In this article we expose some of the general properties of complex
systems that can be symbolically represented by finite models. They are quite simple
models, that consist in a certain number of elements that assume certain states and the
evolution from one state to another is the result of the action of some law. Examples of
this kind of models are genetic algorithms and cellular automata, for instance, the most
known model is the Game of Life. We
begin by presenting them here in some general context, as symbolic entities, because there
exists common properties in a symbolic description that motivate an analysis not related
to the reality they may intend to describe. Thus, through the paper, we refer to models
considered in a scientific perspective, initially not connected to any particular object,
without any intention of relating the model to a particular reality. However, when the
abstraction is sufficiently general, it is natural that similar consequences arise from
certain common characteristics. These analogies and interpretations will be presented in
the final section.
To be more specific, in the first section we present some of the general
properties of the finite models. We will often appeal to the reader's intuition, and
therefore the second section will have as a main goal the computational synthesis of those
properties. We will present computational experiments with a code conceived for that
specific purpose, which will guide the discovery of complex systems and the analysis of
the concept of emergence. In the third section, we will also make some
computational experiments to which the reader is invited and where his intuition and
action lead him to a particular finite model, the model of Schelling and to the
segregation processes. Finally, in the last section, we will present what kind of
substantive hypothesis about the empirical reality can be obtained from the symbolic
description and the computational synthesis of the finite models.
1. Finite Models
The finite models in discrete time processes here analysed are the models
that describe the evolution of a system under successive application of a law. To be more
precise, a model of that type consists in a finite number of elements that admit a finite
number of states. A simple example is to consider that elements admit only two states,
that we can call state 0 and state 1. To the set of states that the model presents in a
certain moment we will call configuration. Obviously, instead of considering only
two possible states, we can consider models where the elements can present a finite number
of states. The following example shows these introductory definitions.
Example. Consider a model with 10
elements, represented by small squares that may assume 4 different states represented by
colours. The Fig. 1.1 presents two possible configurations of that model. The difference
between them can be resumed in the change between the white and the blue colour in the two
central elements. The arrow placed between the two configurations represents the evolution
by the application of some law. In this example, that law caused a change of state in the
Figure 1.1. Example of evolution of a certain
Note that there are several possible interpretations for the evolution
shown in Fig.1.1, and all these interpretations are equivalent. One can either consider
that each element corresponds to the position of a static particle and, in that
interpretation, the evolution was the colour change in the central elements, from blue to
white and vice versa, or one can also see that change as a movement from a blue particle
into a void space, represented by the white colour. The observer is the one responsible
for this kind of interpretations, and they are usually motivated by a connection between
the model and some reality, or is just an interpretation that allows an easier
understanding. One should keep in mind that, in both situations, and with no connection to
the interpretation, the configuration evolution is always the same. This is a
characteristic of these models that we will address again.
The lines that are plotted connecting the small squares define a topology
establishing a proximity relation between the elements. The elements and the lines
connecting them define a graph, a structure that allows the formal analysis of the
finite models as nets. That topological framework can be avoided, but it will be
necessary when we need to distinguish between local and global phenomena, since
neighbourhood notions are present. In the models presented in sections 2 and 3, being
obvious what is the neighbourhood considered, we do not present the lines connecting the
It is worth noting the enormous amount of possible configurations that
exist even in simple examples, like the one presented in Fig.1.1. Having 4 colours for 10
positions, the number of different configurations is quite big, 410 = 1 048
576, which shows the magnitude of these values even in models with a small number of
elements (positions). For instance, bellow in the text, we will present simulations for a
model with 400 positions and only two states, and this gives 2400 possible
configurations - a number with 121 digits. We do not even consider finite models based on
the pixels of a computer screen, since 1024*768 pixels
with 65536 possible colours, generates an astronomical number of combinations, with more
than 3 million digits! This kind of magnitudes may lead to the illusion of almost infinite
quantities, and to careless conclusions.
The evolution of finite models in discrete time, and the dynamics that
leads from one configuration to the next one results by the application of some law.
Each application of the law is called iteration, being
possible to apply the same law to the new configuration, and so on. A characteristic of
the evolution of the configurations on a finite model is the generation of cycles or of fixed points.
When the law determines only one possibility and when after n iterations you
obtain the same configuration, we get a cycle with length n. If the
configuration does not change it is called a fixed point. Fixed points and cycles
are types of what is called an attractor, and besides these two types one also
considers chaotic attractors, that do not occur in finite models.
