Automi Cellulari - Master Equation

Mathematical modelling of complex systems with cellular automata
Statistical mechanics methods based on the use of the Master Equation
Fuzzy logic
Random differential equations
State of the art in our research
Reflections on the concepts of model’s calibration and validation
Computer simulations
Scientific originality and innovations
Expected benefits of the research work
Working papers and preprint


ACME is a Swiss National Science Foundation project started at the Accademia di architettura di Mendrisio (University of Italian Switzerland, USI) on May 2002 under the direction of Prof. Sergio Albeverio focusing on the mathematical models of urban growth. 
Researchers of the project are:

The results of this project will constitute the PhD thesis of D. Andrey in Mathematics at the Bonn International Graduate School of the University of Bonn.        
The project started from the idea to connect cellular automata (CA) based approaches of urban models to statistical mechanics methods based on the use of the master equation (ME). The introduction of a continuum state (stochastic) CA instead of a finite one opened indeed the possibility to construct a CA based model with a detailed and faithful description of the micro-dynamics of agents populations, and at the same time to prove that the state variables of this CA verify a differential equation (DE) for the time step going to zero (both an ordinary DE proved using ME methods, and a new type of random differential equation which permits to study memory effects and random fluctuations). 
In this way two seemingly different modelling paradigms can communicate: the use of a continuum state instead of a finite one seems to improve the modelling flexibility of CA, and the related differential equations allow the study of the CA from a continuum dynamical systems theory perspective, seeking e.g. for bifurcations and complex phenomena. The same type of modelling method seems to be applicable to several complex systems other than urban ones.
Moreover the use of fuzzy set theory methods permits to formalize in a very meaningful way both the intensities of the stochastic processes we are interested in, and the decision dynamics of agents.

Our work is basically founded on four main theoretical topics:

a.      mathematical modelling of complex systems with cellular automata;

b.      statistical mechanics methods based on the use of the master equation;

c.      fuzzy logic;

d.      random differential equations.

Mathematical modelling of complex systems with cellular automata

Cellular automata represent a well known scientific paradigm frequently used to model complex systems using a bottom up approach: complexity emerges as a non linear effect of simple local interactions between a large number of cells. For a general description of CA see [Ma-To 87] and references therein. CA have been used to model, e.g., urban evolution, car traffic, gas dynamics, reaction-diffusion systems, excitable media, several biological systems, with successful results, both analytical and from the point of view of computer simulations. In particular CA modelling of urban growth processes and traffic dynamics is a rather fast increasing field involving researchers in many areas of urban, social, geographical and mathematical sciences (see, e.g., [Na-Sc 92], [La-Phi 97], [Whi-En 97], [Bäk et al. 96], [Sc-Sc 98], [Por 99], [Whi et al 99], [Sem 00], [Whi et al. 00], [O’su-To 00] and references therein). Among their positive features we can mention that they are dynamical models which make use of local transition rules that can be set up rather intuitively, taking into account the real local dynamics of the systems under study. In the case of urban systems one of the most appealing features of CA modeling is that it can take care of the spatial structure, and a real city is indeed characterized by a spatial distribution of different land uses competing and cooperating through interactions depending on the spatial proximity.

In the last few years many problems have been raised about the CA based models of urban systems that require the improvement of the modeling strategies [Tor 01]. These problems are of general interest in the field of the modeling of complex systems with CA. Part of the problems are connected with the fact that the state of a CA is often only a qualitative object, e.g. a categorial map of the urban space, not suitable for powerful analytical investigation methods. For instance, concerning the calibration of the many parameters appearing in the most developed and powerful CA models, as for example that of White and Engelen [Whi et al. 00], the problem of calibration is connected with the possibility to compare simulated configurations of the city with the empirical ones; this aspect is also connected with the important problem of models’ validation. Fuzzy logic methods have been used to compare qualitative or categorial maps, see [Pow et al. 01], [Hag 03]. A more difficult problem is connected with the expected presence of bifurcations, chaotic phenomena and phase transitions in complex systems. These are highly typical phenomena of complex systems that have been extensively studied in the field of dynamical systems, especially those described by differential equations (see e.g. [Haa 86], [Kuz 98], [Wei 00], [Hol et al. 00], [Yeo 92], [Gol 92] and references therein). The dependence of the observed time evolution of the urban system on the initial configuration and the response of the system to an external perturbation are other examples of problems that are hard to be faced in an effective way in the usual conceptual framework of CA. In fact such problems are mainly discussed in the framework of continuous dynamical systems described by differential equations. To try to associate an ordinary differential equation to a CA one must at least have a quantitative description of the configuration of the CA through numerical variables. Furthermore one must be able to take a limit for the discrete time step going to zero.

