Crossover between the extended and localized regimes in stochastic partially self-avoiding walks
in one-dimensional disordered systems
Juliana Militão Berbert
*
Instituto de Física Teórica (IFT), Universidade Estadual Paulista (UNESP), Caixa Postal 70532-2, 01156-970 São Paulo, SP, Brazil
Alexandre Souto Martinez†
Departamento de Física e Matemática (DFM), Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP), Universidade de São Paulo (USP) and National Institute of Science and Technology in Complex Systems (LNCT-SC),
Avenida Bandeirantes 3900, 14040-901 Ribeirão Preto, SP, Brazil
共Received 1 February 2010; revised manuscript received 25 March 2010; published 18 June 2010兲
ConsiderNsites randomly and uniformly distributed in ad-dimensional hypercube. A walker explores this disordered medium going to the nearest site, which has not been visited in the last 共memory兲 steps. The walker trajectory is composed of a transient part and a periodic part 共cycle兲. For one-dimensional systems, travelers can or cannot explore all available space, giving rise to a crossover between localized and extended regimes at the critical memory1= log2N. The deterministic rule can be softened to consider more realistic situations with the inclusion of a stochastic parameterT 共temperature兲. In this case, the walker movement is driven by a probability density function parameterized byTand a cost function. The cost function increases as the distance between two sites and favors hops to closer sites. As the temperature increases, the walker can escape from cycles that are reminiscent of the deterministic nature and extend the exploration. Here, we report an analytical model and numerical studies of the influence of the temperature and the critical memory in the exploration of one-dimensional disordered systems.
DOI:10.1103/PhysRevE.81.061127 PACS number共s兲: 05.40.Fb, 05.60.⫺k, 05.90.⫹m, 05.70.Fh
I. INTRODUCTION
Stochastic optimization algorithms have been proven to be suitable when difficult共nonpolynomial兲problems are ad-dressed. These algorithms have mechanisms of that explore locally and globally the parameter space regions. The local exploration drives the system rapidly to local minima and the global one takes the system away from these minima. Good algorithms are able to effectively combine the amount of local and global exploration. Nevertheless, these algorithms suffer froma prioriparametrization such as initial tempera-ture in simulated annealing algorithm 关1兴 or mutation taxes in the genetic algorithm关2兴. Further, full system information must be furnished to the algorithm before optimization starts. Having optimization algorithms in view, here we consider the situation where the information about the system to be optimized is supplied as it evolves. Also, we consider the worst case hypothesis: no correlation in the parameter space. In this case, one can avoid the initial parametrization and the best results are possibly obtained along a line between two regimes: a nonergodic one, which explores locally the pa-rameter space, and the ergodic one, which drives the system away from the local minima.
A. Exploring disordered media: Deterministic and stochastic tourist walks
Consider a walker exploring a disordered medium withN sites representing, for example, a landscape with feeding
sites 关3兴, such as flowers for insects, trees for birds, water fountains for animals in general etc.关4–10兴. The walker re-members a visited site关11兴during its regeneration time共time for it to become attractive to the walker again兲. After this regeneration time, he/she can, again, revisit this site. If the walker does not know the position of all sites, or, if he/she knows the sites position but have a restrict movement ability, what kind of exploration strategy can emerge from this ex-ploration problem? The tourist walk关12,13兴is an exploration strategy that addresses this question, and it can be either deterministic关14兴or stochastic 关15兴.
Here, the disordered medium is defined by drawing uni-form and randomly each site coordinate along unitary edges of a d-dimensional hypercube, xi共k兲, i= 1 , 2 , . . . ,N and k = 1 , 2 , . . . ,d. The tourist walk movements are based on the distance between sites. One considers the normalized Euclid-ean distance between sites, Dj,i=N兵兺k=1
d 关
xi共k兲−x共jk兲兴2其1/2 , to perform the walk. These normalized distances keep the site density constant, for possible comparison between systems with different values ofNand/ord.
