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Undertanding Complexity

REVIEW OF THE SEMINAR UNDERSTANDING COMPLEXITY: SYSTEMS, EMERGENCE AND EVOLUTION. CASE STUDY: THE CITY.

By Verónica Coca Zancajo, participant as attendee.

The seminar is part of the PRIMER initiative for promoting interdisciplinary methodologies in education and research, and it was held in Leon (Spain) during the days 27^th-30^th of January of 2014.

The lectures were imparted by the professors:

Rainer Zimmerman, from the Munich University of Applied Sciences and Clare Hall of Cambridge;

Gordana Dodig Crnkovic, from the Swedish Mälardalen University;

José María Díaz Nafría, from Hochschule München;

Paz Benito, from Universidad de León.

The topic of the seminar is the Theory of systems applied to understand how complexity emerge, structure itself and evolve; also the relationship between ethics and beautiful according to the principles of urban-planning.

In order to understand the relationship between urban-planning and the theory of systems, we should analyse four main aspects of cities as open, complex and with a hierarchical structure systems:

1.  the urban structure, studied by the classical urban planning, focus in the growing process, the physical and social dimensions and the economic functions;

2. the urban processes directed from public authorities, to the governance and management;

3.  the dynamics of land uses, in processes of expansion or intensification of use;

4. the internal flows (mobility, economic transactions, transports, new information technologies, etc.) and external flows (connectivity with another cities, the quality of interurban spaces, etc.).

Introduction to dynamical systems theory

To begin this study we must have a look back to the origin of the computational methods. Nowadays Ramon Llull can be considerate the father of computing and information science. By using the aristotelian logic Llull posed the principle of computing versus discussing, and he proposed two main ideas: the knowledge as a tree of different science and the combinatory logical machine able of verifying any sentence. The tree of knowledge represents the idea of many different branches of knowledge that get together to one unique trunk that is reality. So there are many different ways to approach the knowledge of reality, but they cannot be in conflict. Llull anticipate the mechanical calculation by symbolic representations, the heuristic methods of artificial intelligence, the study of graph theory and the semantic networks.

This vision of a unique reality that can be known causes the determinism some centuries later. After the Laws of Newton that allow the anticipation of the position of planets (positivism), Pierre-Simon Laplace imagined the existence of a demon, with no supernatural abilities but with the capability of specify the exact position of any particle of the universe in every moment, so this demon could know the past as well as the future. That idea was called determinism and became a problem of free will.

However the Newton’s equations were no able to solve the question of the interaction of a third body in the position and trajectory of celestial bodies.  This was known as the Problem of the Third Body. To solve this problem a new theory was born: the K.A.M. theory.

In the twentieth century many theories changed the idea of a unique reality that can be determinate in every moment. Gödel demonstrated with mathematical expressions that there are axioms and formulations that cannot be verified inside their own logical system. So the Mathematics are incomplete. Also the uncertainty principle of Heisenberg establishes the impossibility of certain and precision in the observation of the movement and position of a particle. The most we know the velocity of a particle, the less we can determine its precise position. So after Heisenberg the tree of knowledge is not unique any more, there are many possible trees of reality. That is the reason why the theory of complex systems is being used in the new researches in different fields of knowledge.

Reality versus modality

Reality itself cannot be known since the appearance of quantum physics. So we are going to take the ideas of Spinoza about substance, attributes and modes. Reality is the substance, because it exists by itself, so Nature is the substance. The attributes are infinite, for example the extension of the matter or the thoughts (the mind), but they settle on modes. Modes are physical or ideal objects. Instead of reality, we will talk about modality, i.e., a representation of reality. This representation is not the same that the reality represented. The modality is the representation of some qualities that we, human beings, can perceive. Those qualities are just a part of the reality, and we must have this in mind in order to evaluate the achievements of Science. These representations are simplifications of a much more complex unattainable reality.

Systems are modalities created by human beings in order to understand Nature. Every system has three inner qualities: composition (collection of components or agents, matter or ideal), structure (links and relations among the agents, i.e., interaction and organization) and mechanism (processes of transformation inside the system, notice that transformation needs energy). Every system has a purpose that has to be accomplished in order to maintain the existence of the system. There is an outer quality: the environment (things out of the composition, if they interact with the components, the system is open, if not, it is close). The systems can contain sub-systems and be part of super-systems. These modalities can be applied in physical science and in social science. The holistic vision of the reality as a set of systems is called systemism.