Statistical mechanics methods based on the use of the Master Equation (ME)

A large amount of work in the direction of the problems sketched above has been done in the last 30 years in the field of synergetics and socio-dynamics and related approaches (see [Hak 78], [Haa 89], [Sch 02a], [Sch 02b], [We-Ha 83], [We-Ha 87], [Wei 91], [Wei 97], [Wei 00] end references therein). In this approach a socio economical system is usually described through a Master equation, governing the time evolution of the probability distribution on the configuration space of the system. Under certain conditions a differential equation for the time evolution of the mean values of the dynamical variables can be derived. Bifurcations, chaotic dynamics, phase transitions and other interesting behaviors are effectively observed in many of these models of socio-economical systems (ibidem and e.g. [Mu-We 90a], [Mu-We 90b], [We-Ha 87], [Pop et al. 98]). Master equations or Markov semigroups methods are particularly useful to model situations where a great number of microscopic agents interact together to give a complex dynamics at the macroscopic level. In our work we used the setting of Markov semigroups methods (in particular the Kolmogorov forward eqution, a method equivalent to the use of the ME) to prove that extensive observables of our system (e.g. any state variable) verify an ordinary differential equation (ODE) if we can assume, as usual e.g. in Socio-dynamics, that the observed quantity has an infinitesimal standard deviation.

Fuzzy logic (See [Haj 02])

 The term "fuzzy logic" emerged in the development of the theory of fuzzy sets by Lotfi Zadeh [Zad 65]. A fuzzy subset  of a (crisp) set  is characterized by assigning to each element  of  the degree of membership of  in . Now if X is a set of propositions then its elements may be assigned their degree of truth, which may be “absolutely true”, “absolutely false” or some intermediate truth degree: a proposition may be more true than another proposition. This is obvious in the case of vague (imprecise) propositions like “this person is old” (beautiful, rich, etc.). In the analogy to various definitions of operations on fuzzy sets (intersection, union, complement, …) one may ask how propositions can be combined by connectives (conjunction, disjunction, negation, …) and if the truth degree of a composed proposition is determined by the truth degrees of its components, i.e. if the connectives have their corresponding truth functions (like truth tables of classical logic). Saying “yes” (which is the mainstream of fuzzy logic) one accepts the truth-functional approach; this makes fuzzy logic to something distinctly different from probability theory since the latter is not truth-functional (the probability of conjunction of two propositions is not determined by the probabilities of those propositions).

Two main directions in fuzzy logic have to be distinguished. Fuzzy logic in the broad sense (older, better known, heavily applied but not asking deep logical questions) serves mainly as apparatus for fuzzy control, analysis of vagueness in natural language and several other application domains. It is one of the techniques of soft-computing. It is in this context that we used fuzzy logic in our model.

Fuzzy logic in the narrow sense is symbolic logic with a comparative notion of truth developed fully in the spirit of classical logic (syntax, semantics, axiomatization, truth-preserving deduction, completeness, etc.; both propositional and predicate logic). It is a branch of many-valued logic based on the paradigm of inference under vagueness.

