Small worlds, mazes and random walks

Transcription

1 EUROPHYSICS LETTERS 1 July 2003 Europhys. Lett., 63 (1), pp (2003) Small worlds, mazes and random walks B. Luque 1 and O. Miramontes 2 1 Departamento Matemática Aplicada y Estadística Escuela Superior de Ingenieros Aeronáuticos, Universidad Politécnica de Madrid Plaza Cardenal Cisneros 3, Madrid 28040, Spain 2 Departamento de Sistemas Complejos, Instituto de Física Universidad Nacional Autónoma de México, Cd Universitaria México DF, México (received 23 December 2002; accepted in final form 2 May 2003) PACS Fb Random walks and Levy flights. PACS r Probability theory, stochastic processes, and statistics. PACS Uu Applications of Monte Carlo methods. Abstract. A parametrized family of random walks whose trajectories are easily identified as graphs is presented. This construction shows small-world like behavior but, interestingly, a power law emerges between the minimal distance L and the number of nodes N of the graph instead of the typical logarithmic scaling. We explain this peculiar finding in the light of the well-known scaling relationships in Random Walk Theory. Our model establishes a link between Complex Networks and Self-Avoiding Random Walks, a useful theoretical framework in polymer science. The Watts-Strogatz model of Small World graphs [1] efficiently interpolates between regular and random graphs thanks to a small number pn of shortcuts (long-range connections) which are superimposed on a regular graph formed by N nodes. In random graphs, the mean minimal distance or diameter between all pairs of nodes in the system scales logarithmically with the system size the so-called Small World (SW) effect while scales linearly in regular graphs. Much attention has been devoted recently to the topological properties of the Watts-Strogatz model and scale-free models (named complex networks in general) and to the effects that a network connectivity may have on the properties of dynamical systems [1 4]. The aim of this work is to show features common to complex networks and Self-Avoiding Random Walks (SAW). There are a number of papers relating Random Walks (RW) and SW phenomenon [5 14]. In [7, 8] the authors examine RW on SW networks, in particular the probability of a random walker of being at the original site at a later time. Their interest stems from the motion of excitons over polymer chains, where steps between spatially close sites can connect regions far apart along the chemical backbone. However, this model follows a standard SW network building [15]. Later, the same authors [9], considering self-avoiding constraints, assume that the probability that two sites far apart along the backbone come close together in space is approximately an inverse power law of their mutual distance. In other studies, RW have c EDP Sciences

2 B. Luque et al.: Small worlds, mazes and random walks Fig. 1 An illustrative example of shortcut by loop in the RW path or maze: (a) The path of N = 21 steps traced by a RW, can be interpreted as a maze of length N = 21. (b) To solve the maze is: to travel starting at 1, and ending at 21. One non-optimal solution is a travel of length 21. (c) At step 18 the path has a selfintersection with step 8, a loop (d) In order to optimally solve the maze, this loop should be avoided. Then, the loop acts as a shortcut in the graph version of the maze: the node 8 is connected with the node 9 and 19. (e) To efficiently solve the maze we use the minimal distance L = 10 between nodes 1 and 21: (f) Then, the length of the maze is N = 21, but using the shortcut, it is solved in L =10. been employed as dynamical nodes to study dynamical SW effects [13, 14]. Furthermore and in another context, RW on the family of SW networks have been addressed, where RW correspond to random spread of information over the network [10]. It has been demonstrated that the average access time between nodes for a SW geometry shows a crossover from regular to random behavior with increasing distance from the starting point of the RW. Average access time is important in any Markov process and is very relevant for the exploration and navigation of the WWW [11] that, as a scale-free network, also shows the SW effect. In this paper we study the properties of two dimensional complex mazes from the point of view of the SW theory. For this, it is important to explore new methods (such as the use of the properties of excitable media [16]) to find minimum-length paths in complex labyrinths. Navigational methods for solving mazes are widely applied in computer sciences for searching through data structures and the so-called depth-first search method is an example [17]. In this work, we generate a maze by using a non-reversing RW in two dimensions [18]. By non-reversing we mean that, at each step, the walker does not jump back to its latest position. Self-intersections of the RW are not avoided and so looping is permitted. Each site reached by the RW at step i is a node labeled by i. The nodes are connected in the step sequence, i.e.: i i + 1. But if a loop exists, for example i-step intersect with j-step, then we connect i j + 1. Therefore, the loops act as shortcuts in the graph. In fig. 1 we show a path constructed from a non-reversing RW. The path is the maze and solving it consists in discovering the minimum number of steps to reach both ends. Thus jumping the loops is the quickest way of reaching the exit. To generate a specific maze we fix the number N of steps of the non-reversing RW and introduce a probability p [0, 2/3]. At each step, the RW changes its direction with probability p (toward the right or the left with probability p/2), and follows forward with probability 1 p. In this manner we can construct a variety of mazes. From p = 0, that produces linear trajectories, to p = 2/3, that gives intricate trajectories with equal probability to continue straight, turn right or turn left. Obviously, the number of self-interactions grows with p, and so does the number of loops.

