Urban Evolution Analysis Method Using Self Organizing Dynamic Model.
Dr. Adolfo Benito Narváez Tijerina.
Facultad de Arquitectura, Universidad Autónoma de Nuevo León, México.
The focus of this work is centered on application of urban ecology analysis to comprehend the urban evolution. Describes the analysis method to know how much the environmental variables on an urban land interact. This information serves as a basis of a simulation model of urban evolution. As a conclusion analyses the philosophical implications of the method in face to deeply understand the correlations between social and physical spaces in the city.
How can be an urban evolution predictive model (UEPM)?
The UEPM can be defined using a variable correlation matrix being this as a system equilibrium. In fact, the matrix shows the state in which the system is found in a given moment of its history. As I. Prigogine  state, we look that the urban system is itself far away from equilibrium, Like the Russian scientist says about dissipative system that rise from the fluctuation of elements that leads the disequilibrium, therefore leads to the beginning of the self organization process.
It is possible to define an effective media to create the evolution of this system in time with the cell automata, from the middle state of the system (the system equilibrium) which is defined by the correlation matrix. Such matrix represents the correlation degree of the environment variables, in this case the path in which different aspects of form and space is used and urban forestation correlate. The equilibrium- non equilibrium can be defined by the matrix in a proportion terms, meaning the different forces of environment interacting upon the cells. From this two questions rises: What and How can be defined as a cells in this particular system? And the other is, how the cells changes toward the system equilibrium?
In the cell automata, the cells are the minor division in which each component of the system works keeping its distinctive characteristics; the microscopic elements define this units, that works on a macroscopic scale and react as unit. As Prigogine  says, in the evolution of this self regulated systems is possible to observe that the time divides the macroscopic components more precisely each time (after a qualitative jump that the system suffers from a fluctuation that rises the critic horizon of equilibrium, that the cell system states), whit this, the interchange process of each cell with it’s environment accelerates the evolution process of the system as a whole, because the fluctuations are more frequents every time. In this way, the system can be absorbed by the environment (recovering the equilibrium “entropic death”) or jumping to another state being the nuclei of a new compound organization in its initial states from a gross cells that will become to states of more complex organizations.
We study the basic cells of the urban system are the individual properties as defined by the Public Property Registry Office. It is possible to state that the morphology, land use and urban forestation are attributes that changes quickly with the time but these changes occur upon a constant pattern.
The cells as we see have an edge (property limits) and some attributes which are the variables that we define before in order to build the matrix correlations these associated to each cell fluctuates according to a interchange of information process with environment, in this case the next neighbor (laterally or frontal properties in the street). What is this interchange? Prigogine  states that the relationships of each cells of the dissipative system with its neighborhood are given in the imitation or non imitation process of the human systems.
In this way we can think that the variables will changes with the time by the imitation of its neighbor cells or react in the opposite manner. Let say that each land use has a different color we will see how in a self regulation process will form groups more or less thick, until few colors dominate in the system, and the system became homeostatic and this can be compared to a thermodynamic death. And less we provoke a fluctuations in the system –architectural development, urban works, public development (lets say a road, transport system, park) and this can will put to play another dissipation process of a new self regulated system that will revitalize the dead urban environment.
This process however is not that simple, lets say: making an urban evolution simulator that only takes the imitation or non imitation process of next neighbor, this is right for a static system that raises the critical horizon of the system. In Monterrey file the correlation matrix reveals the state system upon a time is like a picture of the environment in a moment of its history, that changes very slowly being the portrait of one step in the dissipative system evolution, this defines the subsequent steps. Our hypothesis is that the matrix is like a resistant force of change and will put a measure to the transformation possibilities of environment. The imitation and non imitation can be defined as a two contradictory forces that the cell faces in its evolution, let say in a vector diagram the origin if the two forces is the center of the system and disposed over the same axis there will be another force that will be opposed to both in a perpendicular axis, and the result of this diagram represents the probabilities of the self fluctuations in the dissipative system. Let us complicate this a little bit more. In the environment it is found a specific proportion in the attributes distributions (variables) associated to each cell and we define this by the next formula:
rVB1 = __________________ + W
n[VB1, VB2... VBN] (1)
rVB1; proportional distribution of the attribute
(VB1); is the number of cases in what VB1 appeared in the category B
n [VB1, VB2... VBN]; is the simple arithmetical addition of the cases that have VB1, VB2...VBN in the category B
k is a number to normalize the result upon a base of scale, for example a percent where k=100
W; is the amount of “local energy” and its non homogeneous level on the environment normalized in a percentile level (2 formula).
This proportion states that each cell will fluctuate from one direction to another. Do not forget that is possible that a high proportion of system attributes can mean that the system in this moment it is in a non equilibrium state by an attribute fluctuation associated, this has a potential to impact the whole system and lead to a new equilibrium based on the attribute, if this fluctuation overcomes the critical horizon. Now, let us say that these probabilities are vector forces that are opposite to imitation or non imitation forces and the system state as another force opposite to both we have the components to calculate the probabilities of the self fluctuations.