In closed finite models, when a well-determined law is applied, a simple
and immediate consequence occurs:
After a finite number of iterations, the configurations are repeated and therefore
A justification of this fact is quite elementary. It suffices to think that the number of
possible configurations is finite, equal to EP, where P is the
number of positions and E is the number of states. After a number of iterations
greater than EP there must be a repetition of a configuration and since
the law determines the same evolution, a cycle is formed.
Below on the text we invite the reader to execute simulations where that
conclusion is not immediate. Due to the enormous amount of possibilities, the viewer might
think that the evolution leads always to different configurations. However, when a
well-determined law is applied and the configurations are repeated, we are in the presence
of a cycle or a fixed point. Only in the case of an open system, using external factors,
for instance, random factors, that conclusion may not be true.
Examples. Let us see two examples in
which we obtain all the configurations in the system starting from one initial
configuration. We consider a model with 3 elements (positions) and 2 states. This gives 8
different possible configurations. Initially we present a global cycle:
Figure 1.2. Iterations on a global cycle. The arrows
represent the well-determined evolution, one for each iteration.
Notice that each ball represents a configuration (a set of elements
with precise states) and not an element.
In this first example (Fig.1.2), it is obvious that
starting from any configuration we can arrive to another one after a finite number of
iterations. This defines the length, or period, of the cycle. In other example (Fig.1.3),
the initial configuration (and all the others in blue) are not repeated, because the
evolution ends in a smaller cycle (configurations in green):
Figure 1.3. Iteration
that leads to a smaller cycle. Arriving at the last configuration, the
system evolution returns to a previous configuration and enters in a
cycle (configurations plotted in green).
In these two examples one can see that the iterations
may lead the system from one configuration to all the others. This is not always the case.
In Fig. 1.4, we see that the system evolution may leave aside a part of the
Figure 1.4. Independent evolutions, cycles and fixed points.
As shown in Fig.1.4, the configurations 1, 2, 3 and 4 are inside a
cycle, therefore if we start the evolution in any of these configurations, it is clear
that the configurations 5, 6, 7 or 8 will never occur in the evolution. On the other hand,
if we start in any of the configurations 5, 6 or 7, the result, after a small number of
iterations will always be the configuration 7... also, starting in the configuration 8,
there will be no change. Configurations 7 and 8 are fixed points under the defined law.
There is a major difference between finite and infinite models. In the
finite case, if the law is one to one, starting from a configuration that is not a fixed
point it is not possible to attain a fixed point configuration. In the infinite case, this
is perfectly possible with certain one to one laws. Thus, in a discrete law, we can only
aim to attain fixed points starting from different configurations if the law that we apply
is not one to one. This immediately implies the existence of primordial configurations
for that law, i.e., the existence of configurations that do not result from the evolution
of other configuration.
Example. The existence of cycles and
primordial configurations in a law that it is not one to one.
Figure 1.5. Primordial
configurations (in blue) and a fixed point (in red). Following the arrows, the reader
may see that any initial configuration converges to a fixed point (in
This leaves some initial configurations, plotted in blue, that are
primordial configurations, meaning
that they do not arise from a previous configuration, as the picture
This defines several 'paths' that, by the iteration of the law,
lead from a primordial configuration (in blue) to the fixed point (in red).
As pointed out before, even if fixed points do not exist, the result of the iteration
of a well-determined law ends always in cycles. The configurations that are not included
in those cycles are always preceded by primordial configurations. Therefore we can
conclude that in a finite model the evolution of the configurations can be described by
cycles or paths that end in cycles (these cycles may be fixed points).
2. Emergence of patterns or laws
Previous conclusions were taken in a symbolic level, and at the same time
they were used to introduce some basic concepts used on finite models and on the complex
systems that they might represent. Their computational synthesis will be made by
experiments (to which the reader is invited). By those experiments it will be possible to
synthesise the local/global dialectic. The characteristics of these last two concepts will
be revealed progressively, but we will be more close to their nature if we notice that a
law operating in a finite model may present features not on the element level, but also at
the level of a set of elements, i.e. at the level of a configuration. This level
distinction, between a micro and a macro level will now assume a central
Consider a finite model in which the elements are 20x20 small squares
(pixels). Each element may assume two possible states, red or white. In Fig.2.1 are
represented two configurations for this model, where the configuration presented in the
right picture was the result of the configuration on the left applying 5 times a
(In the simulations that follow, we use circular topology on the
plotted square, in the sense that an element being in the north border is a neighbour of
an element being in the south, and the same goes for the east and west borders of the
square. Thus, when a particle is moving and arrives at the square border it may appear on
the opposite side)
Figure 2.1. Apparent movement of
a segment on the model.
