Stationary Probability and First-Passage Time of Biased Random Walk

Transcription

1 Commun. Theor. Phys Vol. 66, No. 3, September 1, 2016 Stationary Probability and First-Passage Time of Biased Random Walk Jing-Wen Li 李井文, Shen-Li Tang 唐沈立, and Xin-Ping Xu 徐新平 School of Physical Science and Technology, Soochow University, Suzhou , China Received April 13, 2016; revised manuscript received June 27, 2016 Abstract In this paper, we consider the stationary probability and first-passage time of biased random walk on 1D chain, where at each step the walker moves to the left and right with probabilities p and q respectively 0 p, q 1, p + q 1. We derive exact analytical results for the stationary probability and first-passage time as a function of p and q for the first time. Our results suggest that the first-passage time shows a double power-law F N 1 γ, where the exponent γ 2 for N < p q 1 and γ 1 for N > p q 1. Our study sheds useful insights into the biased random-walk process. PACS numbers: Fb, Cd, Jc Key words: random walk, biased random walk, first-passage time, stationary probability 1 Introduction Random walk RW is a powerful tool in studying stochastic processes. As a fundamental model in many miscellaneous areas, the theory of random walk has been successfully applied in computer science, physics, ecology, economics, and a number of other fields. [1 4 Many physical characteristics of RW, such as the dispersal distributions, first-passage times FPT, encounter rates, etc., have been extensively investigated in Refs. [5 7. Here, we focus on the stationary probability SP and first-passage time FPT of biased random walk BRW on 1D chain. The stationary probability SP is the longtime limiting probability and plays an important role in Markov processes. The first-passage time FPT indicates the expected time to hit a target node for the first time for a walker staring from a source node. [6,8 It is a quantitative indicator to characterize the transport efficiency, and carries much information of random walks and many other quantities are related to FPT. [8 10 In this paper, we will study the BRW on the onedimensional 1D chain, which lacks one connection compared to the regular cycle. The left-most and right-most points are topological boundaries of the 1D chain and reflect the walker to their neighbors. Although there are numerous studies of FPT of RWs in Refs. [11 14, BRWs has received little attention due to the difficult analytical calculation. Previous work study FPT of homogeneous/unbiased RWs using the effective resistance method. [15 17 However, the method of effective resistance is only valid for the homogeneous/unbiased RWs, where the probability to each direction at a given vertex is equal. For the homogeneous/unbiased RWs on the 1D chain, the SP is proportional to the degree of the vertex d i / j d j and FPT from the left-most vertex P SP i 1 to the right-most vertex N equals to N 1 2 easily obtained by the method of effective resistance. Here, we consider BRW on the 1D chain, where at each step the walker moves to the left and right with probabilities p and q respectively 0 p, q 1, p + q 1. When p q 1/2, the BRW becomes homogeneous/unbiased RWs, so the BRW defined by parameters p, q is a more general model and its SP and FPT have not been investigated yet. To study BRW parameterized by p, q 0 p, q 1, p + q 1, we consider the SP Pi SP, i 1, 2,..., N and FPT from the left-most vertex 1 to the right-most vertex N T1 N FTP using the spectrum method. The SP and FPT of RWs are related to the spectrum of the normalized adjacency matrix A D 1/2 AD 1/2. [18 19 where A is the adjacency matrix A ij equals to 1 if i and j are connected, and 0 otherwise and D is the degree diagonal matrix diagonal element D ii d i, d i is the degree of vertex i. For the homogeneous/unbiased RWs, the nonzero elements between two connected vertices i and j of the normalized adjacency matrix are A ij 1/d i 1/dj, where 1/d i is the probability from i to j and 1/d j is the probability from j to i. In view of this fact, the nonzero elements between two connected vertices i and j of the normalized adjacency matrix are given by A ij p i j p j i, 1 where p i j is the probability from i to j and p j i is the probability from j to i. For the BRW on 1D chain, p 1 2 p N N 1 1, p i i 1 p and p i i+1 q i 2, 3,..., N 1. Thus, the elements of A ij are sum- Supported by the National Natural Science Foundation of China under Grant No , Shanghai Key Laboratory of Intelligent Information Processing IIPL , and Innovative Training Program for College Students under Grant No. 2015xj070 c 2016 Chinese Physical Society and IOP Publishing Ltd

