diagram
 * ...its position in the diagram is a spatially central one such that a variable's proximity to the center of the diagram is a proxy for its relevance
 * ...it is constituent of many (reinforcing/balancing) loops

Systems thinking

 * Systems thinking takes on complex, dynamic systems and how they behave over time, which calls for a different sort of language
 * A system (in the sense of systems theory)
 * Is an entity
 * Consists of several components that have properties and that are interconnected
 * It has an internal structure, which is determined by the properties of the system components and their interactions
 * The structure gives the system a certain functionality and performance
 * The system's state (at a given time) is defined by the properties of the system components and their interactions
 * Environment: the realm outside the system, into which the system is embedded
 * Boundary: the frontier that delimitates the system and its environment with respect to space and time
 * Distinguish isolated system (neither inputs nor outputs of energy/information), closed (exchanges energy but not matter with its environment) and open (exchanges both energy and matter with its environment)
 * The essential variables characterising the system’s state are called state variables while the essential factors characterising the system’s environment are called parameters - If changes in the state variables have an effect on these state variables themselves, meaning that the state variables are a function of their own lagged values, this is called a feedback loop.
 * Positive (self-reinforcing) vs. negative (extenuating) feedback loops
 * For the system to exhibit complex behaviour, at least one interrelation between the systems component needs to be non-linear, and at least one feedback loop needs to exist - in systems of such a kind, an incidence cannot be traced back to a single cause or even a sequence of causes.

Chaos

 * A system’s choice between different paths at bifurcation points can be understood as a decision between the sphere of influence of different attractors. If the distance from the initial (thermodynamic) equilibrium is increased, new bifurcation points appear that cause the paths to further split up and allow for a multitude of attractor shapes. Eventually, the succession of critical points becomes so tight that the branches intersect and the possibilities for the system’s further evolution become infinite (see bifurcation diagram). This is the realm of so-called (deterministic) chaos, where highly complex, so-called "strange" or "chaotic" attractors govern the system’s progression
 * In the sphere of influence of a chaotic attractor, systems behaviour is still governed by deterministic laws, but even marginal differences in the initial situations can give rise to completely differing evolutions.
 * Complex systems do/can have definite properties
 * Random aperiodicity
 * Life (just as we and our bodies) is incredibly ordered on several scales of magnitude, but irregularly so
 * Though the study of predictable, linear systems makes up the bulk of a classical education, those systems are by far the exception, not the rule. Most natural systems play by the rules of chaos
 * Chaotic systems have extraordinary sensitivity to internal conditions which makes them inherently unpredictable in the long term
 * Another characteristic of chaotic systems is order without periodicity
 * A chaotic system operates according to set rules, but constant feedback, time delays, and tiny changes make the system behave randomly without repetition
 * A system can wear order and chaos like different masks, depending on the situation
 * When the system becomes increasingly unstable, an attractor draws the stress and the system splits and returns to order. This process is called bifurcation. Bifurcation results in new possibilities that keep the system alive and random
 * Empirically observable irregular time series need not necessarily be purely random in nature but can be governed by deterministic, albeit non-linear regularities that generate complex chaotic behaviour → random ≠ chaotic
 * Chaotic sequences are in fact generated deterministically from the dynamical system
 * Random (or stochastic) processes: two successive realizations of a random process will give different sequences, even if initial state is the same
 * Since the random influences that are decisive at the points of bifurcation cannot be reproduced at will, a path, once chosen, cannot be simply reverted. The choice of paths itself, in turn, is crucial for the system’s further possibilities of development as different options present themselves, depending on the path entered. This property is known as path dependency of development: A system’s future evolutionary options are determined by the choices of paths in the past → the existence of several stable paths to be chosen from is at the basis of the hysteresis

Further materials

 * 
 * CLDS
 * Systems thinking
 * Hayek, F. (1964) - The theory of complex phenomena
 * Hayek, F. (1945) - The use of knowledge in society
 * The following math problem illustrates with a few simple steps how complexity is constructed logically/mathematically:


 * According to Google: In mathematics, the fibonacci sequence is analogous to the following simple calculation: 4,2+4^0=7.5*85, then in a second step add the square root of the result and apply the same procedure
 * Add π in a third operation whereby the first two steps are repeatedly continued until you obtain the result of the following operation: 6*4.2=0,88+9,7. The fibonacci result is the ℳ 02≈54-70,75​ and in the least-square form simply the fraction of (26)/734,393
 * The point of the calculation breaks and does not continue forever note: the square root of 7.398 is a sixth of 5590, the same result as indexing 14,09 at the 68-2*2 point.
 * The cascade starts with 5,729. Add π to obtain (rounded to three decimals) 2,152. Subsequently double the result and you get the difference between the initial 4,624 and the 83 to the power of 77
 * Similarly, take the differences between the following numbers 4,015 and 8,475 and note how they come out at the same result as the cumulative difference of the sequence of the following operation
 * Multiply 53*10 and subtract the square root of 7180