In this paper, we study the scaling for the mean first-passage time (MFPT) of the random walks on a generalized Koch network with a trap. Through the network construction, where the initial state is transformed from...In this paper, we study the scaling for the mean first-passage time (MFPT) of the random walks on a generalized Koch network with a trap. Through the network construction, where the initial state is transformed from a triangle to a polygon, we obtain the exact scaling for the MFPT. We show that the MFPT grows linearly with the number of nodes and the dimensions of the polygon in the large limit of the network order. In addition, we determine the exponents of scaling efficiency characterizing the random walks. Our results are the generalizations of those derived for the Koch network, which shed light on the analysis of random walks over various fractal networks.展开更多
Suppose {Xn} is a random walk in time-random environment with state space Z^d, |Xn| approaches infinity, then under some reasonable conditions of stability, the upper bound of the discrete Packing dimension of the r...Suppose {Xn} is a random walk in time-random environment with state space Z^d, |Xn| approaches infinity, then under some reasonable conditions of stability, the upper bound of the discrete Packing dimension of the range of {Xn} is any stability index α. Moreover, if the environment is stationary, a similar result for the lower bound of the discrete Hausdorff dimension is derived. Thus, the range is a fractal set for almost every environment.展开更多
Controls, especially effficiency controls on dynamical processes, have become major challenges in many complex systems. We study an important dynamical process, random walk, due to its wide range of applications for m...Controls, especially effficiency controls on dynamical processes, have become major challenges in many complex systems. We study an important dynamical process, random walk, due to its wide range of applications for modeling the transporting or searching process. For lack of control methods for random walks in various structures, a control technique is presented for a class of weighted treelike scale-free networks with a deep trap at a hub node. The weighted networks are obtained from original models by introducing a weight parameter. We compute analytically the mean first passage time (MFPT) as an indicator for quantitatively measurinM the et^ciency of the random walk process. The results show that the MFPT increases exponentially with the network size, and the exponent varies with the weight parameter. The MFPT, therefore, can be controlled by the weight parameter to behave superlinearly, linearly, or sublinearly with the system size. This work provides further useful insights into controllinM eftlciency in scale-free complex networks.展开更多
Recently a great deal of effort has been made to explicitly determine the mean first-passage time (MFPT) between two nodes averaged over all pairs of nodes on a fractal network. In this paper, we first propose a fam...Recently a great deal of effort has been made to explicitly determine the mean first-passage time (MFPT) between two nodes averaged over all pairs of nodes on a fractal network. In this paper, we first propose a family of generalized delayed recursive trees characterized by two parameters, where the existing nodes have a time delay to produce new nodes. We then study the MFPT of random walks on this kind of recursive tree and investigate the effect of the time delay on the MFPT. By relating random walks to electrical networks, we obtain an exact formula for the MFPT and verify it by numerical calculations. Based on the obtained results, we further show that the MFPT of delayed recursive trees is much shorter, implying that the efficiency of random walks is much higher compared with the non-delayed counterpart. Our study provides a deeper understanding of random walks on delayed fractal networks.展开更多
We study the mixing rate of non-backtracking random walks on graphs by looking at non-backtracking walks as walks on the directed edges of a graph. A result known as Ihara’s Theorem relates the adjacency matrix of a ...We study the mixing rate of non-backtracking random walks on graphs by looking at non-backtracking walks as walks on the directed edges of a graph. A result known as Ihara’s Theorem relates the adjacency matrix of a graph to a matrix related to non-backtracking walks on the directed edges. We prove a weighted version of Ihara’s Theorem which relates the transition probability matrix of a non-backtracking walk to the transition matrix for the usual random walk. This allows us to determine the spectrum of the transition probability matrix of a non-backtracking random walk in the case of regular graphs and biregular graphs. As a corollary, we obtain a result of Alon et al. in [1] that in most cases, a non-backtracking random walk on a regular graph has a faster mixing rate than the usual random walk. In addition, we obtain an analogous result for biregular graphs.展开更多