2 No. 3 Communications in Theoretical Physics 331 marized as, p, if i 1, j 2, q, if i N 1, j N, A ij A ji 2 pq, if i j ± 1, 0, Otherwise. The SP and FPT are closely related to the spectrum of A. [1 4 Suppose the eigenvalues and orthonormalized eigenvectors of A are { i, v i } i 1, 2,..., N. The maximal eigenvalue is 1 and its corresponding eigenvector v 1 determines the SP distribution. The SP at vertex k Pk SP, k 1, 2,..., N equals to the square of the k th component of v 1, i.e., [20 Pk SP v 1 k 2, k 1, 2,..., N. 3 The FPT starting from vertex i to j is related to the other except 1 eigenvalues and eigenvectors by the following equation, [19 20 F i j N 1 [ vk i 2 v kiv k j. 4 1 k q pq k 1 In this paper, we focus on the FPT from the left-most vertex 1 to the right-most vertex N, which can be recasted as, F 1 N N 1 [ vk 1 2 v k1v k N. 5 1 k q pq k 1 In order to calculate SP and FPT in Eqs. 3 and 5, all the eigenvalues and eigenvectors { i, v i } are required. In the following, we will put emphasis on the calculation of the eigenvalues and orthonormalized eigenvectors of A. We use the Chebyshev polynomial method [21 22 to calculate the eigenvalues and orthonormalized eigenvectors of A. The eigenequation is Av v, is the eigenvalues and v is its corresponding eigenvector. Suppose eigenvector v has N components v T {v1, v2,..., vn}, the eigenequation can be decomposed into the following N linear equations, pv2 v1, 6 pv1 + pqv3 v2, 7 pq[vi 1 + vi + 1 vi, i [3, N 2, 8 pqvn 2 + qvn vn 1, 9 qvn 1 vn, 10 Equation 8 can be simplified as vi 1 + vi + 1 / pqvi, which is similar to the recursive relation of the Chebyshev polynomials of the second kind. Using the recursive definition of Chebyshev polynomials and the mapping relation / pq 2x, the arbitrary component vj can be written as a function of v3 and v2, vj U j 3 xv3 U j 4 xv2, j [2, N Combining Eqs. 9 and 10 to eliminate vn, we get pqvn 2+q/ vn 1 0, which is further simplified as a function of v3 and v2 by setting j N 1 and j N 2 in Eq. 11, [ pqun 5 x q U N 4 x v3 [ + q U N 5 x pqu N 6 x v Utilizing Eqs. 6 and 7, we obtain an equation for v3 and v2, p pqv3 + v We have obtained two independent Eqs. 12 and 13 for v3 and v2. The two equations should have nonzero solutions, the determinant of the coefficients equals to zero, which leads to, p + q U N 4 x + pqu N 6 x pq 1 22 U N 5 x 0, 14 where the first term equals to pq/ 2 U N 4 x. After a simple recombination of Eq. 14, we arrive at 2 1 [U N 4 x pqu N 5 x + pq 2 U N 4x + [pqu N 6 x pqu N 5 x Noting that 2 pqx mapping relation and the Chebyshev polynomial recursive relation 2xU i x U i 1 x + U i+1 x, the first term in the square brackets of the above equation equals to pq U N 3 x and the third term in the square brackets equals to pqu N 4 x. Thus Eq. 15 becomes as [ pqu N 3 x pqu N 4 x pqu x 0, which can be written as a simple form, 2 1U x Solving Eq. 16, we obtain two special solutions ±1. The remaining N 2 solutions are determined by U x sinn 1θ/sin θ 0 x cos θ, which leads to x k cos θ k, θ k kπ, k 1, 2,..., N N 1 Thus we get the N eigenvalues for : 1, if k 0, k 2 kπ pq cos, if k 1, 2,..., N 2, 18 N 1 1, if k N 1. Next, we analyze the eigenvectors. First, we consider the eigenvectors of ±1. When ±1, Eq. 13 becomes as v3 p v2 p 1 v1 pq pq p 2 p p q v1 1 p q/pv1, thus Eq. 11 is simplified as q j 2 v1 vj ± ±, j [2, N 1. p p