We see that the vertical segment 'moves' in the northwest direction,
keeping the shape.
Suppose that the law that acts on the model is unknown. We will try to
infer by abduction, using computational simulations, what is the law that explains the
In the example presented in Fig.2.1, we can visually identify a vertical
segment that seems to move in the northwest direction. Starting with the initial
configuration given in the left picture, the reader is invited to press the button Iterate on the dialog box of the online simulation. One sees
that, in fact, the segment moves always in the northwest direction. Undoubtedly, the
program viewer clearly identifies a vertical segment, but is that identification of its
own responsibility? That grouping is even further justified when the viewer sees that by
application of the law (pressing the button Iterate),
there is a similar structure that the only difference to the previous one is the fact that
it has changed position. That is, with no further information the viewer might think that
the only existing reality on the model is a segment that moves in a certain direction. The
movement of the all block is therefore inferred. (In a certain way, one might say that
this is the process that generates the illusion of animation. The identification of shapes
by the association of pixels and the illusion of movement is a consequence of the viewer
It will be used the notion of body to designate any well-defined
set of elements, like vertical or horizontal segments. When a body keeps its structure
invariant under the action of some law, we will say that we have a pattern.
Notice that, at this stage, one is supposed to ignore what is the law
that acts in the system. However, after some computational experiments, similar to the one
presented in Fig. 2.1, for other vertical or horizontal segments, it might be quite
tempting to admit that that law is simply: bodies are moving northwest, keeping shape
and constant 'speed'. This tentative law holds perfectly if we only test the model
with vertical and horizontal segments - the only bodies that we assume that might be
tested/built at this stage. Notice that the process described is analogous to the method
used on the elaboration of scientific theories. In both cases, one infers laws from the
existing observables. In the computational simulations that follow, as in the scientific
method, we will proceed with the construction of new observables.
The first of those simulations consists in testing the
previous conjectured law using different bodies. In the example that follows we built
segments that are almost diagonal, like the one shown in Fig.2.2A):
Figure 2.2. Transformation of an
almost diagonal segment.
Executing the suggested simulation the reader will see that initially the
body will be transformed in a J-like shape, and when that transformation is concluded
, the body will be moving northwest, like the previous tested segments. However
the previously conjectured law is not enough to explain the transformation from the almost
diagonal shape to the J-like shape. Basing ourselves in the new observations, we might be
tempted to introduce auxiliary hypothesis to that law, adding the assumption that almost
diagonal bodies have a limited time of life, being decomposed in a new body which could be
seen as the union of a vertical and an horizontal segment. This kind of auxiliary
hypothesis is frequently used in science. But does this bring new advances in the theory
that explains the evolution of the bodies in the system? Is one closer to the law that
acts in the system? Let us see...
In a third phase, the new experiment consists in 'bombing' the almost
diagonal segment with an elementary particle. Notice that this was not possible before
introducing almost diagonal bodies, because all bodies were moving at the same speed, thus
with no chocks. The result of the simulation is the following:
Figure 2.3. Collision between an
elementary particle and an almost diagonal particle.
, we see that by simulating the chock between an elementary
particle and a diagonal body, in the middle of the transformation process (Fig.2.3B):
the vertical and horizontal segments predicted before, but now they are
disjoint. Additionally a new body was obtained - a 2x2 square. If the reader iterates
further on, it can be seen that that square does not change its position, i.e. the new
square is not affected by the 'northwest attraction law'! This is a bigger exception to
that law, and new explanations were in need.
We will not proceed with a list of new experiments that could be made in
order to find out the law that acts in the system and that is responsible by the several
features of the tested bodies. We can say that with some effort, it could be possible to
state the correct law or an equivalent one, which was simply:
Law: if an element has less than 3
neighbours of its own colour, he changes colour with a neighbour of opposite colour.