Random differential equations (RDE)

RDE are a general probability tool sometimes used to represent the pathwise dynamics of a physical system having a random component (i.e. a random dynamical system, see [Arn 02] and references therein). They are equations of the form   where  and  is a probability space. Thus they are dynamical systems under the influence of randomness because the unknown function  depends on elementary events . Their complete solution amounts to determine the probability distribution of .       
In our work we proved a new type of RDE using the notion of forward mean derivative:


where  and  is the sigma algebra generated by present and past states of  (so  is a way to consider the past history of the system see [Nel 85]). This notion of derivative of a stochastic process has been introduced in Stochastic Mechanics, an alternative but equivalent way to study Quantum Mechanics and this is the first time that it is used in a completely different context.

More precisely our RDE is a system of differential equations, one for each moment of the random observable so that its solution gives the whole probability distribution of  (not only the mean value), and the complete stochastic dynamics of  can be considered. The complete study of this type of RDE and its numerical solution will be performed in a future project, even if it seems clear that the discovery and proof of this RDE opens new interesting possibilities, e.g. the study of chaotic behaviour induced by random fluctuations.


State of the art in our research

The researchers of this project has developed a CA for the study of urban dynamics in which the state of a cell is not described using a finite set, but by means of continuum variables (volumes and surfaces of different types of land uses; see [Van et al 04] for a detailed description of the model and an informal but meaningful way to introduce the ODE).       
A system of ODE has been derived from the Kolmogorov forward equation (a method essentially equivalent to the use of the ME) associated to the CA in the limit


Use the following link to see a working paper with a rigorous proof of the ODE.        
Use the following link to see some computer simulations.        
Use the following link to see a detailed description of the urban system used for these simulations.

The type of model we constructed has been appreciated both by scientists in the field and by national and international city planners. In fact, the next step will consist in the use of the model in a real case study and an important and fruitful collaboration with the institute for the Contemporary Urban Project (iCUP) of Prof. Arch. J. Acebillo at the Accademia di architettura has already been started, concerning data collection, data management and adaptation of the model with respect to specific urban problems relevant for the considered real case. We plan, among other things, to have a joint study of the urban problems connected with the development of the “Nuova Lugano”.

More precisely our CA model is formed by the following components:

               i.      A cellular space  constituted by the set of all the cell of the system.

             ii.      A continuum state space  (usually a finite dimensional vector space) describing the state of a cell . The components of a vector   associated to a cell  are built volume end related surfaces for different land uses.

            iii.      A finite set  representing elementary events like for instance “building of new residential volumes” or  “conversion from  residential use to offices” or “occupation of  free already existing residential volume”.

           iv.      A finite dimensional vector space  for each . A given  represents the set of continuum parameters describing the event  quantitatively. For example for  = ”building of new residential volume” we have  , where  is the total surface used by the building,  the covered surface and  the built volume. The parameters  , representing what is produced by the event , are called “goods”. The production of goods requires the consumption of spatial resources. So the urban dynamics can be seen as a process of consumption of resources with a related production of goods.

             v.      We call “  -event” an event of the kind  taking place in the cell  and with continuum parameters . An  -event is described as a Poisson process taking place with (“density of”) intensity . This means that the probability  to have   -events in the cell  during a time step and with continuum parameters included in a measurable subset  of  is given by:


where .

The hypothesis on the Poisson distribution has a clear urban meaning, see [Van et al 04] (note that the intensity  depends on the present state of the automaton and thus it is time dependent, so that generally speaking it changes at every time step  ). In order to build a model for urban dynamics in the conceptual framework summarized above one has to decide the intensity  of the processes. We face this task using fuzzy logic methods (see e.g. [Du-pr 80], [Got 93], [Ba-Go 95]). Fuzzy logic methods have been used in the last years in CA models for urban growth in order to deal with problems connected with the vague knowledge about cell states (see [Li-Ph 01] and references therein). We have applied fuzzy logic in order to model explicitly the decision making processes of the agents producing the time evolution of the urban systems. An   -event is the result of a decision process made by an   -agent , that is an agent interested to  -processes. One has first to specify a set  of criteria relevant for the action of  using the natural language (hence they are propositions in English, and this is very useful to perform an efficient interdisciplinary work) and then to compose them through fuzzy logic operator like AND, NOT and OR with the aim to produce a global criterion  driving the behaviour of the agent . Following fuzzy logic, the truth value  of this sentence can take continuum values between 0 and 1 depending on the truth values  of the compounding propositions . The elementary propositions  refers to a universe of the discourse given by set of indicators which are nothing but functions of the state vectors of the cell  in a suitable set of neighbourhoods. The higher is , the higher is the intensity   of the process.