In the deterministic tourist walk共DTW兲, at each discrete time step, the walker follows the rule of hopping to the near-est site that has not been visited in the last steps. The resulting trajectory is partially self-avoiding, limited by the memory. The memory can be understood as the character-istic regeneration time, so that the walker is aware of the nearest neighbors, besides its own position to make move-ment decisions. Although the rule is simple, nontrivial trajec-tories emerge from this strategy关16,17兴. They present a tran-sient part sizedt, where the walker explores共not necessary兲 new sites; and a final part, composed by a periodic cycle sized pⱖ+ 1, where the walker is trapped in a region vis-iting only the same sites. The transient time t and the cycle *[email protected]
periodpdepends on the medium configuration and the initial condition. A review of the statistical studies of t and p are detailed in Ref.关14兴. From those statistical studies, we point out that for = 0 and 1, the walker is rapidly trapped in cycles and presents a localized exploration behavior; and for ⬎1, the distribution of p presents a whole spectrum of values with possible power-law decay关12,13,16–18兴. Appli-cations of these walks are given in Refs.关19–21兴.
Therefore, to visit all sites, a walker needs a sufficient memoryto escape from attractors and extend the explora-tion. In one-dimensional systems, a crossover between local-ized 共trapping in cycles兲and extended 共visiting many sites兲 exploration walker behavior occurs at a critical memory
关16,22兴: 1= log2N. The critical memory is much smaller than the system size and with only local information a walker can present good exploration of the whole system.
The stochastic tourist walk共STW兲 关3,23兴is a generaliza-tion of the DTW关12兴. In this case, the walker movements are defined by a growing cost function E共Dj,i兲, by a probability density function 共pdf兲dependent on the inverse of a formal temperature T⬎0 共the stochastic parameter is = 1/T兲 and on the walker memoryⱖ0. The temperature characterizes the walker exploring ability, namely, the exploration region. For low temperatures, this pdf favors hops to nearest sites. For high temperatures, the walker is able to hop to distant sites. The memory defines an avoidance window with the last visited sites. The pdf, that defines the probability of departing from site Si to site Sj, is written as: Pj,i
共,兲
=e−E共Dj,i兲/Zi共,兲, whereZi共,兲=兺k=1
N
⬘
e−E共Dk,i兲 is the normal-ization coefficient. We call attention to the summation 兺⬘
, which excludes the forbidden sites from the avoidance win-dow.In Ref. 关3兴, the dynamics of STW for walkers, with null memory 共= 0兲, exploring disordered d-dimensional systems is described. In this case, the order parameter is the mean residence time 具tr共兲典 and for a cost function as E共Dj,i兲=Dj,i
d
, the 具tr共兲典and the tail of the nearest distances distribution compete, and a glass transition 关24–29兴 occurs in a finite value: d=Ad, where Ad is the volume of a
d-dimensional hypersphere with unitary radius. It means that the walker can explore the medium while the stochastic pa-rameter⬍Ad. After=d=Adthe walker is trapped in
re-gions for long time and the system enters in an out-of-equilibrium regime and an aging effect emerges. For one-dimensional systems, A1= 2, it means that the critical temperature is T0= 1/2. A detailed review is found in Ref. 关15兴.
B. Combining stochasticity and memory
The results for DTW and STW lead us to believe that the tourist walk problem may present two distinct regimes: a nonergodic one, where the walker is trapped in localized exploration; and an ergodic regime, where the walker can explore the whole medium. We conjecture that to maximize the exploration efficiency it is necessary to choose a cost function that allows a glass transition. Besides, the optimum exploration performance should occur for a specific tempera-ture and memory.
Here, we examine the stochastic tourist walk with memory. We consider the influence of the stochastic param-eter and the critical memory in the exploration of one-dimensional disordered systems. As discussed below, we have found a transition diagram between localized and ex-tended regimes of exploration. Further, we have also found a relation between the system parameters, which reveals a way to optimize the exploration. The presentation is organized as follow. In Sec. II, in order to find a pdf dependent on both temperature and memory, without restrictions on the summa-tion, we develop an analytical model. In Sec.III, we imple-ment a numerical algorithm based on the developed model. In Sec. IV, the numerical results are discussed. Finally in Sec. V, the final remarks are presented.