In this paper we study the asymptotic behavior of the maximal position of a supercritical multiple catalytic branching random walk(X_(n))on Z.If M_(n) is its maximal position at time n,we prove that there is a constan...In this paper we study the asymptotic behavior of the maximal position of a supercritical multiple catalytic branching random walk(X_(n))on Z.If M_(n) is its maximal position at time n,we prove that there is a constantα>0 such that M_(n)/n converges toαalmost surely on the set of infinite number of visits to the set of catalysts.We also derive the asymptotic law of the centered process M_(n)-αn as n→∞.Our results are similar to those in[13].However,our results are proved under the assumption of finite L log L moment instead of finite second moment.We also study the limit of(X_(n))as a measure-valued Markov process.For any function f with compact support,we prove a strong law of large numbers for the process X_(n)(f).展开更多
We study a counterbalanced random walkS_(n)=X_(1)+…+X_(n),which is a discrete time non-Markovian process andX_(n) are given recursively as follows.For n≥2,X_(n) is a new independent sample from some fixed law̸=0 wit...We study a counterbalanced random walkS_(n)=X_(1)+…+X_(n),which is a discrete time non-Markovian process andX_(n) are given recursively as follows.For n≥2,X_(n) is a new independent sample from some fixed law̸=0 with a fixed probability p,andX_(n)=−X_(v(n))with probability 1−p,where v(n)is a uniform random variable on{1;…;n−1}.We apply martingale method to obtain a strong invariance principle forS_(n).展开更多
Random walks are a standard tool for modeling the spreading process in social and biological systems But in the face of large-scale networks, to achieve convergence, iterative calculation of the transition matrix in r...Random walks are a standard tool for modeling the spreading process in social and biological systems But in the face of large-scale networks, to achieve convergence, iterative calculation of the transition matrix in random walk methods consumes a lot of time. In this paper, we propose a three-stage hierarchical community detection algorithm based on Partial Matrix Approximation Convergence (PMAC) using random walks. First, this algorithm identifies the initial core nodes in a network by classical measurement and then utilizes the error function of the partial transition matrix convergence of the core nodes to determine the number of random walks steps. As such, the PMAC of the core nodes replaces the final convergence of all the nodes in the whole matrix. Finally based on the approximation convergence transition matrix, we cluster the communities around core nodes and use a closeness index to merge two communities. By recursively repeating the process, a dendrogram of the communities is eventually constructed. We validated the performance of the PMAC by comparing its results with those of two representative methods for three real-world networks with different scales展开更多
The second-order random walk has recently been shown to effectively improve the accuracy in graph analysis tasks.Existing work mainly focuses on centralized second-order random walk(SOW)algorithms.SOW algorithms rely ...The second-order random walk has recently been shown to effectively improve the accuracy in graph analysis tasks.Existing work mainly focuses on centralized second-order random walk(SOW)algorithms.SOW algorithms rely on edge-to-edge transition probabilities to generate next random steps.However,it is prohibitively costly to store all the probabilities for large-scale graphs,and restricting the number of probabilities to consider can negatively impact the accuracy of graph analysis tasks.In this paper,we propose and study an alternative approach,SOOP(second-order random walks with on-demand probability computation),that avoids the space overhead by computing the edge-to-edge transition probabilities on demand during the random walk.However,the same probabilities may be computed multiple times when the same edge appears multiple times in SOW,incurring extra cost for redundant computation and communication.We propose two optimization techniques that reduce the complexity of computing edge-to-edge transition probabilities to generate next random steps,and reduce the cost of communicating out-neighbors for the probability computation,respectively.Our experiments on real-world and synthetic graphs show that SOOP achieves orders of magnitude better performance than baseline precompute solutions,and it can efficiently computes SOW algorithms on billion-scale graphs.展开更多
In this paper, we give a general model of random walks in time-random environment in any countable space. Moreover, when the environment is independently identically distributed, a recurrence-transience criterion and ...In this paper, we give a general model of random walks in time-random environment in any countable space. Moreover, when the environment is independently identically distributed, a recurrence-transience criterion and the law of large numbers are derived in the nearest-neighbor case on Z^1. At last, under regularity conditions, we prove that the RWIRE {Xn} on Z^1 satisfies a central limit theorem, which is similar to the corresponding results in the case of classical random walks.展开更多
