random walks on graphs ideas techniques and results random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology in recent years relevant mathematical results have been obtained in this field and new ideas have been introduced which can be fruitfully extended to different areas and disciplines here we aim at giving a brief but comprehensive perspective of these progresses with a particular emphasis on physical aspects

mapping between hopping on hierarchical structures and diffusion on a family of fractals the authors present a simple analytical approach to studying diffusion on hierarchical and fractal structures they show that hopping on one dimensional hierarchical structures can be mapped onto diffusion on a family of loopless fractals

approximations on the peano river network application of the horton strahler hierarchy to the case of low connections a network analysis is used to investigate the low connections of natural river channels at the basin scale the river networks are analyzed according to the horton strahler hierarchy we propose a quantitative criterion for the average junction degree as a function of a fixed hierarchical order of the network and independent of the usual scaling laws the numerical results of this analysis are compared with exact results of the peano river network showing differences of the order of 10 ensuremath 3 this aspect is especially relevant for the characterization of transport and diffusion processes at the basin scale

global mean first passage times of random walks on complex networks we present a general framework applicable to a broad class of random walks on complex networks which provides a rigorous lower bound for the mean first passage time of a random walker to a target site averaged over its starting position the so called global mean first passage time gmfpt this bound is simply expressed in terms of the equilibrium distribution at the target and implies a minimal scaling of the gmfpt with the network size we show that this minimal scaling which can be arbitrarily slow is realized under the simple condition that the random walk is transient at the target site and independently of the small world scale free or fractal properties of the network last we put forward that the gmfpt to a specific target is not a representative property of the network since the target averaged gmfpt satisfies much more restrictive bounds

determining global mean first passage time of random walks on vicsek fractals using eigenvalues of laplacian matrices the family of vicsek fractals is one of the most important and frequently studied regular fractal classes and it is of considerable interest to understand the dynamical processes on this treelike fractal family in this paper we investigate discrete random walks on the vicsek fractals with the aim to obtain the exact solutions to the global mean first passage time gmfpt defined as the average of first passage time fpt between two nodes over the whole family of fractals based on the known connections between fpts effective resistance and the eigenvalues of graph laplacian we determine implicitly the gmfpt of the vicsek fractals which is corroborated by numerical results the obtained closed form solution shows that the gmfpt approximately grows as a power law function with system size number of all nodes with the exponent lies between 1 and 2 we then provide both the upper bound and lower bound for gmfpt of general trees and show that the leading behavior of the upper bound is the square of system size and the dominating scaling of the lower bound varies linearly with system size we also show that the upper bound can be achieved in linear chains and the lower bound can be reached in star graphs this study provides a comprehensive understanding of random walks on the vicsek fractals and general treelike networks

the class of suboptimal and optimal solutions to the discrete nehari problem characterization and computation simple and explicit formulae in a linear fractional transformation lft form for all suboptimal and optimal solutions to the discrete time version of the one block nehari problem are derived under no restrictive conditions imposed on the initial data using the so called signature condition a generalized popov theory type argument general solvability conditions in terms of a certain type of discrete time kalman szego popov yakubovich system are obtained for the class of optimal approximations derived via singular perturbation techniques the lft coefficients are all of mcmillan degree n r where n and r are the mcmillan degree and the multiplicity of the largest hankel singular value of the system to be approximated respectively the results also lead directly to a simple numerical algorithm

technical note analysis of a preference order traveling salesman problem application is made to the preference order dynamic programming solution procedure proposed by kao for a stochastic traveling salesman problem although the procedure is flawed from the myopic interpretation of the monotonicity condition it may be used as a convenient heuristic tool for solving stochastic problems