II. ANALYTICAL MODEL
In this section, we develop an analytical formalism to describe the dynamics of STW when the temperatureT and memoryvary simultaneously. Our aims are to comprehend the STW dependence onTandand obtain an algorithm for a computational implementation.
The walker movements are defined by a pdf parameter-ized by a temperature, an avoidance window 共characterized by the walker memory兲and a cost function共growing on the distances between sites兲favoring hops to nearest sites in the low temperature limit. This formalism allows the calculation
共at each time interval兲 of the hopping probability from one site to another of a walker with memory and temperature T.
A. Formalism
Consider N sites drawn by a spatial Poisson distribution along a unitary line segment. Also, consider a walker with memory , who has hopped Np sites. First, define two
sets: 共a兲 one with all sites: T=兵S1,S2, . . . ,SN其, where the
indexes label the sites, and 共b兲 other with the forbid-den sites forming the avoidance window for step t: F共t兲 =兵S共t−共−1兲兲,S共t−共−2兲兲, . . . ,S共t兲其, where the indexes stand for the site position along the walker trajectory. Note that
F共t兲傺T. Second, it is convenient to define another set K共t兲, which reveals the correspondence between the indexes of
F共t兲 andT such that K共t兲=兵k兩Sk苸F共t兲其. Each site is repre-sented by a unitary value in the state vector of the system:
兩S1典=
冢
1 0]
0
冣
, 兩S2典=
冢
0 1]
0
冣
, . . . ,兩SN典=
冢
0 0]
1
冣
.The transposed vectors are: 具S1兩=共10. . . 0兲, …, 具SN兩 =共00. . . 1兲. It is easy to show the following properties:
具Sj兩Si典=␦j,i,兺i=1
N 具
Si兩Si典=Nand兺k=1
N 兩
Sk典具Sk兩=1, where1is the
identity matrix of sizeN⫻N.
We also define the operators: memory, cost, density and hopping probability. The memory operator M共兲共t兲=1
−兺k苸K兩Sk典具Sk兩 is represented by a diagonal N⫻N matrix,
with unitary elements, if Sj苸F共t兲⇒Mj,j
共兲共t兲
elements vanish 关Sj苸F共t兲⇒Mj,j
共兲共t兲
= 0兴. As the matrix elements are updated, at each step t, according to the se t F共t兲 evolution, this operator is time dependent. When this
operator is applied to 兩Si典,M共兲共t兲is updated. After the pro-jection on 具Sj兩, only the elementM共兲j,j共t兲 is important, since
it indicates whether the site具Sj兩can or cannot be visited. We can express the elements of the memory operator as
具Sj兩M共兲共t兲兩Si典=M共兲j,j共t兲␦j,i, where we emphasize that Mj,j
共兲共t兲
= 0, for prohibited sites, or 1, for allowed ones.
The cost operator is not time dependent. One has
具Sj兩H兩Si典=E共Dj,i兲, such that E共Dj,i兲 is a monotonic growing function of the Euclidean normalized distance Dj,i. This
op-erator is represented as e−H, so that 具Sj兩e−H兩Si典=e−E共Dj,i兲 The density operator is defined by 共,兲共t兲=M共兲共t兲e−H
. It depends on the avoidance window, through M共兲共t兲, on  = 1/Tand on the hopping cost function. Using具Sj兩M共兲共t兲兩Si典 and具Sj兩H兩Si典, we obtain具Sj兩共,兲共t兲兩Si典=M
j,j
共兲共t兲e−E共Dj,i兲. The normalization coefficient is Zi共,兲共t兲 =兺k=1
N 具
Sk兩共,兲共t兲兩Si典=兺k=1
N
Mk共兲,k共t兲e−E共Dk,i兲. It is important to notice that the summation on kexclude the forbidden sites through the avoidance window with length , because Mk共兲,k共t兲 is null in this case. This exclusion occurs naturally when we use the operatorM共兲共t兲, avoiding restrictions in the summation.