We consider the state-dependent reflecting random walk on a half- strip. We provide explicit criteria for (positive) recurrence, and an explicit expression for the stationary distribution. As a consequence, the ligh...We consider the state-dependent reflecting random walk on a half- strip. We provide explicit criteria for (positive) recurrence, and an explicit expression for the stationary distribution. As a consequence, the light-tailed behavior of the stationary distribution is proved under appropriate conditions. The key idea of the method employed here is the decomposition of the trajectory of the random walk and the main tool is the intrinsic branching structure buried in the random walk on a strip, which is different from the matrix-analytic method.展开更多
We consider a branching random walk on N with a random environment in time (denoted by ξ). Let Zn be the counting measure of particles of generation n, and let Zn(t) be its Laplace transform. We show the converge...We consider a branching random walk on N with a random environment in time (denoted by ξ). Let Zn be the counting measure of particles of generation n, and let Zn(t) be its Laplace transform. We show the convergence of the free energy n-llog Zn(t), large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale Zn(t)/E[Zn(t)ξ].展开更多
The authors consider the simple random walk on the infinite cluster of the Bernoulli bond percolation of trees, and investigate the relation between the speed of the simple random walk and the retaining probability p ...The authors consider the simple random walk on the infinite cluster of the Bernoulli bond percolation of trees, and investigate the relation between the speed of the simple random walk and the retaining probability p by studying three classes of trees. A sufficient condition is established for Galton-Watson trees.展开更多
In this paper, we study strong laws of large numbers for random walks in random sceneries. Some mild sufficient conditions for the validity of strong laws of large numbers are obtained.
Inspired by Benjamini et al.(2010) and Windisch(2010),we consider the entropy of the random walk ranges Rn formed by the first n steps of a random walk S on a discrete group.In this setting,we show the existence of hR...Inspired by Benjamini et al.(2010) and Windisch(2010),we consider the entropy of the random walk ranges Rn formed by the first n steps of a random walk S on a discrete group.In this setting,we show the existence of hR:=limn→∞H(Rn)/n called the asymptotic entropy of the ranges.A sample version of the above statement in the sense of Shannon(1948) is also proved.This answers a question raised by Windisch(2010).We also present a systematic characterization of the vanishing asymptotic entropy of the ranges.Particularly,we show that hR=0 if and only if the random walk either is recurrent or escapes to negative infinity without left jump.By introducing the weighted digraphs Γn formed by the underlying random walk,we can characterize the recurrence property of S as the vanishing property of the quantity limn→∞H(Γn)/n,which is an analogue of hR.展开更多
We consider laws of iterated random walks in random environments. logarithm for one-dimensional transient A quenched law of iterated logarithm is presented for transient random walks in general ergodic random environm...We consider laws of iterated random walks in random environments. logarithm for one-dimensional transient A quenched law of iterated logarithm is presented for transient random walks in general ergodic random environments, including independent identically distributed environments and uniformly ergodic environments.展开更多
Let be the simple random walk in zd, and SupPers f(n) is an integer-valued function and increases to infinity as n tends to infinity, and In this paper,a necessary and sufficient condition to ensure or 1 is derived fo...Let be the simple random walk in zd, and SupPers f(n) is an integer-valued function and increases to infinity as n tends to infinity, and In this paper,a necessary and sufficient condition to ensure or 1 is derived for d=3,4. This problem was first studied by P. Erdos and S.J. Taylor.展开更多
Efficient cell migration is crucial for the functioning of biological processes, e.g., morphogenesis, wound healing, and cancer metastasis. In this study, we monitor the migratory behavior of the 3D fibroblast cluster...Efficient cell migration is crucial for the functioning of biological processes, e.g., morphogenesis, wound healing, and cancer metastasis. In this study, we monitor the migratory behavior of the 3D fibroblast clusters using live cell microscopy,and find that crowded environment affects cell migration, i.e., crowding leads to directional migration at the cluster’s periphery. The number of cell layers being stacked during seeding determines the directional-to-random transition. Intriguingly,the migratory behavior of cell clusters resembles the dispersion dynamics of clouds of passive particles, indicating that the biological process is driven by physical effects(e.g., entropy) rather than cell communication. Our findings highlight the role of intrinsic physical characteristics, such as crowding, in regulating biological behavior, and suggest new therapeutic approaches targeting at cancer metastasis.展开更多