To choose among the possible opposite elements, he starts with the
northwest one. If that one is not opposite, he chooses the west one, otherwise the
(This law is applied starting with the northwest positions and finishing
with southeast ones)
It is important to notice that this is a local law. Each
element has short sight, since the evolution of his state is only determined by the states
of its closest 8 neighbours. He does not see anything behind that horizon, he does not
'know' anything about the global configuration of the net. This type of law,
defined using a neighbourhood with 8 elements, is usually used in cellular automata (from
which this model is an example). An aspect of this law is shown in Fig.2.4.
Figure 2.4 In the left
picture one sees the red element rounded by its 8 white neighbours.
Since that element has less than 3 neighbours of its own colour (in
this case, it has none), the application of the law stated above
is reflected on the change of position between the red and the
northwest white element.
Consider now the Fig.2.5. All coloured squares have 3 or more coloured
neighbours, and this would imply no changes. However changes occur due to the fact that
the white square in the centre has only two white neighbours. In fact, in this example,
one can see two cross-shaped bodies glued together. The cross shape is the only one that
enables immobility when we consider only 5 elementary particles, under the action of the
stated law. One might expect that being cross shapes immobile under the action of the law,
these two glued together would also be stable. However, since the white particle must
change position, some of the coloured ones will also change.
Figure 2.5. A white square
caught inside two stable cross-shaped bodies.
What is mainly important to conclude from these simulations is that one
can infer complex laws of behaviour, at a macroscopic level, from any law that determines
the system evolution with local statements, applied to each element. Thus, the previously
called 'northwest attraction law' is a completely true law for the observables available
in the first stage of our experiment. It is a completely true law at that level the
macroscopic level, and we do not have to know the real underlying micro law that generates
it. In subsequent stages one can infer other laws that are equally true and that have
nothing relating them to the local law applied in the model. In that law, it was never
said that one could never have an immobility situation (or a fixed point) with less than 4
coloured elements. One never stated that exactly with 5 coloured elements, the only
immobility situation (and fixed point) would be cross shape. One never stated that the
elements aligned in horizontal and vertical segments move together northwest, and so on.
The law does not even predict the grouping of elements, it is only applied to each element
in its neighbourhood.. Thus, and answering a previous question, one can say that the
interpretation that the grouping of elements and patterns exist is made by the viewer
under its own interpretation and responsibility.
However, we also notice that by stating that there is an interpretation
made by the viewer, this does not mean that the macroscopic state is not real, that it is
only an illusion! That state depends on the observer, but it is real, in the sense that it
is an autonomous state to the microscopic dynamics underlying. We ask the reader to recall
the simulation made in Fig.2.1. In the language
of complex systems theory, one can say that the pattern (the segment moving northwest) emerges.
It is a macroscopic pattern that emerges from the microscopic local law that, in some
point of view, it is the only real dynamic acting on the model. As it was previously
stated, in what concerns the law, it makes no sense talking about a vertical segment, or
even of bodies. However, in the observer's mind, it makes sense to say that it exists and
that it is real a macroscopic state that consists on grouping the elements, even if the
pattern is generated by the local law. It is this subtle difference between a
micro and a macro level that raises the interest on the emergence
phenomena. The confirmation that these two levels exist and that they are real is that the
pattern feature persists although the elements and their states are changing in each
iteration. In the case of the simulation presented in Fig.2.1, we can consider that the
segment in Fig.2.1B):
is made by different elements from the ones considered initially in
. One can even say that this is the confirmation of the existence of a level
that, being completely generated by a local macroscopic law, is however autonomous.
Another important conclusion concerns stability, the effect of small
local changes in the global configuration. All the cases presented so far are
deterministic, being perfectly possible to know the state of all elements after any number
of iterations; however, if we are deprived of the knowledge of some part of the system, it
may be not possible to predict the evolution in the known part. This fact is due to an
interdependence of the local law in the neighbourhood, the behaviour of each one depends
on the others. This interdependence among the elements of a model characterizes complex
systems. This is illustrated in the example shown in Fig.2.6.
Figure 2.6. Propagation of
instability - impossibility of local prevision.
In the situation shown in Fig.2.6A):
, the elements of the rectangle placed above the big
chess-square seem stable according to the 3 neighbour law. In fact, making a prediction of
the system evolution only using the elements that are in the neighbourhood of the
rectangle, we hardly could suspect the evolution that will occur after a few iterations.