The probability operator provides the probability of a walker to hop from siteSitoSj. Therefore, we can define the
probability operator: Pi共,兲共t兲=共,兲共t兲/Zi共,兲共t兲 =M共兲共t兲e−H
/Z i
共,兲共t兲
. Thus, the hopping probability from the siteSito site Sjat time step tis:
具Sj兩Pi共,兲共t兲兩Si典= Mj,j
共兲共t兲e−E共Dj,i兲
兺
k=1N
Mk共兲,k共t兲e−E共Dk,i兲
=Pj共,兲,i 共t兲. 共1兲
Notice two properties of this operator: 兺i=1
N 具Sj兩P i
共,兲共t兲兩Si典
= 1 and 兺i,j=1
N 具
Sj兩P
i
共,兲共t兲兩Si典
=N. It is worth noting that Eq.
共1兲 binds T and in the same expression, without restric-tions. For= 0, this definition allows the walker to remain in the same site Sj=Si. In this case, the probability does not
vanish only ifMi共兲,i共t兲⫽0, which occurs for= 0.
B. Hopping probability
Consider just one trajectory with length Np hops 共steps兲,
connecting the initial and final sites. The probability of a walker, departing from Siand arriving atSjpassing through
fixed sitesSk
a,a= 1 , 2 , . . . ,共Np− 1兲, is written as
P ˜
Sj←Si 共Np兲
=Pj,k Np−1
共,兲 共Np兲
冉
兿
r=1
Np−2
Pk r+1,kr 共,兲 共r
+ 1兲
冊
Pk1,i
共,兲共 1兲. 共2兲
Notice that, due to the walker memory, this probability van-ishes if one of the sites is prohibited to be visited. Therefore, for different values, the walker finds different restrictions about the following site to be visited.
In the high temperature limit 共→0兲, the cost operator e−Hdependence on the distance between sites is negligible. Thus, the hopping probability depends only on the walker memory. Using Eq. 共2兲, the hopping probability between sites Si and Sj, passing through Np− 1 nonforbidden
inter-mediary sites 关just when Mk共兲,k共t兲= 1兴 is P˜S j←Si 共Np兲
=共N−兲−Np
, where all sites, which do not belong to the avoidance win-dow, have the same probability to be visited. In the particular case = 0 共P˜S
j←Si
共Np兲 =N−Np兲, the system is ergodic, since the
(a)
(b)
(c)
FIG. 1. Number of visited sites Nv as a function of walker memoryforN= 128 and different lengthNpsteps:共a兲Np/N= 10, 共b兲Np/N= 102and共c兲Np/N= 103. Observe thatNvincreases asT,
walker can be found at any site共the system can be found in any of its states兲 with the same probability. Further, in this case, the system is in equilibrium, since the detailed balance is satisfied on the master equation, Pj共,0兲,i =Pi共,0兲,j , indepen-dently of the number of steps betweenSiandSj. For the low
temperature limit 共Ⰷ1兲, the exponential factor of the Eq.
共1兲, is dominated, at each step, by the hop to the nearest allowed site, and the DTW is recovered.
III. COMPUTATIONAL IMPLEMENTATION
To understand the microscopic properties of the STW, we have implemented a numerical simulation code using the Monte Carlo method for STW in one-dimensional systems. The developed algorithm is based on the analytical develop-ment共Sec.II兲and it is composed, essentially, by three parts:
共i兲 the map generation module sets up the disordered me-dium, composed byNsites distributed by a Poisson process, and each map generates a distance matrix;共ii兲the calculating probability module, first, calculates the exponentials of the Eq. 共1兲, using E共Dj,i兲=Dj,i; then it calculates the
prob-ability at each step and共iii兲the STW module account for the walker decision, calculatingPj共,兲,i , and following the rule of not returning to the lastvisited sites, forNp steps.
To understand the STW dynamics, we have calculated the density =Nv/N of distinctly visited sites Nv, for several trajectories over several maps. The quantity is the order parameter. In this way, we aspire to find a relation between T, andNp that allows the optimization of the walker
ex-ploration.