Two new solutions of the homogeneous diffusion equation in 1D are derived in the presence of losses and a trigonometric profile for a profile of density. A simulation for the ankle in the energy distribution of cosmic...Two new solutions of the homogeneous diffusion equation in 1D are derived in the presence of losses and a trigonometric profile for a profile of density. A simulation for the ankle in the energy distribution of cosmic rays (CRs) is provided in the framework of the fine tuning of the involved parameters. A theoretical image for the overall diffusion of CRs in galactic coordinates is provided.展开更多
Cell migration plays a significant role in physiological and pathological processes.Understanding the characteristics of cell movement is crucial for comprehending biological processes such as cell functionality,cell ...Cell migration plays a significant role in physiological and pathological processes.Understanding the characteristics of cell movement is crucial for comprehending biological processes such as cell functionality,cell migration,and cell–cell interactions.One of the fundamental characteristics of cell movement is the specific distribution of cell speed,containing valuable information that still requires comprehensive understanding.This article investigates the distribution of mean velocities along cell trajectories,with a focus on optimizing the efficiency of cell food search in the context of the entire colony.We confirm that the specific velocity distribution in the experiments corresponds to an optimal search efficiency when spatial weighting is considered.The simulation results indicate that the distribution of average velocity does not align with the optimal search efficiency when employing average spatial weighting.However,when considering the distribution of central spatial weighting,the specific velocity distribution in the experiment is shown to correspond to the optimal search efficiency.Our simulations reveal that for any given distribution of average velocity,a specific central spatial weighting can be identified among the possible central spatial weighting that aligns with the optimal search strategy.Additionally,our work presents a method for determining the spatial weights embedded in the velocity distribution of cell movement.Our results have provided new avenues for further investigation of significant topics,such as relationship between cell behavior and environmental conditions throughout their evolutionary history,and how cells achieve collective cooperation through cell-cell communication.展开更多
基金Project supported by the Research Foundation of Hangzhou Dianzi University,China (Grant Nos. KYF075610032 andzx100204004-7)the Hong Kong Research Grants Council,China (Grant No. CityU 1114/11E)
文摘In this paper, we study the scaling for the mean first-passage time (MFPT) of the random walks on a generalized Koch network with a trap. Through the network construction, where the initial state is transformed from a triangle to a polygon, we obtain the exact scaling for the MFPT. We show that the MFPT grows linearly with the number of nodes and the dimensions of the polygon in the large limit of the network order. In addition, we determine the exponents of scaling efficiency characterizing the random walks. Our results are the generalizations of those derived for the Koch network, which shed light on the analysis of random walks over various fractal networks.
基金Project supported by NNSF of China (10371092)Foundation of Wuhan University
文摘Suppose {Xn} is a random walk in time-random environment with state space Z^d, |Xn| approaches infinity, then under some reasonable conditions of stability, the upper bound of the discrete Packing dimension of the range of {Xn} is any stability index α. Moreover, if the environment is stationary, a similar result for the lower bound of the discrete Hausdorff dimension is derived. Thus, the range is a fractal set for almost every environment.
基金Supported by the National Natural Science Foundation of China under Grant Nos 61173118,61373036 and 61272254
文摘Controls, especially effficiency controls on dynamical processes, have become major challenges in many complex systems. We study an important dynamical process, random walk, due to its wide range of applications for modeling the transporting or searching process. For lack of control methods for random walks in various structures, a control technique is presented for a class of weighted treelike scale-free networks with a deep trap at a hub node. The weighted networks are obtained from original models by introducing a weight parameter. We compute analytically the mean first passage time (MFPT) as an indicator for quantitatively measurinM the et^ciency of the random walk process. The results show that the MFPT increases exponentially with the network size, and the exponent varies with the weight parameter. The MFPT, therefore, can be controlled by the weight parameter to behave superlinearly, linearly, or sublinearly with the system size. This work provides further useful insights into controllinM eftlciency in scale-free complex networks.