The interior of the chess-square is not stable, because some of the elements with white
states have only two white neighbours. This instability will be quickly propagated to
other elements, generating a sort of 'explosion' of the chess-square. In Fig.2.6B):
already possible to predict that the instability generated inside the chess-square will
produce the destruction of the northwest part of the square and will then change also the
rectangle, previously considered stable (Fig.2.6C):
. It is interesting to notice that in
this example, one could predict that no change could ever occur in the rectangle in the
next two or three iterations, by an analysis of its neighbourhood. The instability
verified inside the chess-square could never be propagated immediately to the rectangle.
There is a maximum propagation speed of one element, by iteration, due to the fact that
the local interactions, defined by the system law, apply only to the immediate neighbours.
This is in some context analogous to what is assumed in the relativity theory, where a
maximum speed of propagation of physical interactions is assumed - the speed of light.
Notice that if the law acting in this system was not a local one, the last considerations
could not be stated. In fact, finite models with local laws clearly verify the locality
principle assumed in the physical models.
3. Schelling's Model
Having pointed out some analogies with Physics, we now state that the
model that was presented is similar to a model introduced by Thomas Schelling in the
seventies (Schelling, T. (1971). "Dynamic models of segregation". Journal of
Mathematical Sociology 1, pp.143-186). It is probably the first complex system model
with the objective of studying social phenomena. Schelling gave a particular
interpretation to a model that is almost equivalent to the one presented above. Schelling
was searching sufficient conditions to the existence of segregation processes (between two
races, for instance). Its model will be present below. First, we ask the following
Let us suppose that the individuals in a certain population are tolerant persons, and
do not feel the need to move even if they are in minority... for instance, suppose that
they only move if they are surrounded by more than 62.5% of persons of another race.
This tolerant behaviour generates segregation phenomena?
Later on we will answer this question, first we state a formulation of the Schelling
A formulation of Schelling's Model:
There exist elements (individuals) that may assume two states (they belong to one of two
races), and the behaviour of each one of them is determined by the state of their
neighbours (in the number of eight). The following law gives the evolution of the model:
· If an individual has at
least three (among the eight) neighbours of its own state, it does not move. Otherwise it
So far, the model is similar to the one previously presented. The
difference is that we will not assume a privileged direction of movement, we will now
· The movement is made in a
random direction (thus, the northwest tendency does not occur).
In what will follow, we will use the term deterministic law when
the individual moves along a prescribed fixed direction (northwest, for instance), and we
will speak of a random law when the individuals choose randomly one of the
eight possible directions.
Starting with an initial random population like the one
presented in Fig.3.1A):
, where each square represents an individual, we will first apply
the deterministic law, considering (as before) that the movement is made in northwest
Figure 3.1. Deterministic
evolution to a fixed point, starting from an initial random state population.
Starting with the initial random states, after some iterations, we
already see that there exist a concentration of individuals, forming groups, and only a
few are remaining, that in their movement (along the northwest direction) will be fixed in
the bigger groups already formed, increasing the dimension. In the picture Fig.3.1C):
can see the final result, where the iteration was stabilized, which is a fixed point
of the law. The reader may continue with new simulations and it will be seen that the
model will always converge to a fixed point, which is an invariant that 'attracts' any
In the next simulations (Fig.3.2) we will present the result of
evolutions made with a random direction law, and we will see again the generation of fixed
points. Notice that in this random situation the law changes. The direction of the
movement of the particles may be different even if at some iteration it was produced the
same configuration. Thus cycles do not occur, only fixed points. When a particle has three
neighbours it still does not move and therefore fixed point situations occur.
We begin with the simulation of law where the rule of the
number of neighbours is decreased from 3 to 2. The reader may verify executing the
simulation that the evolution is similar to the one presented in Fig.3.2B):
configuration is a fixed point to the 2 neighbours' law. Afterwards, the reader might
change the option to the 3 neighbours' law and the previous structure is no longer
stable... new movements occur and the system evolution leads to a configuration similar to
the one presented in Fig.3.2C):
, which is now a fixed point to the 3 neighbours' law. It is
clear that the segregation feature is more evident in the 3 neighbours' law, leading to
the constitution of more compact groups.
Figure 3.2. Evolution according
to two different local laws. Fixed points for the 2 neighbours and 3 neighbours law.
