Do you like mazes?. To establish a relationship between the Small-World behavior found in complex networks and a family of Random Walks trajectories using,

Presentación del tema: "Do you like mazes?. To establish a relationship between the Small-World behavior found in complex networks and a family of Random Walks trajectories using,"— Transcripción de la presentación:

1 Do you like mazes?

2 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.

3 This is a simple maze traced by one of this random walks: We can use non-reversal random walks for generating mazes.

4 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 N=20 and 21 nodes.

5 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.

6 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.

7 To solve efficiently the maze we use the minimal distance, the chemical distance L=10 between nodes 1 and 21: 1-2- 3-4-5-6-7-8-19-20-21. Then, the length of the maze is N=20, but using the short-cut, we can solve it in L=10.

8 Non-reversal biased Random Walk 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. 1-p p/2

9 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 N=1024 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.

10 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.

11 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

12 We can try to collapse the numerical experiments by the scaling form: L N*F 1 (N/N*) p -1 F 1 (Np)

13 Or by the scaling form: L N F 2 (N/N*) N F 2 (pN)

14 Then: For Np << 1, L(N,p) N For Np >> 1, L(N,p) p -1/3 N 2/3 For classical SW we expected L scaling as Log(N), in this system we have L scaling as a power-law : N 2/3.

15 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 N 1/2

16 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 N R

17 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 L 3/4 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

18 pR (pN) 1/2 pR (pL) 3/4 L(N,p) p -1/3 N 2/3 R L

19 Diccionarios Bartolillo: 1. m. Pastel pequeño en forma casi triangular, relleno de crema o carne. Triangular: 1. adj. De figura de triángulo o semejante a él.

20 Bartolillo Pastel pequeño forma casitriangular relleno crema carne Descartando en de o y a: figura triángulo semejante él ¡¡¡¡Muchas complicaciones para construirlo!!!!

21 Pero mucha información interesante: (1) ¿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?

22 (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. ComplejidadTamaño de los loops

23 ¿Cómo es la distribución de tamaños de loops? ¿Es diferente en distintos idiomas?

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26 Networks / Pajek Program for Large Network Analysis http://vlado.fmf.uni-lj.si/pub/networks/pajek/default.htm