3 10 EUROPHYSICS LETTERS p=0.001 p=0.01 p= L/N N=512 N=1024 N=2048 N= p Fig. 2 Upper figures: three examples of mazes generated by RWs with p =0.001, p =0.01, and p =0.1 with a proper choice of the scale. Main figure: the effect of growth p on the minimal distance L normalized by N in the path graph for several sizes N and log-linear axes. For all simulations in this article, 31 points separated logarithmically between p = and p = 2/3 were considered. In all cases each point represents the mean of 1000 numerical experiments. We notice a size-effect in L similar to the one in the classic SW model that predicts a transition at p 0andN. In the upper insets of fig. 2 we show three non-reversing RW (properly rescaled) of N = 1024 steps. A very small value of p such as p =0.001 can produce a maze without any loops. A value of p =0.01 generates mazes with a moderate number of loops. Finally, a value of p = 0.1 generates intricate mazes. Because increasing p implies an increase of the number of loops or, equivalently, shortcuts as described above, we expect SW behavior in the model. Here, the SW effect means that the maze can be walked from the start to the end within much fewer steps (L) than the total number of steps N. The main plot of fig. 2 illustrates the SW effect. If the number of path-intersections are regarded as shortcuts, then the minimal distance L is dramatically reduced with small increments of p. Similarly to the crossover length in SW models, it is possible to introduce in the present context a persistence length N, defined as the mean number of steps needed for producing the first loop or shortcut. In fig. 3, a law N p 1, is observed, which can be explained by the average number of steps that a RW must perform before turning for the first time. In fig. 4, we observe that the following scaling relationship is satisfied: L p 1 F 1 (pn). (1) If p 1 is proportional to the mean number of RW steps necessary for producing one loop, F 1 (pn) can be interpreted as the mean number of loops in the system. The scaling function F 1 (x) behaves linearly for values of x<1. That is, while pn the average number of turns is less than one, there are few shortcuts and L N. When x>1 the presence of shortcuts

4 B. Luque et al.: Small worlds, mazes and random walks N = 512 N = 1024 N = 2048 N = 2048 N* τ = p Fig. 3 Log-log graph showing the scaling of the number of RW steps N that it takes to reach the first shortcut or the first loop as a function of p. The line with slope 1 is a guide for the eye. The linear fitting of the scaling region gives τ = ± has an impact and the distance L is reduced, following a power law with an exponent value α equal to 2/3. We display in fig. 5 a collapse plot of the scaling function F 2 (x) defined as L NF 2 (pn). (2) F 2 (x) can be interpreted as the reduction factor of the total length as a function of the size N and the turning rate p. Again, for values of x<1, L and N take the same value. However, as N or p increase and as soon as pn > 1, the ratio L/N decreases as a power law with an exponent value β equal to 1/3. Both results confirm that L(N,p) N, (3) Lp α = 1 α = 2/3 N = 512 N = 1024 N = 2048 N = pn Fig. 4 L/N 10-1 Fig. 5 N = 512 N = 1024 N = 2048 N = 4096 β = 1/ pn Fig. 4 Log-log graph showing the behavior of the scaling function F 1. The lines with slope 1 and slope 2/3 are a guide for the eye. The linear fitting on the scaling regions gives α =1.004 ± and α =0.668 ± for pn < 1andpN > 1, respectively. Fig. 5 Collapse of the scaling function F 2. The line with slope 1/3 isaguidefortheeye. The linear fitting of the scaling region Np > 1givesβ = ±