High or low level of correlation of the system variables state a superior force of resistant to a change, meanwhile a level near to the middle of the correlation scale will mean a low level of resistance of the environment, in this way each attribute should be compared first in correlation to others attributes of the cell, and then calculated the transformation probability level. A global calculus that takes to account the statistical toward the changes or non changes of the total system can indicate the system evolution potentiality. The figure 1 shows the way of this calculation upon the environment transformation probabilities. R represents the system resistance, which is a measure of system variable correlation value minus a W factor, which is a constant in (1) and (2) equations that defines the “local energy” and its non homogeneous level in the environment. W is an empiric quantity, it is possible to imagine W as a differentiate land value in every locations of the urban system. Each cell in the process of change, take W from the environment and translate it to another cells of its proximity. It is possible to think that when this process occur, merge a new type of W, produced by each cell in it. In this case, W transforms in an additive factor in the interchange equation of system evolution. Because now we are already searching about this phenomenon, we do not have an adequate idea of this process.
To know the amount of system resistance you can apply the next formula:
R= PH1 [A UVB1...BN] – W (2)
R; is the system resistance to change
PH1 [A U VB1...VBN]; is the correlation hypothesis distribution in A field into VB1...VBN when Ho is normalized.
The imitation or non imitation represents the conflict probabilities rVB1…rVBN and is a function that varies with its magnitudes in a mirror like fashion being the probabilities the positive and the back probabilities the negative aspect, being exact in its measurements. In this way it is possible to calculate the system fluctuation tendency until you have the critical horizon of the system that leads to a major non equilibrium. The back probability stabilize the tendencies establishing that the cell go for the non imitation leading to another pattern or fluctuation.
Now, with these attributes proportions in the environment that you can graph as a two dimension diagram you have to add the system state for each cell type and represents the level of variable correlation. Then with the correlations scale now is possible to get a new scale in which the odds would be equivalents in size and the middle of the scale is equal to 0.
With this data you can have a tendency for each cell system attribute and the measure can be obtained by the next formula:
T= (rVBN)2 + (R)2 (3)
T; equal probability of attribute transformation in the given cell
The transformation probability in this case always is a higher number than the system resistance which is the way where the systems fluctuates to equilibrium, in other words when rVBN values get equal (unique variable) and R, reducing the matrix to a single component, in other words 1x1 matrix where T is equal to any value, the system will meet equilibrium in an entropic dead fashion. This is because the m value defines the tendency toward changes, because when the magnitude is equal to 1 the system will meet equilibrium: no attribute can overcome the critical horizon and therefore it would not get an isolated state, but when the value is greater than 1 the system will have an strong tendency to change. You can have horizon level fluctuations. This level, defined by m, will appear by the system inherent characteristics and is self regulated. The fluctuations act by them, so the fluctuations will appear by the simulation itself and not from previously established parameters, m only points a tendency. When m is less than 1, you will have a situation where transformation is less like, and there is a great resistance to a change induced by the attributes correlations of the environment.
m= is the way where the tendencies goes
m is a value that is helpful to define the attributes of cell fluctuations. We can use m value to point out the attributes with great tendencies to a change. This calculation will be done in the way of imitation and not imitation as a mirror like values defined by rVBN. In a dynamic system each new change is associated with a new system of equilibrium and the T, m, R and rVBN values vary reciprocally these are strongly dynamic values chained by mathematic operators. So the simulator will have to do global calculations that will interact with local calculations.
We have to mention that there are some exotic attributes that merge from uncommon patterns in the system. What is the tendency of the exotic patterns of the system? How much is bounded the isolated system with its environment (cosmopolitan city, citizens with lots of information not related with their environment)? If we transform this variable to a number that means global chaos-order, we can generate a simulation with exotic attributes. This number can be related with the level of variety in the environment meaning that the value is equal or greater than 1. 1 means all the cells are equal and greater than 1 all the cells are different from each other.
A system like this can be helpful when you are evaluating a new architectural or urban project toward its environment, put to interact with the rest of cell systems analyzing any merging variable that can be produced, what effects will interact in other subsystems. Also it came be helpful to understand the equilibrium or non equilibrium state that acts on a urban system in any moment of time, so you can act in order to revitalize the system. In many cases you can avoid a miscalculation before the things are done and prevent mayor financial looses. And finally this is what you have to avoid.
The method we developed opened a set of questions to the research that nowadays it has become search axis. The stronger question has to be with the conceptual articulation of the physical and social space. We mentioned at the beginning of this article how the correlation of the physical and social space seemed to emerge of the metaphor, which was created by means of the language respect to a parallel world with the space of action in order to the new conceptual world would live there, claiming objectivity. So, the site in what something is in the world would belong to the social localization, hierarchy of the building or zone to social rank. It seems to each physical space attribute could belong another one which would define it (almost in the same semantic way) some social space attribute.
So, we assume that the localization is a parallel value and were watched directly; also we built an instrument which would measure the correlation of the physical and social space, subordinated to this shared attribute. The creation of the analysis instruments that look for the correlation of the physical and social components through of other attributes shared by these without a doubt will be the new task of our research team in the future.
 PRIGOGINE, Ilya (1997). ¿Tan sólo una ilusión? Una exploración del caos al orden. Barcelona, TusQuets.
 The T value is always positive because this is one of the dissipative system characteristics and this one defines time for it and is irreversible, this means a permanent tendency to change in the self organized systems.