Agents are grouped in populations , , possibly subdivided in subpopulations characterized by different fuzzy rules of behaviour. Agents belonging to different subpopulations of  can be characterized  by a different social or cultural status or by different attitudes towards behaviour. In our implementation of the model we have distinguished for instance the population not belonging to the city in two subpopulations: not active individuals, only potentially interested in moving towards the city and in active ones, that can be seen as individuals actively involved in the decision process to find a house in the city. The first class of individuals look at the city as a whole and can pass in the active category with an intensity which depends only on global features of the urban system; individuals in the second class are actively comparing configurations of the cells and can either become part of a population of a cell or return in the passive state if they do not find an advantageous location in the system inside a given time. Thus in our model agents are not represented as individuals (as in ordinary multi agent Information Science based models) but as populations with elements acting following fuzzy rules. This is an essential step to use Statistical Mechanics methods.

The use of a continuum state space on the one hand enables one to realize new ideas about the model’s calibration using meaningful distances on the configuration space of the whole CA (see this document for an extended reflection about calibration and validation for these type of models).  On the other hand it also permits to write a system of differential equations for the time evolution of the automaton and thus to study the system from a dynamical systems theory perspective. This makes possible, in particular, to look systematically for bifurcations in urban systems.      

Two types of DE describe the dynamics of the state variables: the first one is of ordinary type and can be used for urban processes without memory (Markovian) and with first order infinitesimal standard deviation (and hence zero variance, see [Gio 04] for the rigorous theory of actual nilpotent infinitesimals we used in this work), and is derived from the Kolmogorov forward equation (these results will be published in [Van et al], see ODE for a working paper about this topic). As usual using Master equation methods the unknown function of the ODE is the mean value of the observable. This permits on the one hand to use fuzzy logic sentences to construct meaningful observables and on the other hand to solve the ODE to obtain the dynamics of the mean value.

The second one is a system of RDE, one for each moment of the state variable. It describes non-Markovian processes too because of the use of the forward mean derivative and enables us to remove the above mentioned zero-variance assumption, because we have one RDE for each moment  (these results will be published in [Gio et al]).    
Use the following link to see a working paper with the proof of the RDE.



The first results has been presented to the International Congress of Industrial and Applied Mathematics ICIAM, in Sydney, July 2004. Moreover during the development of this first SNF project we started several contacts with leading groups of researchers in the area. This has led to the preparation of the workshop “The dynamics of complex urban systems - An interdisciplinary approach” which has been held in Ascona, Monte Verità, from 4th to 6th November 2004, and will enjoy the presence of practically all international specialists in this area.


Reflections on the concepts of model’s calibration and validation


To work correctly these kind of models usually requires the evaluation of a great number of parameters. It is the classical problem of model’s calibration, faced by several authors using optimization techniques, statistical analysis, fuzzy logic methods, etc. Usually the problem can be schematized in the following way in case of urban systems:

-          suppose to have empirical configurations of the system at different times;

-          start the dynamical model taking as initial configuration the oldest one;

-          use “some method to compare” the simulated configurations (outputs of the model) with the real ones;

-          “adjust the parameters” so as to minimize the differences between this two types of configurations.

The problem of model validation (i.e. “what is the precise meaning we give to the sentence: the model describes the real system?”) is frequently faced in a similar way: instead of “adjust the parameters” we consider different empirical configurations and, running the model as above, we compare again simulated and real outputs.