group theoretic reduction of laplacian dynamical problems on fractal lattices discrete forms of the schr umlt o dinger equation the diffusion equation the linearized landau ginzburg equation and discrete models for vibrations and spin dynamics belong to a class of laplacian based finite difference models real space renormalization of such models on finitely ramified regular fractals is known to give exact recursion relations it is shown that these recursions commute with lie groups representing continuous symmetries of the discrete models each such symmetry reduces the order of the renormalization recursions by one resulting in a system of recursions with one fewer variable group trajectories are obtained from inverse images of fixed and invariant sets of the recursions a subset of the laplacian finite difference models can be mapped by change of boundary conditions and time dependence to a diffusion problem with closed boundaries in such cases conservation of mass simplifies the group flow and obtaining the groups becomes easier to illustrate this the renormalization recursions for green functions on four standard examples are decoupled the examples are 1 the linear chain 2 an anisotropic version of dhar close quote s 3 simplex similar to a model dealt with by hood and southern 3 the fourfold coordinated sierpi acute n ski lattice of rammal and of domany ital et al and 4 more a form of the vicsek lattice prospects for applying the group theoretic method to more general dynamical systems are discussed copyright ital 1997 ital the american physical society less

determining mean first passage time on a class of treelike regular fractals relatively general techniques for computing mean first passage time mfpt of random walks on networks with a specific property are very useful since a universal method for calculating mfpt on general graphs is not available because of their complexity and diversity in this paper we present techniques for explicitly determining the partial mean first passage time pmfpt i e the average of mfpts to a given target averaged over all possible starting positions and the entire mean first passage time emfpt which is the average of mfpts over all pairs of nodes on regular treelike fractals we describe the processes with a family of regular fractals with treelike structure the proposed fractals include the t fractal and the peano basin fractal as their special cases we provide a formula for mfpt between two directly connected nodes in general trees on the basis of which we derive an exact expression for pmfpt to the central node in the fractals moreover we give a technique for calculating emfpt which is based on the relationship between characteristic polynomials of the fractals at different generations and avoids the computation of eigenvalues of the characteristic polynomials making use of the proposed methods we obtain analytically the closed form solutions to pmfpt and emfpt on the fractals and show how they scale with the number of nodes in addition to exhibit the generality of our methods we also apply them to the vicsek fractals and the iterative scale free fractal tree and recover the results previously obtained

explicit determination of mean first passage time for random walks on deterministic uniform recursive trees the determination of mean first passage time mfpt for random walks in networks is a theoretical challenge and is a topic of considerable recent interest within the physics community in this paper according to the known connections between mfpt effective resistance and the eigenvalues of graph laplacian we first study analytically the mfpt between all node pairs of a class of growing treelike networks which we term deterministic uniform recursive trees durts since one of its particular cases is a deterministic version of the famous uniform recursive tree the interesting quantity is determined exactly through the recursive relation of the laplacian spectra obtained from the special construction of durts the analytical result shows that the mfpt between all couples of nodes in durts varies as n text text ln text n for large networks with node number n second we study trapping on a particular network of durts focusing on a special case with the immobile trap positioned at a node having largest degree we determine exactly the average trapping time att that is defined as the average of fpt from all nodes to the trap in contrast to the scaling of the mfpt the leading behavior of att is a linear function of n interestingly we show that the behavior for att of the trapping problem is related to the trapping location which is in comparison with the phenomenon of trapping on fractal t graph although both networks exhibit tree structure finally we believe that the methods could open the way to exactly calculate the mfpt and att in a wide range of deterministic media

global first passage times of fractal lattices the global first passage time density of a network is the probability that a random walker released at a random site arrives at an absorbing trap at time t we find simple expressions for the mean global first passage time for five fractals the d dimensional sierpinski gasket t fractal hierarchical percolation model mandelbrot given curve and a deterministic tree we also find an exact expression for the second moment and show that the variance of the first passage time var t scales with the number of nodes within the fractal n such that var t similar to n 4 d where d is the spectral dimension