SIMULATION (2 neighbours' law) ---------- EXECUTE SIMULATION (3
Suppose again that the law is deterministic, prescribing the northwest
tendency with the 3 neighbours law. As it was pointed out in Section 1, if there is not a
convergence to a fixed point the evolution will lead to the generation of cycles. An
example of that situation can be achieved with the initial configuration plotted in Fig.
Figure 3.3. An initial
configuration case where local stability (almost everywhere)
generates an extra difficulty for the global stability.
The particle that is not aggregated to one of the 9 groups that we see in
can never be placed in a stable situation, unless we change the position of the
others. If the law is deterministic the evolution will enter in a cycle. The same
situation occurs in Fig.3.3B):
if the evolution law is the one with the northwest tendency.
The two isolated particles will enter in a cycle and they will never be with 3 red
neighbours. Since we have a global vision of the model it would be easy to place the
remaining 2 particles in a stable configuration, as presented in Fig.3.3C):
. This is also
possible if the random direction law is prescribed. The configuration of Fig.3.3C):
corresponds to one of the possible stable situations and it was found after a large number
of iterations. Notice that if there was only one particle like in Fig.3.3A):
, it would be
impossible a stable configuration, even with the random direction law.
By introducing the random direction condition we are no longer with the
deterministic case and it will be impossible to predict the system evolution. In the
random case, the evolution of the same configuration may lead to different results and
cycle situations will be no longer possible, the only type of attractors that may exist
are fixed points. In fact, if one excludes initial configurations that can never converge
to a fixed point (for instance, starting with an initial configuration like Fig.3.3A or
with a configuration with less than 4 red elements), the system evolution will stop in a
fixed point if one assumes that the distribution of probabilities is normal.
4. Complex Systems and Distributed Causation
The model proposed by Thomas Schelling is one of the first examples of agent-based
modelling. It is a domain of research that is growing fast, and a considerable amount
of information can be found in these useful links. A good web site
to get software is Artificial Life Online, and one
of the main research centers on the subject is http://www.santafe.edu/.
As it was referred the model may be seen as a model of social dynamics of segregation. The
initial idea of Schelling was that the model was to be interpreted as a sufficient
condition of that kind of processes. Obviously, such a simple model can not by itself
explain the dynamics of segregation, and it is not our intention to suggest otherwise.
Nevertheless, we should note that its simplicity is really a virtue because it suggests a
general type of explanation in the social sciences, which may be developed on the basis of
the construction of models that have more and more empirical plausibility. This type of
models is being proposed, and Sugarscape is
the most ambitious project. It is important to notice that the general philosophy of these
models is almost completely present in Schelling's Model, and therefore it may be a
powerful heuristic tool that enables us to get common conclusions in any agent-based
modelling. In fact, as it has been suggested on this paper, the computational model
presented in the previous Sections exhibits generic properties. In the following, our aim
will be to guide the formulation of substantive hypothesis about the reality using the
What are the sufficient conditions to the existence of segregation
phenomena, racial, urbanistic or social ones? If such a question was addressed to someone
that effectively belongs to a segregated group, probably the answer would be of this kind:
I prefer to be in a neighbourhood where my group is in majority.
Of course, this rule should be applied to everyone that wants to be in a same group.
Below, we will analyse this type of answer, but we notice that the answer to this question
obtained by an interpretation of Schelling's Model is not the same.
Recall that the law defining the evolution of the model is the following.
Each individual counts the number of neighbours that have the same colour. If that number
is less or equal to three (ie. more than 62.5% are of different colour), he moves in a
random direction, otherwise he does not move.
We will now give an answer to the question formulated in Section 3, and
recalled in the beginning of this section. This model states that even if the individuals
have a tolerant behaviour in their neighbourhood, segregation phenomena occur. In fact,
starting with any random distribution of individuals, we already saw what is the result of
the consecutive application of that law. For instance, one of the possible results was
presented in Fig.3.1C):
The iteration led the system to an invariant final configuration, a fixed point. That
configuration shows clearly a segregation situation: groups with red elements clearly
separated by groups of white elements. Notice that, following the statement of the law,
each individual does not mind to be in a minority situation, one can say that each one of
them is non-segregationist, however the final invariant result of such a behaviour is a
segregation configuration. If one thinks that the individuals follow such a rule one may
conclude that they cause the global state, locally non intentional, of complete
One can go even further, and we suggest the reader to execute some iterations of this simulation, where the
law was changed, increasing the criteria of the number of neighbours from 3 to 4. This
means that each individual demands to be in majority in its neighbourhood (i.e. 5 in 9,
including himself), otherwise he will move. This is the situation stated above; when we
supposed that the answer would be that each member should be in a neighbourhood with a
majority group. Notice that this is behaviour is not completely intolerant: each member
admits the presence in its neighbourhood of members from another group. However, after a
considerable number of iterations it is easy to conclude that the evolution of the system
will not produce a stable configuration! The majority rule does not usually produce stable
configurations... there is a generalized unhappiness! In fact, even with the more tolerant
rule of the three neighbours, segregation configurations were already present, although
each individual was not segregationist. This means that each one of the elements has no
representation of the large-scale consequences that their local action is producing. In
this example, the global state is even opposed to the premises of each individual.
