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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,

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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 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

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: 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 = 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?

24 SISTEMA DE ANÁLISIS DE BALANCES IBÉRICOS SABI es el único sistema del mercado de análisis financiero, donde podrá encontrar más de empresas españolas y portuguesas con todos sus datos financieros, con un exclusivo software de búsqueda y tratamiento de datos SABI es una útil herramienta que le permite realizar múltiples funciones como búsquedas multicriterio, análisis empresariales, comparativas entre empresas o de grupos empresariales, rankings, análisis de concentración, segmentaciones, estudios sectoriales etc. Contenido de S.A.B.I.: Cuentas anuales de más de empresas con históricos desde Razón Social. Dirección, localidad, provincia. Código Nif. Teléfono. Códigos de actividad (IAE,CNAE93, CSO). Descripción de actividad. Forma Jurídica Fecha de Constitución. Número de empleados. Consejo de Administración. Ratios de Informa. Ratios Europeos. Tasas de Variación. Auditores. Bancos. Cotización en Bolsa. Accionistas, Participaciones(%). Cuentas Consolidadas.

25

26 Networks / Pajek Program for Large Network Analysis


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