mean first passage time for random walk on dual structure of dendrimer abstract the random walk approach has recently been widely employed to study the relations between the underlying structure and dynamic of complex systems the mean first passage time mfpt for random walks is a key index to evaluate the transport efficiency in a given system in this paper we study analytically the mfpt in a dual structure of dendrimer network husimi cactus which has different application background and different structure contains loops from dendrimer by making use of the iterative construction we explicitly determine both the partial mean first passage time pmft the average of mfpts to a given target and the global mean first passage time gmft the average of mfpts over all couples of nodes on husimi cactus the obtained closed form results show that pmfpt and emfpt follow different scaling with the network order suggesting that the target location has essential influence on the transport efficiency finally the impact that loop structure could bring is analyzed and discussed

analysis of diffusion and trapping efficiency for random walks on non fractal scale free trees abstract in this paper the discrete random walks on the recursive non fractal scale free trees nfsft are studied and a kind of method to calculate the analytic solutions of the mean first passage time mfpt for any pair of nodes the mean trapping time mtt for any target node and mean diffusing time mdt for any starting node are proposed furthermore we compare the trapping efficiency and diffusion efficiency between any two nodes of nfsft by using the mtt and the mdt as the measures of trapping efficiency and diffusion efficiency respectively and find the best or worst trapping sites and the best or worst diffusion sites the results show that the two hubs of nfsft are not only the best trapping site but also the worst diffusion site and that the nodes which are the farthest nodes from the two hubs are not only the worst trapping sites but also the best diffusion sites furthermore we find that the ratio between the maximum and minimum of mtt grows logarithmically with network order but the ratio between the maximum and minimum of mdt is almost equal to 1 the results imply that the trap s position has great effect on the trapping efficiency but the position of starting node has little effect on diffusion efficiency finally the simulation for random walks on nfsft is done and it is consistent with the derived results

first passage times for random walks in bounded domains we present a novel computational method of first passage times between a starting site and a target site of regular bounded lattices we derive accurate expressions for all the moments of this first passage time validated by numerical simulations their range of validity is discussed we also consider the case of a starting site and two targets in addition we present the extension to continuous brownian motion these results are of great relevance to any system involving diffusion in confined media

fractal models for diffusion controlled aggregation a two dimensional fractal model is constructed for diffusion controlled deposition on a surface the fractal geometry of the deposit and the power law behaviour of the quantities characterising the non equilibrium cluster size distribution are shown to be consequences of the competition generally present in a nonlinear growth process a qualitative agreement with previous numerical results is found and the scaling laws for the critical exponents of the problem are shown to be satisfied exactly

exact mean first passage time on the t graph we consider a simple random walk on the t fractal and we calculate the exact mean time taug to first reach the central node i0 the mean is performed over the set of possible walks from a given origin and over the set of starting points uniformly distributed throughout the sites of the graph except i0 by means of analytic techniques based on decimation procedures we find the explicit expression for taug as a function of the generation g and of the volume v of the underlying fractal our results agree with the asymptotic ones already known for diffusion on the t fractal and more generally they are consistent with the standard laws describing diffusion on low dimensional structures

extended vicsek fractals laplacian spectra and their applications extended vicsek fractals evf are the structures constructed by introducing linear spacers into traditional vicsek fractals here we study the laplacian spectra of the evf in particularly the recurrence relations for the laplacian spectra allow us to obtain an analytic expression for the sum of all inverse nonvanishing laplacian eigenvalues this quantity characterizes the large scale properties such as the gyration radius of the polymeric structures or the global mean first passage time for the random walk processes introduction of the linear spacers leads to local heterogeneities which reveal themselves for example in the dynamics of evf under external forces

mean first passage time for random walks on generalized deterministic recursive trees we describe a technique that allows the exact analytical computation of the mean first passage time mfpt for infinite families of trees using their recursive properties the method is based in the relationship between the mfpt and the eigenvalues of the laplacian matrix of the trees but avoids their explicit computation we apply this technique to find the mfpt for a family of generalized deterministic recursive trees the method however can be adapted to other self similar tree families