We emphasize that we do not state that Schelling's model is a necessary
and sufficient condition capable of explaining the empirical reality of segregation. The
emergence of the state of segregation in this model is not even generated by social
behaviour constraints, it is not only a consequence of the local law but also of the
topological/geometrical features that characterise the connections between the elements.
This topological constraint exists in any model in which the interactions occur and is
determined by the number of local connections that were established, defining a
neighbourhood. Thus, Schellings model has generic properties that are shared by any
dynamic based in local action principles. We can render this idea more plausible if we
point out some of the generic situations of Schellings model.
Suppose that a model similar to Schelling's model (perhaps more
sophisticated, with other parameters - that is not important here) is a model of the
empirical reality, i.e. the individuals are really myopes and they only act based in some
local rules. In the model this situation was analysed in Section 2 with the example in Fig.2.6. Analysing
the system locally, one could not predict the evolution, due to the interdependence of all
elements. The real agents, that the model may represent, they also do not have the
perception of all interactions, and since they are myope they only represent isolated
parts of the system. Therefore, one concludes that a real agent does not have the
capability of anticipation of the local actions produced, and one can only predict the
future configuration after an evaluation of the whole. Schelling's model reveals its
heuristic capability of thinking some phenomena. It points to one fundamental hypothesis
that each individual is in fact myope, has an extremely bounded rationality, and does not
have the perception of the global state to which he is contributing in a non-intentional
Such a simple model can also suggest other heuristic lines of thought. In
fact, it raises epistemological questions such as the need of rethinking a fundamental
category of thought, the category of causation.
We now formulate the question of segregation differently. What was the
cause of the segregation feature exhibited in Schelling's model? It is certain that it is
not any of the four great causes mentioned by Aristotle. If we leave aside the formal
cause, with an interpretation that is not always very clear, we are left with the
Material causation that from which one thing
comes and that makes it persist, i.e. the material from which one thing is made.
Efficient causation the primary cause of
rest and change, i.e. the thing or agent responsible by the change in the form of a
certain body, as in the case of the sculptor and the statue. This definition does not
necessarily mean that the change must occur by direct physical contact.
Final causation the purpose for which one thing
is made, as when a knife is used to cut some desired food.
These causes may be resumed in the conception of an individual
(considered almost in isolation) that is the cause that modifies some object (efficient
causation), eventually as a mean to an end (final causation). Notice that, among the three
causes, the efficient causation is the primordial type of causation, and it is a kind of local
So, again, which type of causation is present in Schellings model?
Which one is responsible by the final state of segregation that can be mathematically
described as a fixed point? In a certain sense, at least in the Aristotelian concept of
causation, that cause does not exist. The real cause is a distributed property that
it is not present in any isolated part of the system. We will name it: distributed causation. It is no more than the result of the
multiple non-linear interactions between the elements of the system. As it was shown in
the simulations, it is not present in any individual taken separately, and therefore it
can not be represented or identified by any of the individuals.