3 332 Communications in Theoretical Physics Vol. 66 Combining Eqs. 6 and 10, we express the arbitrary component vj as a function of v1, q j 2 v1 ± ±, if j [2, N 1, p p vj ±1 q v1, 19 ± ± if j N. p Noting that N j1 vj 2 1 normalization condition, we obtain 1 q/p v1 2[1 q/p N 1. Thus we have successfully determined eigenvectors of the eigenvalues ±1. For the other eigenvectors of ±1, we apply the same technique, where the normalization factor depends on the eigenvalues. To obtain the eigenvectors of ±1, we write Eqs. 6 and 7 as v2 v1/ p and v3 [ 2 p/p qv1. Thus Eqs. 10 and 11 can be rewritten as a function of v1, [ Uj 2 x U j 3x v1, if j [2, N 1, p q vj p q U N 3xv1, if j N. 20 Similarly, according to the normalization condition N j1 vj 2 1 and after some algebraic calculations, we obtain v1 2q1 x 2 N 11 4pqx 2. Thus we have obtained the remaining N 2 eigenvectors when the variable x in Eq. 20 is replaced to x k in Eq. 17. In the rest of this paper, we will use the exact solutions of eigenvalues and eigenvectors to calculate the SP and FPT. In accordance to Eq. 3, the SP is equal to the square of the eigenvector of 1, which is calculated using the expression vj 1 in Eq. 19 as follows, 1 q/p 2[1 q/p N 1, if k 1, Pk SP 1 q/p q k 2, if k [2, N 1, 2p[1 q/p N 1 21 p 1 q/p 2[1 q/p N 1 q/p, if k N, When p q 1/2 and taking the limit of q/p 1, the above equation is simplified as 1, if k 1, N, Pk SP 2N N 1, if k [2, N 1, which is consistent from the calculation Pk SP d k / j d j for the homogeneous/unbiased RWs. We now proceed to calculate the FPT in Eq. 5. The summation of contribution in Eq. 5 is composed by two parts: one is the contribution from 1 and the others from k 2 pq cos kπ/n 1 k 1, 2,..., N 2, i.e., F 1 N F 1 N 1 + k ±1 F 1 N k, 23 where the contribution of 1 can be calculated using vj 1 in Eq. 19 F 1 N 1 N 11 q/p[1 + 1N q/p N 1/2 4q[1 q/p N 1, 24 and contribution of k 2 pq cos kπ/n 1 can be simplified using the eigenvectors in Eq. 20, which leads to F 1 N k N 1v2 1 [ xk 1 1 k q 1 p pq q U N 1 v 2 1 xk N 3x k [1 + U N 3 x k k q Noting that v 2 1 xk 21 x 2 k q N 11 4pqx 2 k, U N 3x k sinn 2θ k 1 k+1, sin θ k Eq. 25 can be simplified as F 1 N k 21 x2 k [1 + 1k+1 2[1 + 1 k+1 sin 2 [kπ/n 1 1 k 1 4pqx 2 k 1 2 pq cos[kπ/n 11 4pq cos 2 [kπ/n Thus, the FPT in Eq. 23 can be written as the following explicit form: F 1 N N 11 q/p[1 + 1N q/p N 1/2 4q[1 q/p N 1 2[1 + 1 k+1 sin 2 [kπ/n pq cos[kπ/n 11 4pq cos 2 [kπ/n Particularly, when p q 1/2, the first term of Eq. 27 is simplified as F 1 N 1 1/2[1 + 1 N by taking the limit of q/p 1. In this case, the second term of Eq. 27 is calculated to be F S 1 N k k ±1 F 1 N k where the series summation formula 1 NN 2, 1 cos[kπ/n 1 3 2[1 + 1 k+1 1 cos[kπ/n 1 NN [1 1N, 28 1 k+1 NN cos[kπ/n [1 1N

4 No. 3 Communications in Theoretical Physics 333 are applied in the above calculations. Hence, the FPT for the homogeneous/unbiased RWs is F 1 N pq [1 + 1N + NN [1 1N N 1 2, 29 which is consistent with the expectation using the method of effective resistance. Now we analyze the two extreme cases: p 0, q 1 and p 1, q 0. For these two extreme cases, the second summation in Eq. 27 is simplified as 2[1 + 1 k+1 sin 2 kπ N 1 N 1, while the first term See Eq. 24 approaches to 0 for p 0, q 1 and for p 1, q 0. This result is consistent with our intuition. When p 0, q 1, the walker moves to the right at each step, the FPT is identical to the length of the 1D chain N 1; When p 1, q 0, the walker moves to the left at each step and the walker can never reach the right-most vertex, thus the FPT is infinite. Fig. 1 Color online F 1 N 1 vs. N for q/p > 1 a and q/p < 1 b. F 1 N 1 shows an exponential decay for q/p > 1 see a p 0.45, p 0.4, p 0.3 and p 0.2. In contrast, F 1 N 1 increases linearly for q/p < 1 see b q 0.45, q 0.4, q 0.3, and q 0.2. Finally, we investigate the dependence of FPT on the system size N. In Eq. 27, the first term F 1 N 1 see Eq. 24 shows different behavior for q/p > 1 and q/p < 1. Figure 1a shows F 1 N 1 vs. N for p 0.45, p 0.4, p 0.3, and p 0.2 q/p > 1, which shows an exponential decay and approaches to 0 for large N. On the contrary, F 1 N 1 increases linearly for q/p < 1, as shown in Fig. 1b. Particularly, when q 0, F 1 N 1 is close to the analytical expression N 1p/4q. The main contribution of FPT is determined by the second term of Eq. 27 F1 N S k see Eq. 28. Figure 2 shows F1 N S k as a function of N for different values of p, q. As we can see, F1 N S k only depends on the product of p and q; when pq 0, F1 N S k N 1 is a complete Fig. 2 Color online F1 N S k as