基金Project supported by the National Natural Science Foundation of China (Grant Nos.61203155 and 11232005)the Natural Science Foundation of Zhejiang Province,China (Grant No.LQ12F03003)the Hong Kong Research Grants Council under the GRF Grant CityU (Grant No.1109/12)
文摘Recently a great deal of effort has been made to explicitly determine the mean first-passage time (MFPT) between two nodes averaged over all pairs of nodes on a fractal network. In this paper, we first propose a family of generalized delayed recursive trees characterized by two parameters, where the existing nodes have a time delay to produce new nodes. We then study the MFPT of random walks on this kind of recursive tree and investigate the effect of the time delay on the MFPT. By relating random walks to electrical networks, we obtain an exact formula for the MFPT and verify it by numerical calculations. Based on the obtained results, we further show that the MFPT of delayed recursive trees is much shorter, implying that the efficiency of random walks is much higher compared with the non-delayed counterpart. Our study provides a deeper understanding of random walks on delayed fractal networks.
文摘We study the mixing rate of non-backtracking random walks on graphs by looking at non-backtracking walks as walks on the directed edges of a graph. A result known as Ihara’s Theorem relates the adjacency matrix of a graph to a matrix related to non-backtracking walks on the directed edges. We prove a weighted version of Ihara’s Theorem which relates the transition probability matrix of a non-backtracking walk to the transition matrix for the usual random walk. This allows us to determine the spectrum of the transition probability matrix of a non-backtracking random walk in the case of regular graphs and biregular graphs. As a corollary, we obtain a result of Alon et al. in [1] that in most cases, a non-backtracking random walk on a regular graph has a faster mixing rate than the usual random walk. In addition, we obtain an analogous result for biregular graphs.
基金supported in part by the National Natural Science Foundation of China (No.12271374)。
文摘In this paper we study the asymptotic behavior of the maximal position of a supercritical multiple catalytic branching random walk(X_(n))on Z.If M_(n) is its maximal position at time n,we prove that there is a constantα>0 such that M_(n)/n converges toαalmost surely on the set of infinite number of visits to the set of catalysts.We also derive the asymptotic law of the centered process M_(n)-αn as n→∞.Our results are similar to those in[13].However,our results are proved under the assumption of finite L log L moment instead of finite second moment.We also study the limit of(X_(n))as a measure-valued Markov process.For any function f with compact support,we prove a strong law of large numbers for the process X_(n)(f).
基金Supported by the National Natural Science Foundation of China(11671373).
文摘We study a counterbalanced random walkS_(n)=X_(1)+…+X_(n),which is a discrete time non-Markovian process andX_(n) are given recursively as follows.For n≥2,X_(n) is a new independent sample from some fixed law̸=0 with a fixed probability p,andX_(n)=−X_(v(n))with probability 1−p,where v(n)is a uniform random variable on{1;…;n−1}.We apply martingale method to obtain a strong invariance principle forS_(n).