In such hypothesis, what is the condition of the myopic individuals that
we are? Our condition is a given condition, with unknown origin and historic
evolution. The past and future are both beyond our horizon of accessibility. We emphasize
again that these statements do not contain any metaphysics of history, and their
intelligibility is instantly rendered clear by the simple models studied here. In fact,
they are the leitmotiv of a very precise research project in sociology, as
documented by the great work of Michael
New ideas may now arise. The individuals live in realities that are
given, and the evolution, the dynamic, is beyond accessibility. None of the individuals
has the perception of the whole distributed causation. However, it is known that
intelligibility is always searched. How? A detailed answer is not possible here, but we
suggest that a certain sort of intelligibility is obtained when the distributed causation
is replaced by rational causes of control, in particular by the efficient
causation and by the final causation. Obviously, we are not saying that this is made in a
conscious way. We have said that the individuals can not represent a distributed
causation, because it is beyond the horizon of accessibility. As a matter of fact, it is
not really a replacement but rather the quest for explanations using a type of causation
that is accessible to an individual. The form of the single causes: some entity that is
the absolutely first and radiant center responsible by the dynamic (following the
efficient causation), which drives towards a certain final result (following the final
causation). In other words, we are arguing that in one hand the linear causation, as an
independent sum of the parts, is intelligible. On the other hand, the non-linear causation
can not be in the perception of an individual tangled in a web of non-linear
We could give many examples supporting the last statement, with the help
of different branches of knowledge such as the theory of financial markets (see http://www.unifr.ch/econophysics/) or the
mythology. Nevertheless, we can present the main idea using again Schellings model.
In the beginning of this Section we raised the question about the conditions thought to be
sufficient for a segregation process. Let us suppose again that a local and distributed causation led to a state of segregation.
To the ones attached to a cohesive group, segregation appears as given. We could ask a
What do you think it was in the origin of this segregation?. Surely the
reader will agree that it will be implausible an answer of the type:It seems to
have been generated by the accumulation of many interactions of individuals, all
It is more plausible to consider other answers: our community as decided so,
or each one of us has decided so because we dont like them around and they
dont like us. Notice that the last answer assumes a majority
rule which in fact was not present. We also point out here the use of the pronouns our,
In the first case this shows the replacement of a local and distributed causation (which
no one can experience) by a single and global cause (the colectivity, in this
case) which completely satisfies the intelligibility, because that cause works in the same
way that a local cause works, i.e. as an efficient causation. In the second case the
answer shows the replacement of non-linear interactions by an explanation based on linear
ones, using the independent sum of segregation behaviours.
Thus, two processes of searching intelligibility are:
(i) The use of a single global cause.
This is in the origin of myths.
(ii) The use of linear independent local causes.
This is a classical scientific approach.
In fact, when a non-intelligible cause is replaced by a single global
cause, this satisfies each individual with an intelligible explanation. Explaining the
segregation by a community decision (a single and efficent act of will) gives the idea of
a central and coordinated order obeyed by everyone. This could be a plausible
justification to the final invariant state, but it could hardly explain the whole
evolution. However, since the individuals would not have access to that detailed
information, the use of a single global cause could be kept as an explanation to the final
On the other hand, in the second case, the local law produces a global
state from which one may infer a segregational behaviour. The projection of this global
behaviour to a local law simplifies the model using an intelligible explanation. In
Schelling's model this would correspond to consider a different number of admissible
neighbours, in such a way that the majority rule would hold. Curiously, such a deduction
would lead to instable situations, as it was seen in the 4-neighbour law simulation.
The non-intelligible cause is replaced by the sum of independent local causes, simplifying
the rule, assuming that the final result is in fact the result of a similar local law. In
the case of Schelling's model, this would not even justify the stability on the border,
where some individuals are in minority, but this could be seen as an exception to the
rule, for instance, saying that some of the individuals (the ones that are on the border)
are more tolerant than others.
(i) Why do we say that the first one is in the origin of myths?
In fact, arguing on the existence of one unique central cause is in the genesis of most of
the myths. The evolution to the final state is no longer important, the result is what
matters. There is also the possibility that a class of final states can be attributed to a
cause, and other classes to other causes. This induces the generation of several distinct
myths. However it is perfectly clear that we can easily substitute the several different
causes by a single cause.
(ii) Why do we say that the second one is a scientific approach?
From the observation of global states a local law may be inferred. To check the validity
of the law, one planifies an experiment. Isolating a part of a system, controlling it,
seeing how it works, corresponds to an experiment that it is intelligible.We then argue
that the same experiment can be repeated with similar results. If we then assume that the
observed evolution is the result of the sum of all these parts, one may get an approximate
model of what happens in more complex situations. This model is intelligible, but it only
constitutes a good approximation in some favorable conditions. Usually, in transition
situations these explanations (given by the sum of independent actions) have some serious
exceptions. These exceptions are often at the origin of new breakthroughs on science