But these solution to the calibration and validation problems seem to be incorrect if faced using the above mentioned ideas, based only on the comparison of configurations. In fact they reflect exactly the following idealized situation:

-          suppose to have a loaded die (the urban system);

-          the only knowledge about this die is that it started in the past with the face ‘1’ up, and after a throw it ended with the face ‘6’ up (the empirical configurations);

-          suppose to have a detailed physical model of a generic loaded die, with several parameters;

-          the problems are: how can I calibrate model’s parameters without any knowledge about the probability of the die’s path going from ‘1’ to ‘6’? How can I precisely say that my model describe the given real loaded die?

It is important to consider that our urban models (generally speaking models of complex systems) usually are necessarily stochastic models so the problem is even more great: what simulated configuration do I have to compare with the real ones? The mean configuration? But what does it happen if the path of my urban system (the empirical configurations) doesn’t reflect a mean behaviour? The risk is hence to calibrate our models to work with a mean dynamics corresponding to a real situation which is not “mean”. It seems indeed really difficult to evaluate the probability associated to the path covered by the real city.

These problems generally affect every model of a complex system to which is not possible to apply the old paradigm of Physics about the repeatability of a physical experiment.


For these reasons the use of fuzzy logic in our model permits to found both calibration and validation on a different base, explained in the following document.


Computer simulations


In our model each cell has a state represented by 11 random variables. For this reason is not possible to show, like in ordinary finite state CA, “the dynamics of the whole city”, but some kind of indicator has to be chosen. Obviously each state variable can be a possible indicator, but using these numerical quantities and fuzzy logic it is possible to define a great set of meaningful indicators, three of which are reported below. Obviously there is also the possibility to define a criterion to loose information and obtain a categorical map instead of a continuous one so as to have a more classical simulation. This last work is in progress.


Centres: fuzzy indicator able to detect high density zones with a high degree of mixing of different kind of activities (“centres”).


Small centres: fuzzy indicator able to detect lower density zones with a high degree of mixing of different kind of activities (“small centres”).


Population: The amount of occupied residential volume in the cell (giving an estimate of the resident population).


For a complete notes about these indicators and the simulations see this document.


Scientific originality and innovations


In this research work theoretical and practical aspects are strongly connected. The theoretical topics, both in the theory of stochastic processes and in the modelling of complex systems, are interesting aims in themselves and are suited to be further developed beyond the specific objectives of this research project.

Scientific originality and innovations can be sketched as follow:

1.     These type of CA seems to be a new kind of modelling tools, similar to classical CA but with a continuum state space and stochastic evolution rules (usually governed by a continuum probability unlike what happen in standard stochastic CA). Like CA have been applied to a wide range of complex situations, they seems to have potential useful applications to the study of several complex systems. The use of continuum state variables seems to facilitate models construction, conducting to a more detailed and natural mathematical description of the studied system.

2.     The use of a continuum state space enables to compare simulated configurations of the system with empirical data through meaningful metrics in the configuration space, e.g. the relative percentage deviation. That enables to deal with the problem of parameters calibration using optimization techniques. This approach is not possible using standard CA models because of the use of a finite state space (where anyway powerful techniques of comparison of categorial maps are available). The evaluation of the probability associated to a simulation permits to quantify the meaningfulness of future forecasts obtained from the model. For example it is possible to evaluate the probability to have a forecast in a given neighbourhood of a suitable configuration and hence to answer questions like: what is the probability that our system will end to less than 15% of the average development? In this way we have a scientific meaningful way to talk about models validation. Besides the use of standard metrics on the configuration space, it is possible to compare configurations in the frame of fuzzy programming and this enable to select explicitly the feature considered as relevant in the calibration of parameters and in problems connected with the model validation. Indeed a key factor is that the continuum configuration space enable to base these evaluations by means of meaningful and intuitive indicators of the urban configuration, explicitly connected via fuzzy logic methods to criteria of evaluation of experienced people.