基金supported by the National Natural Science Foundation of China(Nos.61272422,61572260,61373017,and 61572261)
文摘Random walks are a standard tool for modeling the spreading process in social and biological systems But in the face of large-scale networks, to achieve convergence, iterative calculation of the transition matrix in random walk methods consumes a lot of time. In this paper, we propose a three-stage hierarchical community detection algorithm based on Partial Matrix Approximation Convergence (PMAC) using random walks. First, this algorithm identifies the initial core nodes in a network by classical measurement and then utilizes the error function of the partial transition matrix convergence of the core nodes to determine the number of random walks steps. As such, the PMAC of the core nodes replaces the final convergence of all the nodes in the whole matrix. Finally based on the approximation convergence transition matrix, we cluster the communities around core nodes and use a closeness index to merge two communities. By recursively repeating the process, a dendrogram of the communities is eventually constructed. We validated the performance of the PMAC by comparing its results with those of two representative methods for three real-world networks with different scales
文摘The second-order random walk has recently been shown to effectively improve the accuracy in graph analysis tasks.Existing work mainly focuses on centralized second-order random walk(SOW)algorithms.SOW algorithms rely on edge-to-edge transition probabilities to generate next random steps.However,it is prohibitively costly to store all the probabilities for large-scale graphs,and restricting the number of probabilities to consider can negatively impact the accuracy of graph analysis tasks.In this paper,we propose and study an alternative approach,SOOP(second-order random walks with on-demand probability computation),that avoids the space overhead by computing the edge-to-edge transition probabilities on demand during the random walk.However,the same probabilities may be computed multiple times when the same edge appears multiple times in SOW,incurring extra cost for redundant computation and communication.We propose two optimization techniques that reduce the complexity of computing edge-to-edge transition probabilities to generate next random steps,and reduce the cost of communicating out-neighbors for the probability computation,respectively.Our experiments on real-world and synthetic graphs show that SOOP achieves orders of magnitude better performance than baseline precompute solutions,and it can efficiently computes SOW algorithms on billion-scale graphs.
基金the Natural Science Foundation of Anhui Province (No. KJ2007B122) the Youth Teachers Aid Item of Anhui Province (No. 2007jq1117).
文摘In this paper, we give a general model of random walks in time-random environment in any countable space. Moreover, when the environment is independently identically distributed, a recurrence-transience criterion and the law of large numbers are derived in the nearest-neighbor case on Z^1. At last, under regularity conditions, we prove that the RWIRE {Xn} on Z^1 satisfies a central limit theorem, which is similar to the corresponding results in the case of classical random walks.
基金Acknowledgements The authors would like to thank Drs. Hongyan Sun and Ke Zhou for their stimulating discussion. Also they would like to express their gratitude to the referees for their careful reading of the first version of paper and useful suggestions for revising the paper. This work was partially supported by the National Natural Science Foundation of China (Grant No. 11131003), the 985 Project, and the Natural Sciences and Engineering Research Council of Canada (Grant No. 315660).
文摘We consider the state-dependent reflecting random walk on a half- strip. We provide explicit criteria for (positive) recurrence, and an explicit expression for the stationary distribution. As a consequence, the light-tailed behavior of the stationary distribution is proved under appropriate conditions. The key idea of the method employed here is the decomposition of the trajectory of the random walk and the main tool is the intrinsic branching structure buried in the random walk on a strip, which is different from the matrix-analytic method.
基金Acknowledgements The authors would like to thank the anonymous referees for valuable comments and remarks. This work was partially supported by the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (HIT. NSRIF. 2015102), the National Natural Science Foundation of China (Grant Nos. 11171044, 11101039), and by the Natural Science Foundation of Hunan Province (Grant No. 11JJ2001).
文摘We consider a branching random walk on N with a random environment in time (denoted by ξ). Let Zn be the counting measure of particles of generation n, and let Zn(t) be its Laplace transform. We show the convergence of the free energy n-llog Zn(t), large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale Zn(t)/E[Zn(t)ξ].
基金Supported in part by Grant G1999075106 from the Ministry of Science and Technology of China
文摘The authors consider the simple random walk on the infinite cluster of the Bernoulli bond percolation of trees, and investigate the relation between the speed of the simple random walk and the retaining probability p by studying three classes of trees. A sufficient condition is established for Galton-Watson trees.
基金the National Natural Science Foundation of China(No.10401037)Doctoral Program Foundation of the Ministry of Education of China(No.20060269016)
文摘In this paper, we study strong laws of large numbers for random walks in random sceneries. Some mild sufficient conditions for the validity of strong laws of large numbers are obtained.
