Diapositiva 1

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Do you like mazes?
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To establish a relationship between the
Small-World behavior found in complex networks
and a family of Random Walks trajectories using,
as a linking bridge, a maze iconography.
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We can use non-reversal random walks for
generating mazes.
This is a simple maze traced by one of this
random walks
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Up we can see an illustrative example of
short-cut by loop in the random walk path or
maze. The path 1-2-3-4-5-6-7-8-9-10-11-12-13-14
-15-16-17-18-19-20-21 of 20 steps traced by a RW,
can be interpreted as a maze of length N20 and
21 nodes.
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To solve the maze is to travel starting at 1,
and ending at 21. One non-optimal solution is a
travel of length 20. At step 18 the path has a
self-intersection with step 8, a loop
8-9-10-11-12-13-14-15-16-17-18.
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We can avoid this loop to solve optimally the
maze. Then, the loop acts as a short-cut in the
graph version of the maze the node 8 is
connected with the node 9 and 19.
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To solve efficiently the maze we use the minimal
distance, the chemical distance L10 between
nodes 1 and 21 1-2-3-4-5-6-7-8-19-20-21. Then,
the length of the maze is N20, but using the
short-cut, we can solve it in L10.
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Non-reversal biased Random Walk
1-p
p/2
p/2
In each step the RW will vary his direction
with probability p at right with probability
p/2 or at left with the same probability. And
with probability 1-p the RW no turn. In this
manner we can construct a variety of mazes. From
p 0 that produce linear trajectories to p 2/3
with intricate trajectories with equal
probability to continue right, turn right or
turn left.
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To generate a specific maze we fix the number N
of the RW steps and the probability p. Obviously
the number of self-interactions grows with p.
Each time we have a self-interaction we have a
loop. In the figure we shown tree cases (zoomed
properly) for a RWs with N1024 steps. A very
little value of p as p 0.001 produce a maze
without loops. A value as p 0.01 generate a
mazes with a moderate number of loops. And a
value as p 0.1 generate a very intricate maze.


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Because grow in p imply grow in the number of
loops or equivalent short-cuts, we expect
Small-World (SW) behavior in the model. In the
upper figure we can see the effect of grow p onto
chemical distance end-to-end L normalized by N in
the path graph for several sizes N. Each point is
the mean of 1000 numerical experiments. We point
out two differences with standard SW model Here
the links are direct, and the short-cuts are
created by the system dynamics.
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We have a persistence length N (the number of
random walk steps needed to the first loop is
produced). As we can expected N ? p-1
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We can try to collapse the numerical experiments
by the scaling form
L ? NF1(N/N) ? p-1F1(Np)
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Or by the scaling form
L ? N F2(N/N) ? N F2(pN)
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Then For Np ltlt 1, L(N,p)? N For Np gtgt 1,
L(N,p)? p-1/3N2/3 For classical SW we expected
L scaling as Log(N), in this system we have L
scaling as a power-law N2/3.

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Why the model presents this SW effect in a power
law form? Usually, for a two-dimensional RW,
the end-to-end distance R is defined as the mean
Euclidean distance separating both ends of the RW
of length N. In two dimensions, for a classical
RW, R scales with N as R ? N1/2
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This fact holds also for a classical non-reversal
RW that corresponds to our model when p 2/3. A
variation of p is equivalent to a change of scale
of the RW trajectory. We expect that the above
scaling relationship between R and N will hold
when re-scaling both variables by p pR
?(pN)1/2
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Self-Avoiding Random Walks (SAW) are RW where
self-intersections are avoided. In our model once
the RW trajectories finished, deleting the loops
produces a SAW of length L. It is known that SAWs
of L steps in two dimensions obey the scaling
relationship
R ? L3/4
This implies, after rescaling R and L properly as
pR and pL, that we can expect the following
scaling relation to hold here
pR ? (pL)3/4
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pR ? (pN)1/2 pR ? (pL)3/4
L(N,p)? p-1/3N2/3
R
L
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Diccionarios

• Bartolillo
• m. Pastel pequeño en forma casi triangular,
• relleno de crema o carne.

Triangular 1. adj. De figura de triángulo o
semejante a él.
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Descartando en de o y a
Pastel
carne
Bartolillo
pequeño
crema
forma
relleno
casi
triangular
figura
él
triángulo
semejante
Muchas complicaciones para construirlo!!!!
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• Pero mucha información interesante
• Es el grafo conexo? O el diccionario
• se puede editar a trozos auto-contenidos?
• (2) Las palabras recién admitidas, muchos
• barbarismos y palabras jóvenes serán
• cul-de-sacs
• Cuántas habrá? Podemos correlacionar
• la antigüedad de las palabras con su
  conectividad?

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(3) Toda definición del diccionario es circular.
Partiendo de una palabra, tarde o temprano
volveremos a ella. Entidad Lo que constituye
la esencia o la forma de una cosa. Cosa Todo lo
que tiene entidad, ya sea corporal o espiritual,
natural o artificial, real o abstracta.
Complejidad
Tamaño de los loops
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Cómo es la distribución de tamaños de loops? Es
diferente en distintos idiomas?
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Networks / Pajek    Program for Large Network
Analysis
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ult.htm