We have validated our algorithm throughout the analytical considerations in the high and low temperature limits 共Sec. II兲. We have calculated some probabilities for the high tem-perature limit 共→0兲, for different values of the walker
memory and found that the hopping probability at each step isPj共=0.1,兲,i ⬇1/共N−兲; afterNpsteps, the probability of the
trajectory was P˜S j←Si
共Np兲 ⬇关1/共N−兲兴Np. For low temperature limit, we have observed that the walker hops to the nearest allowed neighbors. With the validated algorithm, we have calculated for 103 independent trajectories. We have con-sidered the temperature interval T=关0.1; 4兴, walker memory interval=关0 , 20兴, number of sitesN=兵64, 128其and number of steps Np=兵10, 102, 103, 4000其N.
IV. NUMERICAL RESULTS
Here, we report the numerical results for STW in one-dimensional systems. From each trajectory, we have obtained the density of distinctly visited sites, for different values of T, , and Np. In Fig. 1, we show how the number of
dis-tinctly visited sites Nv increases as T, , and Np increase. The exploration dependence onNpsuggests that STW has an aging effect. For specific values of T and Np, Nv shows a crossover between the localized 共few visited sites兲 and ex-tended共many visited sites兲exploration regimes. This kind of crossover is also found in the deterministic tourist walk关14兴. The 共兲 inflection point 共Fig. 2兲 gives us the critical memory 1, where the mentioned crossover occurs. To de-termine this point we have fitted data with,
= min−max 1 + exp关k共−1兲兴
+max, 共3兲
wherekis a scale factor, min andmax are the maximum and minimum values, respectively. In the numerical experi-ments, max= 1, namely, after certain value of memory, the walker visits all sites, at least once. We have tried to fit data with other models, such as the hyperbolic tangent, however, Eq. 共3兲gives better results.
We have plotted T共1兲, for maps with N= 128 sites and Np/N= 10, 102and 103, and found a linear behavior as shown
in Fig. 3. Besides, as Np increases, the curve T共1兲 approaches to the curve given by Eq.共4兲,
FIG. 2. Density of visited sites as a function of the walker memoryfor maps withN= 128 and withNp= 10Nsteps, for
tem-perature valuesT= 0.44; 0.50; 0.62; 0.75; 1.0 and 1.5. The symbols stand for numerical data, and lines are the curves fitted by Eq.共3兲. From these fits one obtains the critical memory1. The numerical data have been obtained with 103 independent trajectories, bars stand for standard error.
T
T0
= 1 −1 0
, 共4兲
whereT0 is the transition temperature when= 0, and0is the critical memory when T= 0. This equation just connects the purely determinist case 共T= 0 and the critical memory is 0= log2N, for maps with N= 128, 0= 7兲 and purely sto-chastic case 共= 0, and a glass transition at temperatureT0 = 1/2兲. The connection represented by Eq. 共4兲 also
corre-sponds to the hypothesis that as the walker exploration time Npincreases, the curve connecting the extreme cases of the
tourist walk,共0 ,T0兲and共0, 0兲, is achieved.
To corroborate our hypothesis, we have looked for a gen-eral expression, that共i兲depends onNp,共ii兲fits the numerical
data and,共iii兲when extrapolated to the maximumNpleads to
Eq.共4兲. For each value ofNp, we have found scale factorsTE
and E to multiply T and 1, respectively. This procedure give us a data collapse 共see Fig. 4兲 for three different Np
values. In this way, one obtains a crossover diagram between localized and extended exploration regimes governed by a general expression as function of T,, N, and Np. Plotting
TE and E as function of Np/N 共Fig. 5兲, we obtain their
dependence on the ratio共Np/N兲,
TE=
1 10
冉
Np
N
冊
5␥and E=
3 2
冉
Np
N
冊
−␥, 共5兲
where ␥= 1/18. Notice that TE⬀E
−5
. Jointing Eqs. 共4兲 and
共5兲, we obtain
T
T0
TE= 1 −
1 0
E, 共6兲
and we are able to discriminate the localized and extended exploration regimes as function of T,,NandNp,
1 10
T
T0
冉
NpN
冊
5/18= 1 −3 2
1 0
冉
Np
N
冊
−1/18. 共7兲
Moreover, using TE= 1 共or E= 1兲, one finds the ratio
Np/N that provides the hypothesis in Eq. 共4兲. Thus:
TE=共Np/N兲5␥/10= 1,共Np/N兲5␥= 10 andNp/N= 101 /5␥,
冉
NpN
冊
max⬇4000. 共8兲
In this way, Eq. 共7兲 共i兲 gives a transition diagram between localized and extended exploration regimes, and 共ii兲 fits the numerical data. Its resolution is meaningful for a walker ex-ploring a disordered medium with N sites for Np= 4000N steps共exploration time兲.