3.     It is important to note that our model enables to put together the advantages offered by usual CA, with the construction of  simple local evolution rules, with the powerful mathematical and physical background of the approach of Synergetics. In Synergetics and Econophysics usually the ME is not exactly solvable for non trivial systems. One is interested to obtain from the ME an ODE about the mean values of the dynamical variables. To perform this an approximation is frequently used which is equivalent to say that the stochastic motion of the state variables has an infinitesimal standard deviation. The system of infinite RDE, one for each moment of the state variable, describes the stochastic dynamics without approximations. This opens the possibility to study bifurcations induced by random fluctuations. This is important to study complex phenomena intuitively describable as bifurcations using either the ODE or the RDE related to the CA which models the system. In this sense continuum state CA go beyond both the techniques used in Synergetics (where the ODE derived by the ME is only approximated and do not enable a full investigation of phenomena connected with the stochasticity of the system) and in the theory of CA (where the analytical tools needed to study the stochastic phenomena mentioned above are neither so extended nor so powerful). E.g. in our model for urban growth we consider a set of elementary processes with a local dynamics which share with CA concepts like that of neighbourhood and of evolution rule (the rules can be seen as simple, being possible to define them in the intuitive form  ) : as a consequence of these definitions we prove the related RDE and this open the possibility to study the CA as a  (stochastic) dynamical system.

4.     In the standard approach of Sociodynamics the behaviour of agents is modelled trough a “utility function” expressing the degree of  affinity of an agent for a given configuration of the system. But this is a much more complicated task than just specifying in a verbal form a set of rules of the form . Furthermore fuzzy logic is especially suited to manage situations where human individuals have to interact with machines. Thus the use of fuzzy logic in setting up models for urban growth dynamics could be a first meaningful step toward the creation of a flexible software to support decision making and planning strategies. Also the interdisciplinary collaboration can profit from this, in the sense that the rather natural way the evolution rules can be constructed using fuzzy logic, can contribute to reduce the interdisciplinary gap between the involved parts.


Expected benefits of the research work


We expect that the results of this research work will concern both the scientific community and many users interested in practical issues as management and administration.

Concerning the scientific community the following point seem to be especially relevant:


        Continuum valued CA are a rather innovative approach to the modelling of complex systems. From a general point of view they generalize ordinary CA, which have already proven to be paradigmatic examples of complex systems and very powerful modelling tools. The possibility to define these type of CA as stochastic dynamical systems driven by differential equations increases the arsenal of analytical and computational tools at disposal to investigate CA-like complex systems. So we expect that the results of the project will represent a step forward in the knowledge about complex systems.

        The use of a CA-like approach to the study of complex systems together with the related employment of RDE and ME bring together rather different field of research in Mathematics and Physics. We expect that the synergy among these parts will be advantageous for all the involved disciplines. In particular we expect that the use of RDE with the forward mean derivative in the study of complex systems opens a new interesting field of applications to this mathematical tool that until now has been employed mainly in the field of Stochastic Mechanics.

        The research about urban dynamics is a typical example of interdisciplinary approach to a problem. Researchers from Mathematics, Physics, Economics, Geography, Urban Planning, Sociology and Information Sciences are usually involved in a strongly interdisciplinary collaboration. As already mentioned we have developed a modelling approach based on fuzzy logic which may contribute to enhance the interdisciplinary collaboration. So we expect with our work to perform some steps toward the reduction of the gap between people more involved in a Mathematics and Physics based approach and people more experienced in approaches based on Geography, Sociology and Urban Planning.

        The study of phase transitions, bifurcation and chaotic behaviour, are considered very important inside the scientific community for further development of the study about urban dynamics. We expect that our approach, putting together CA based models of urban dynamics with the theory of stochastic dynamical systems will represent a step forward toward the solution of this problems. We expect to going forward with the very fruitful approach of Sociodynamics and Synergetics enriching it with the new tools from the theory of RDE.


Working papers and preprint


Van et al 04

Preprint submitted to Environment and Planning B


Working paper: proof of the ODE


Working paper: proof of the RDE

Case study

Detailed description of the urban system used for simulations

Computer simulations

Notes on computer simulations presented in this web page


Working paper: calibration and validation for complex systems’ models





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