Continuous model by Keller and Segel

Besides the conception of systems as a set of transformations in the structure, used in linguistics by Saussure and in epistemology by Jean Piaget (they both used structure instead system many times), there is a mathematical definition of system that includes the concepts of domain and range. The dynamical system describes the range of change (transformations) produced in the position in a very small portion of time:

dx/dt = f(x, t)                                   when x = (x₁, x₂, …, x_n)

The analyse of this expression can get high levels of complexity, when the functions cannot be solved because they are non-linear and the solutions are infinite. That is the reason why we will study the stability, by studying the critical points of the system instead of the transformations. The expression to study the critical points takes this form:

f(x, t) = 0

The points where the function is equal to zero can be referred as bifurcation points, because the possible evolution of the system can take different signs. To understand better this idea of bifurcation points, a review of the dissipative structures theory by Ilya Prigogine is required. This theory introduces the non-equilibrium and the asymmetry as main concepts to understand the universe in opposition to the classical view. In the nineteenth century Clausius introduced the concept of entropy in order to express the grade of disorder in a system. He formulated the second law of thermodynamics, usually expressed by dS>0, that means: the entropy always increases inside an isolated system until the non-equilibriums become equal, i.e., the Universe is evolving in direction to the maximum disorder and this process is irreversible. Prigogine modified the second law of thermodynamics in a way only applicable for open systems. The global entropy is the result of this addition:

dS= d_FS + d_IS;

d_IS is the level of entropy inside the system, that is always increasing (positive);

d_FS is the level of entropy that flows out the system, so it is decreasing inside the system (negative).

If the flow of entropy out of the system is higher than the inner entropy, the global disorder inside the system is lower; hence a new order is emerging. This process is natural, produced by the system itself, so it receives the name of self-organization. This new point of view for open systems has changed the bases of Science, because it provides us with new mechanism to study the unstable sides of Nature -unfortunately Nature cannot be describes appropriately by differential equations of dynamic systems, because Nature has not the quality of continuity-.  Prigogine also defeats the determinism, since new structures can emerge spontaneously. Some of these processes have a cyclical behaviour; others appear in a random way (chaos theory). An example of a cyclical process is the evolution of a colony of amoebae, studied by Keller and Segel.

The critical points of the function, or bifurcation points as we saw, do not give us enough information, in order to understand the context we need to study the neighbourhood of the critical points. When we describe the whole system by a critical point plus a very small variation, we call it linearization process. The critical points are unstable points where the symmetry is broken and a new structure can emerge – innovation process -.

We will use a matrix to express in a clearer way a process with many similar functions. Then one matrix will be one state in the process of transformation. The initial state (stable) can be the matrix E. Now we can express the transformations in terms of negation and negation of negation:

E (stable state) transforms into E*(unstable);

E* is the negation of E:                                                E*=N(E)

E*(unstable) transforms into E** (stable);

E** is the negation of negation of E:     E**=N(E*)=NN(E)=N²(E)

We can represent this transformations as  a diagram of evolution.

Networks and spaces

Talking about systems we always must talk about space, as a domain of influence or free paths for interactions. It is defined by describing the boundary of the system (where the rules or attributes of the space cease to exist, i.e., the domain of influence is equal to zero). The space can be physical or virtual (abstract). The geographical space, for example, is a projection of the social space - in order to go into detail about the construction of social space it is essential to read the works of Michel Foucault as well as Henri Lefebvre -.

A Network is the dynamical core of interactions, the transport of information. For example, a social network is the connectedness through interpersonal relations.

The objects of the system are always operating agents. The agents can be human beings, groups, elementary particles, etc. The nature of the agents does not change the result of the system.

Nowadays the study of social networks has become very important to understand the changes in the society, because of the complexity of these changes and because of the new technologies. The studies are based in the computation of information for social levels plus some intuitions for individual levels. The computation of social information can be approached from two different perspectives: focus in human aspects or focus in computational aspects. This last approach will try to create models of automatize calculations (based originally on Hilberg program in 1900 and Turing Machine in 1936) for a better understanding of the mechanisms that cannot be perceived in a simple view. Computing social networks, we see that human groups are information processing networks, but these groups are also information generators (human groups are sub-systems and self-organized). Besides we can see that every system is always self-referenced, because there is always a reaction to being observed, and also every interpretation of another system is a bit of self-interpretation. Information processing is a physical process of morphological change in the informational structure, where the information is always relative to the observer. For human connection as a multi-agent system we create this relation:

information>computation>cognition>information

For the complexity of these social networks a new field of research was born: dynamic networks analyses (see the work of Lazslo Barabási). Looking for patterns of behaviour in complex social networks, a fractal pattern has been found, and it makes possible the existence of a unique model of growing independently of the scale (scale-free networks and self-similarity as a quality of complex networks).

We will define a category as a class of objects and a class of representation or morphism: the relation among agents of a system under condition of every interaction has a unique agent-source and a unique agent-target. When we represent the agents as a node or vertex, we can talk about isomorphism.

We use networks to represent the transport of information among the agents of the system. The transport of information attaches the concepts of time and space (motion), but we will construct an abstraction just to clarify the main structure of interactions inside the system.  

We will take the Graph Theory of Euler as base line for the abstract representation of the structure of the system (the relationships among agents and its interaction) and use this network of vertex and links as a formalization that allows us to provide different meanings to the same structure of vertex and links. This theory is connected with topology. Topoi refers to location, but in Mathematical sense, topos is a category plus an additional structure that allow as to operate “inside” that “space”.