a function of N for different values of p, q. When q/p 1 pq 1/4, F1 N S k is close to the upper bound N 1 2 for p q 1/2. F1 N S k shows a double power-law behavior. For N < N c p q 1, F1 N S k N 1 2 while F1 N S k N 1 for N > N c p q 1. The black solid line indicates the scaling N 2 and the dashed line implies linear behavior of N. linear function of N. However, when q/p 1, F1 N S k is close to the upper bound N 1 2 for p q 1/2. Nevertheless, F1 N S k displays two kinds of behaviour for q/p 1. For small system N < N c p q 1, F1 N S k N 2 see the black solid line in Fig. 2; For large system N > N c p q 1, F1 N S k N see the black dashed line in Fig. 2. F1 N S k shows different scaling behavior below and above the critical value of

5 334 Communications in Theoretical Physics Vol. 66 system size N c p q 1. Fig. 3 Color online FPT F1 N S as a function of q for a 1D-Chain of N 10. FPT has a minimal value at q c 0.2 in the region q 0, 0.5. The peak around q corresponds to the FPT of homogeneous/unbiased RWs. We find that, the FPT of q q is the same as that of the homogeneous/unbiased RWs N 1 2 see the vertical red lines in the figure. The critical value q 0 having the same FPT depends on the chain size N. In the inserted plot, we numerically investigate the dependence of the critical value q 0 on the system size N, which shows a power-law decay q 0 N 1. Figure 3 shows the FPT as a function of q for a 1D- Chain of N 10. Interestingly, FPT has a minimal value at q c 0.2 between q 0, 0.5. For q < q c, the FPT demonstrates a power-law decay. For q c < q < 1, the FPT has a maximal value N 1 2 corresponding to the homogeneous/unbiased RWs of q q We note that, in the region q 0, q c, the FPT for q is equal to N 1 2 FPT of the homogeneous/unbiased RWs q q 0 0.5, see Fig. 3. The value of q 0 for identical FPT depends on the system size N. In the inserted log-log plot, we show the critical value q 0 versus different system size N. We find that q 0 displays a power-law decay q 0 N 1. In summary, we obtain exact analytical results for the stationary probability and first-passage time of biased random walk as a function of p and q for the first time. Our results indicate that the first-passage time shows a double power-law behavior F N 1 γ, where the exponent γ 2 for N < p q 1 and γ 1 for N > p q 1. Furthermore, we find that there is a critical value q q 0 whose FPT is the same as the homogeneous/unbiased RWs of q q We investigate the dependence of such critical value q 0 on the system size N and find a power law relationship q 0 N 1. References [1 N. Guillotin-Plantard and R. Schott, Dynamic Random Walks: Theory and Application, Elsevier, Amsterdam [2 W. Woess, Random Walks on Infinite Graphs and Groups, Cambridge University Press, Cambridge [3 F Spitzer, Principles of Random Walk, Springer, Berlin [4 G.H. Weiss, Aspects and Applications of the Random Walk, North-Holland, New York [5 Brian H. Kaye, A Random Walk Through Fractal Dimensions, VCH Publishers, New York [6 S. Redner, A Guide to First-Passage Processes, Cambridge University Press, Cambridge [7 X.K. Zhang, J. Wan, J.J. Lu, and X.P. Xu, Commun. Theor. Phys [8 Z.Z. Zhang, et al., Phys. Rev. E [9 R. Metzler and J. Klafter, Phys. Rep [10 S. Havlin and D. ben-avraham, Adv. Phys [11 J.J. Kozak and V. Balakrishnan, Phys. Rev. E [12 J.J. Kozak and V. Balakrishnan, Int. J. Bifurcation Chaos Appl. Sci. Eng [13 E. Agliari, Phys. Rev. E [14 Z.Z. Zhang, W.L. Xie, S.G. Zhou, M. Li, and J.H. Guan, Phys. Rev. E ; Phys. Rev. E [15 A.K. Chandra, P. Raghavan, W.L. Ruzzo, and R. Smolensky, in Proceedings of the 21st Annnual ACM Symposium on the Theory of Computing, ACM Press, New York 1989 pp [16 P. Tetali, J. Theor. Probab [17 P.G. Doyle and J.L. Snell, Random Walks and Electric Networks, The Mathematical Association of America, Oberlin, OH [18 A. Garcia Cantu and E. Abad, Phys. Rev. E [19 H.Y. Chen and F.J. Zhang, Discrete. Appl. Math [20 M.A. Pinsky and S. Karlin, An Introduction to Stochastic Modeling, Fourth Edition, Academic Press, New York [21 X.P. Xu, Y. Ide, and N. Konno, Phys. Rev. A [22 T.J. Rivlin, Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory, Wiley- Interscience, 2 ed. June, New York 1990.