基金National Natural Science Foundation of China (Grant Nos. 11790273, 11771286, 11531001, 11371317 and 11271077)the Laboratory of Mathematics for Nonlinear Science, Fudan Universitysupported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ18A010007)。
文摘Inspired by Benjamini et al.(2010) and Windisch(2010),we consider the entropy of the random walk ranges Rn formed by the first n steps of a random walk S on a discrete group.In this setting,we show the existence of hR:=limn→∞H(Rn)/n called the asymptotic entropy of the ranges.A sample version of the above statement in the sense of Shannon(1948) is also proved.This answers a question raised by Windisch(2010).We also present a systematic characterization of the vanishing asymptotic entropy of the ranges.Particularly,we show that hR=0 if and only if the random walk either is recurrent or escapes to negative infinity without left jump.By introducing the weighted digraphs Γn formed by the underlying random walk,we can characterize the recurrence property of S as the vanishing property of the quantity limn→∞H(Γn)/n,which is an analogue of hR.
基金The author would like to thank the referees for comments on conditions (C1) and (C2). This work was supported in part by the National Natural Science Foundation of China (Grant No. 11171262) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20130141110076).
文摘We consider laws of iterated random walks in random environments. logarithm for one-dimensional transient A quenched law of iterated logarithm is presented for transient random walks in general ergodic random environments, including independent identically distributed environments and uniformly ergodic environments.
文摘Let be the simple random walk in zd, and SupPers f(n) is an integer-valued function and increases to infinity as n tends to infinity, and In this paper,a necessary and sufficient condition to ensure or 1 is derived for d=3,4. This problem was first studied by P. Erdos and S.J. Taylor.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 51927804 and 12174306)the Natural Science Basic Research Program of Shaanxi Province of China (Grant No. 2023-JC-JQ-02)。
文摘Efficient cell migration is crucial for the functioning of biological processes, e.g., morphogenesis, wound healing, and cancer metastasis. In this study, we monitor the migratory behavior of the 3D fibroblast clusters using live cell microscopy,and find that crowded environment affects cell migration, i.e., crowding leads to directional migration at the cluster’s periphery. The number of cell layers being stacked during seeding determines the directional-to-random transition. Intriguingly,the migratory behavior of cell clusters resembles the dispersion dynamics of clouds of passive particles, indicating that the biological process is driven by physical effects(e.g., entropy) rather than cell communication. Our findings highlight the role of intrinsic physical characteristics, such as crowding, in regulating biological behavior, and suggest new therapeutic approaches targeting at cancer metastasis.
文摘Two new solutions of the homogeneous diffusion equation in 1D are derived in the presence of losses and a trigonometric profile for a profile of density. A simulation for the ankle in the energy distribution of cosmic rays (CRs) is provided in the framework of the fine tuning of the involved parameters. A theoretical image for the overall diffusion of CRs in galactic coordinates is provided.
基金Project supported by the National Natural Science Foundation of China(Grant No.31971183).
文摘Cell migration plays a significant role in physiological and pathological processes.Understanding the characteristics of cell movement is crucial for comprehending biological processes such as cell functionality,cell migration,and cell–cell interactions.One of the fundamental characteristics of cell movement is the specific distribution of cell speed,containing valuable information that still requires comprehensive understanding.This article investigates the distribution of mean velocities along cell trajectories,with a focus on optimizing the efficiency of cell food search in the context of the entire colony.We confirm that the specific velocity distribution in the experiments corresponds to an optimal search efficiency when spatial weighting is considered.The simulation results indicate that the distribution of average velocity does not align with the optimal search efficiency when employing average spatial weighting.However,when considering the distribution of central spatial weighting,the specific velocity distribution in the experiment is shown to correspond to the optimal search efficiency.Our simulations reveal that for any given distribution of average velocity,a specific central spatial weighting can be identified among the possible central spatial weighting that aligns with the optimal search strategy.Additionally,our work presents a method for determining the spatial weights embedded in the velocity distribution of cell movement.Our results have provided new avenues for further investigation of significant topics,such as relationship between cell behavior and environmental conditions throughout their evolutionary history,and how cells achieve collective cooperation through cell-cell communication.