distinct scalings for mean first passage time of random walks on scale free networks with the same degree sequence in general the power law degree distribution has profound influence on various dynamical processes defined on scale free networks in this paper we will show that power law degree distribution alone does not suffice to characterize the behavior of trapping problems on scale free networks which is an integral major theme of interest for random walks in the presence of an immobile perfect absorber in order to achieve this goal we study random walks on a family of one parameter denoted by q scale free networks with identical degree sequence for the full range of parameter q in which a trap is located at a fixed site we obtain analytically or numerically the mean first passage time mfpt for the trapping issue in the limit of large network order number of nodes for the whole class of networks the mfpt increases asymptotically as a power law function of network order with the exponent obviously different for different parameter q which suggests that power law degree distribution itself is not sufficient to characterize the scaling behavior of mfpt for random walks at least trapping problem performed on scale free networks

first passage properties of mortal random walks ballistic behavior effective reduction of dimensionality and scaling functions for hierarchical graphs we consider a mortal random walker on a family of hierarchical graphs in the presence of some trap sites the configuration comprising the graph the starting point of the walk and the locations of the trap sites is taken to be exactly self similar as one goes from one generation of the family to the next under these circumstances the total probability that the walker hits a trap is determined exactly as a function of the single step survival probability q of the mortal walker on the nth generation graph of the family this probability is shown to be given by the nth iterate of a certain scaling function or map q f q the properties of the map then determine in each case the behavior of the trapping probability the mean time to trapping the temporal scaling factor governing the random walk dimension on the graph and other related properties the formalism is illustrated for the cases of a linear hierarchical lattice and the sierpinski graphs in two and three euclidean dimensions we find an effective reduction of the random walk dimensionality due to the ballistic behavior of the surviving particles induced by the mortality constraint the relevance of this finding for experiments involving travel times of particles in diffusion decay systems is discussed

analytic expression for the mean time to absorption for a random walker on the sierpinski gasket the exact analytic expression for the mean time to absorption or mean walk length for a particle performing a random walk on a finite sierpinski gasket with a trap at one vertex is found to be t n 3 n 5 n 1 4 5 n ensuremath 3 n 3 n 1 1 where n denotes the generation index of the gasket and the mean is over a set of starting points of the walk distributed uniformly over all the other sites of the gasket in terms of the number n n of sites on the gasket and the spectral dimension stackrel ifmmode tilde else fi d of the gasket the precise asymptotic behavior for large n n is t n ensuremath rightarrow 1 3 2n n 2 stackrel ifmmode tilde else fi d ensuremath sim n 1 464 this serves as a partial check on our result as it is a intermediate between the known results t ensuremath sim n 2 d 1 and t ensuremath sim n mathrm ln n d 2 for random walks on d dimensional euclidean lattices and b consistent with the known result for the asymptotic behavior of the mean number of distinct sites visited in a random walk on a fractal lattice

species survival and scaling laws in hostile and disordered environments in this work we study the likelihood of survival of single species in the context of hostile and disordered environments population dynamics in this environment as modeled by the fisher equation is characterized by negative average growth rate except in some random spatially distributed patches that may support life in particular we are interested in the phase diagram of the survival probability and in the critical size problem i e the minimum patch size required for surviving in the long time dynamics we propose a measure for the critical patch size as being proportional to the participation ratio of the eigenvector corresponding to the largest eigenvalue of the linearized fisher dynamics we obtain the extinction survival phase diagram and the probability distribution function pdf of the critical patch sizes for two topologies namely the one dimensional system and the fractal peano basin we show that both topologies share the same qualitative features but the fractal topology requires higher spatial fluctuations to guarantee species survival we perform a finite size scaling and we obtain the associated scaling exponents in addition we show that the pdf of the critical patch sizes has an universal shape for the 1d case in terms of the model parameters diffusion growth rate etc in contrast the diffusion coefficient has a drastic effect on the pdf of the critical patch sizes of the fractal peano basin and it does not obey the same scaling law of the 1d case