Besides information, the system needs energy to make the transformation. Applying the system to different models we can see than the role of the agents or nodes can be very different form one model to another. When the agents wait until the action of an external interaction happens, then those agents are passive and the representation is mapping potentialities - classical geographic map -. When the interactions happen among internal agents, these agents are active and the representation is mapping actualities - dynamic processes -. See more about graphs and networks studying the model of the small world and isomorphic matrix.

Stochastic Petri Nets

Petri nets are another kind of networks that allow us to discern the differences between objects and morphisms. Objects will be named now species (represented by circles) and morphisms will be transitions (represented by squares). The relation between boxes will be define by vector (see Hamiltonian function).These networks are active always and they show one benefit: we see directly the result of the transformation as a new specie. So the Petri nets are represented by input boxes and output boxes. The interaction can take place among different species and with different transitions or changes of state. Because of this, we define Petri nets as a set of species and a set of transitions, with the functions that connect them. When the model is not deterministic and there is some uncertainty about the result, then we said that the network is stochastic, and we should take in consideration the probabilities of certain result or another. By iterations of the probabilistic calculation, we can have very close results in very different fields.

We can represent a matrix of all the spaces in a subspace. The entrances will represent the interaction between species, zero for no interaction and one for interaction. That matrix (called Hamiltonian matrix) will represent all the probabilities of interactions and its expression is quite near the Schrödinger equation.

And important point to define the processes of transformation is their character of reversible or irreversible. A network is weakly reversible if the transformation in one direction, from A to B, is also possible in the way back, from B we can get A.

For these Petri networks, the existence of an equilibrium solution depends on their character of weakly reversible or not weakly reversible, and so their grade of deficiency - positive or zero - (deficiency is the number of vertex minus the number of connected objects minus the order of the stoichiometric subspace). In occasions the solution depends on time.

Applications of systems theory on urban space

The relation between cities and networks seems to be evident nowadays, but when did the urban planning meet the system theory at the first time? Thanks to Jane Jacobs and her book Death and Life of Great American Cities, the classical urban-planning practically collapse and a new point of view was required. This change has two important consequences: in one hand, the relation between the complexity of the city and the complexity of natural processes was born (the idea of ecosystem was applied for urban spaces), and in the other hand, the planning process had to be inverted, becoming a processes bottom-up instead of top-down (democratization in the decision process, citizens participation, etc.). The professor Bettencourt from Santa Fe Institute notices that Jacobs refers to the city problems as organized complexity, and nowadays the challenge is still the creation of a new urban –planning, more scientific and with better complex adaptive models. In that sense, Kevin Lynch contemplated in the ‘80s a very important question: is the city an analogous model to other natural systems? It is very interesting that the planning critics has placed a value on vernacular architecture and spontaneously born settlements as slums, set against the modern big designs of nineteenth and twentieth century. This consideration entails two ideas: first, the emergent aspects of urban space when official planning is missing and second, the importance of the transformation scale in urban space.

Bettencourt has established two main measurable properties: population density and the use of urban material infrastructures. He also considers the distance travelled, the transportation of goods and people, etc. , and has made the comparison between cities and stars in terms of attraction nodes, dissipation, etc. This metaphor is valid for some properties of urban space, such as luminosity-activity, but it cannot be a urban model for general studies, because many characteristic magnitudes of stars are not valid for cities, for example gravitation. In order to maintain this metaphor, we need to find how to change those magnitudes to others that we could use for cities.

Fractality of cities

Michael Batty and his partners were the first to publish a work identifying the self-organized structure of British coast with the Koch curve. They demonstrated the fractal structure across different scales for cities.

Fractality is present in the structure of quartiers, districts, etc. The classical example of fractal space is the model of roman encampment that was extended to the new colonies. The case of Bolonia is good example.

Hodological space

Hodos means routs or ways. Taking the ideas of the school of Gestalt, we can create a representation of the hodological space of neighbours. This hodological space will be the living space of every individual, recognizing morphological spaces in the personal routs, with clearly a subjective connotation by the association of locations and feelings. This hodological space put the geographical space in a secondary order. The map of living space will be the addition of geographical space plus individual connotations. The superposition of the hodological spaces of all the inhabitants will result the social hodological space that can contribute to complete the other objective data analyses.

Final remaks

The observation of the reality cannot be separated of our human capacities.

Modelling the reality is re-creating the result of our observation, including processes of interpolation and extrapolation.

Models are set in theories. Theories are used to understand what is not human itself.

The whole knowledge is just a human perception of reality, a human modality.

The concepts of space (time), networks and systems are characteristics of human knowledge.

Those three concepts are related to another triade, cognition, communication and cooperation.

The ancient idea of harmony posed the relation between adequate form and adequate content. So harmony means the sum of ethics and aesthetics, resumed in the word kalokagathía.

Harmonic systems according to Heraclitus are constantly in motion, always stirring the mixture, always in conflict.