5 12 EUROPHYSICS LETTERS γ = 1/2 ν = 3/4 pr 10 0 pr 10 0 N = 512 N = 1024 N = 2048 N = pn Fig pl Fig. 7 N = 512 N = 1024 N = 2048 N = 4096 Fig. 6 Scaling relation between the end-to-end distance R and N, the number of the RW steps rescaled both by p. The line with slope 1/2 is a guide for the eye. The scaling exponent obtained by fitting of the scaling region is γ =0.499 ± Fig. 7 Scaling relation between the end-to-end distance R and L scaled both by p. The line with slope 3/4 is a guide for the eye. Scaling exponent obtained of the scaling region by fitting is ν =0.75±0.01. for no loops or pn < 1. On the other hand L(N,p) p 1/3 N 2/3, (4) for pn > 1, when loops appears. Let us recall that in the classic SW model, the mean distance scales as log N, while in the present system, it scales as a power law with exponent 2/3. 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 [19]. 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. Thus pr (pn) 1/2. (5) By numerical experiments we measured R for several values of N and p in our model. In fig. 6 it is shown how pr scales with pn as a power law with exponent 1/2 as is predicted by eq. (5). This result shows that the maze model acts as a typical RW model of N steps in two dimensions, where N and R are scaled by p. Self-Avoiding Random Walks (SAW) are RW where self-intersections are avoided. In our model once the RW trajectory is 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 [19]. 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, (6) as confirmed in fig. 7. Combining eqs. (5) and (6), we recover eq. (4). The SW effect in our two-dimensional model as a power law scaling between N and L, with exponent 2/3 for values pn, is a direct consequence of a well-known exact scaling relation in classical RW and SAW models.

6 B. Luque et al.: Small worlds, mazes and random walks 13 RW were proposed 60 years ago by Kuhn for describing polymer chains [19]. Then Flory assumed that the polymer chains cannot self-intersect and proposed SAW as a more realistic model. We think that our model can have interesting applications in polymer sciences or excitonic energy problems because it links SAW and SW effect in graphs. For example, in order to solve the computational problems of SAW, other RW variants have been introduced. One is the Loop-Erased Random Walk model (LERW) discussed first by Lawler [20,21] as an approximation of a SAW more tractable analytically. In a LERW a simple RW is used, but the final non-intersecting path is obtained by erasing each loop as soon as it is formed. LERW is a model clearly related to the one introduced here and has been extremely fruitful [22]: it has been applied to problems of Laplacian SAW, q-potts model, spanning trees, Abelian sand-piles and Self-Organized Critical systems. Now, we can relate LERW to complex networks as well. We would like to thank U. Bastolla, D. Boyer, J. Erler and two anonymous referees for their valuable opinions. OMhas been supported by CONACYT (32453-E and G32723-E) and DGAPA-UNAM(IN ) and BL by CICYT BFM REFERENCES [1] Watts D. J. and Strogatz S. H., Nature, 393 (1998) 440. [2] Albert R. and Barabási A.-L., Rev. Mod. Phys., 74 (2002) 47. [3] Strogatz S. H., Nature, 410 (2001) 268. [4] Dorogovtsev S. N. and Mendes J. F. F., Adv. Phys., 51 (2002) [5] Jasch F. and Blumen A., Phys. Rev. E, 63 (2001) [6] Lahtinen J., Kertész J. and Kaski K., Phys. Rev. E, 64 (2001) [7] Jespersen S., Sokolov I. M. and Blumen A., Phys. Rev. E, 62 (2000) [8] Jespersen S., Sokolov I. M. and Blumen A., J. Chem. Phys., 113 (2000) [9] Jespersen S. and Blumen A., Phys. Rev. E, 62 (2000) 5. [10] Pandit I. M. and Amritkar R. E., Phys. Rev. E, 63 (2001) [11] Tadic B., Eur. Phys. J. B, 23 (2001) 221. [12] Almaas E., Kulkarni R. V. and Stroud D., Phys. Rev. Lett., 88 (2002) [13] Manrubia S. C., Delgado J. and Luque B., Europhys. Lett., 54 (2001) 1. [14] Miramontes O. and Luque B., Physica D, 168 (2002) 379. [15] Moore C. and Newman M. E. J., Phys. Rev. E, 61 (2000) [16] Steinbock O., Tóth A. and Showalter A., Science, 267 (1995) 868. [17] Hofstadter D. R., Gödel, Escher, Bach: An Eternal Golden Braid (Basic Books, New York) [18] Pickover C. A., Random and Markov Approach to Provide a Mechanism for Generating Mazes in Computers, Pattern, Chaos and Beauty (Alan Sutton Publishing, UK) [19] de Gennes P. G., Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca) [20] Lawler G. F., J. Phys. A, 20 (1987) [21] Lawler G. F., Intersections of Random Walks (Birkhauser, Boston) [22] Dhar D., Physica A, 263 (1999) 4.