We have also performed the STW over a medium with N= 64 sites and let the walker to explore it for Np/N
= 10, 102, 103, and 4000 steps. Again, the numerical data were well fitted by Eq. 共7兲 共see Fig. 6兲. Moreover, for Np/N= 4000 steps, the asymptotic values of Eq. 共4兲 are
reached. FIG. 4. We have found a data collapse simply by multiplying the
temperature byTEand the memory byE. The scale factorsTEand
E depend onNp/N, as one can see in Fig.5. The data collapse
reveals that it is possible to find a relation betweenT,,N, andNp.
FIG. 5. The scale factorsTE共square兲andE共circles兲as a
func-tion ofNp/Nin log-log scale. The fits lead toTE共dash-line兲andE
共continuous-line兲of Eq.共5兲.
FIG. 6. The dependence of temperatureT as a function of1, for N= 64. The symbols stand for numerical results and the seg-mented lines have been calculated by Eq.共7兲, forNp/N= 10共dash兲,
Np/N= 102 共dot兲 and Np/N= 103 共dash-dot兲. The continuous line
The maximum exploration timeNp= 4000Nis an
optimi-zation time for the walker exploration. In other words, this is the minimum time that allows a walker, with short memory, to visit all sites, at least once, spending minimum energy
共E共Dj,i兲兲. In this way, a walker, with temperature T and memory1who explores a medium of sizeN for Np steps,
has its optimum performance when T, 1, N, and Np are related by Eq. 共7兲 and the maximum exploration time is given by Eq.共8兲.
Therefore, the main numerical results that can be pointed out are:共i兲numerical data collapse, that leads to a universal-ity in out-of-equilibrium systems, as the STW problem; 共ii兲 correction factor for the finite-size effect共dependence onN兲 and aging effect共dependence onNp兲;共iii兲transition diagram
共Fig.7兲between localized and extended exploration regimes
and共iv兲relation betweenT,1,N, andNp, that reveals a way
to optimize the exploration of a disordered medium.
V. CONCLUSIONS
We have investigated the stochastic tourist walk in disor-dered one-dimensional systems, varying simultaneously the medium temperatureTand walker memory. An analytical formalism has been developed to obtain a hopping probabil-ity as function ofTand, following the rule of not returning to the lastvisited sites. The order parameter is the density of distinctly visited sites . This quantity has allowed us to relate the quantitiesT,, andNpthroughout a numerical data
collapse. This collapse implies in universality. Thus, the sto-chastic tourist walk is an example that systems out of ther-modynamic equilibrium can present universality. The rela-tionship between T,, andNp 关Eq.共7兲兴assures an optimal walker medium exploration. Exploration time is given by Eq.
共8兲. Finally, the fundamental result is the transition line, which discriminates the localized and extended exploration regions. In these regions, a walker explores few 共localized兲 or many 共extended兲 sites, performing the partially self-avoiding stochastic tourist walk. We believe that a consistent choice of values for temperature and memory 共along the transition line兲 gives the best compromise between maxi-mum exploration and minimaxi-mum trajectory共exploration time兲. For this reason, a thorough analysis of the STW problem in higher dimensionalities is interesting to develop an optimi-zation algorithm.
ACKNOWLEDGMENTS
The authors thank Rodrigo Silva González, Marco Anto-nio Alves da Silva, and Roberto Andre Kraenkel for useful comments and nice discussion. A.S.M. acknowledges the support of the Brazilian agency CNPq 共Grants No. 303990/ 2007-4 and No. 476862/2007-8兲. J.M.B. acknowledges the support of the Brazilian agencies CNPq 共Grant No. 134461/ 2007-0兲and FAPESP共Grant No. 2009/11567-6兲.
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