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

the average distances in random graphs with given expected degrees abstract random graph theory is used to examine the small world phenomenon any two strangers are connected through a short chain of mutual acquaintances we will show that for certain families of random graphs with given expected degrees the average distance is almost surely of order log n log d where d is the weighted average of the sum of squares of the expected degrees of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1 k for some fixed exponent for the case of 3 we prove that the average distance of the power law graphs is almost surely of order log n log d however many internet social and citation networks are power law graphs with exponents in the range 2 3 for which the power law random graphs have average distance almost surely of order log log n but have diameter of order log n provided having some mild constraints for the average distance and maximum degree in particular these graphs contain a dense subgraph which we call the core having nc log log n vertices almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core

two step estimation for inhomogeneous spatial point processes summary the paper is concerned with parameter estimation for inhomogeneous spatial point processes with a regression model for the intensity function and tractable second order properties k function regression parameters are estimated by using a poisson likelihood score estimating function and in the second step minimum contrast estimation is applied for the residual clustering parameters asymptotic normality of parameter estimates is established under certain mixing conditions and we exemplify how the results may be applied in ecological studies of rainforests

a deterministic small world network created by edge iterations abstract small world networks are ubiquitous in real life systems most previous models of small world networks are stochastic the randomness makes it more difficult to gain a visual understanding on how do different nodes of networks interact with each other and is not appropriate for communication networks that have fixed interconnections here we present a model that generates a small world network in a simple deterministic way our model has a discrete exponential degree distribution we solve the main characteristics of the model

exact solution for mean first passage time on a pseudofractal scale free web the explicit determinations of the mean first passage time mfpt for trapping problem are limited to some simple structure e g regular lattices and regular geometrical fractals and determining mfpt for random walks on other media especially complex real networks is a theoretical challenge in this paper we investigate a simple random walk on the the pseudofractal scale free web psfw with a perfect trap located at a node with the highest degree which simultaneously exhibits the remarkable scale free and small world properties observed in real networks we obtain the exact solution for the mfpt that is calculated through the recurrence relations derived from the structure of psfw the rigorous solution exhibits that the mfpt approximately increases as a power law function of the number of nodes with the exponent less than 1 we confirm the closed form solution by direct numerical calculations we show that the structure of psfw can improve the efficiency of transport by diffusion compared with some other structure such as regular lattices sierpinski fractals and t graph the analytical method can be applied to other deterministic networks making the accurate computation of mfpt possible

random spread on the family of small world networks we present analytical and numerical results of a random walk on the family of small world graphs the average access time shows a crossover from regular to random behavior with increasing distance from the starting point of the random walk we introduce an independent step approximation which enables us to obtain analytic results for the average access time we observe a scaling relation for the average access time in the degree of the nodes the behavior of the average access time as a function of p shows striking similarity with that of the characteristic length of the graph this observation may have important applications in routing and switching in networks with a large number of nodes

random walks in modular scale free networks with multiple traps extensive empirical investigation has shown that a plethora of real networks synchronously exhibit scale free and modular structure and it is thus of great importance to uncover the effects of these two striking properties on various dynamical processes occurring on such networks in this paper we examine two cases of random walks performed on a class of modular scale free networks with multiple traps located at several given nodes we first derive a formula of the mean first passage time mfpt for a general network which is the mean of the expected time to absorption originating from a specific node averaged over all nontrap starting nodes although the computation is complex the expression of the formula is exact moreover the computational approach and procedure are independent of the number and position of the traps we then determine analytically the mfpt for the two random walks being considered the obtained analytical results are in complete agreement with the numerical ones our results show that the number and location of traps play an important role in the behavior of the mfpt since for both cases the mfpt grows as a power law function of the number of nodes but their exponents are quite different we demonstrate that the root of the difference in the behavior of mfpt is attributed to the modular and scale free topologies of the networks this work can deepen the understanding of diffusion on networks with modular and scale free architecture and motivate relevant studies for random walks running on complex random networks with multiple traps

constrained spin dynamics description of random walks on hierarchical scale free networks we study a random walk problem on the hierarchical network which is a scale free network grown deterministically the random walk problem is mapped onto a dynamical ising spin chain system in one dimension with a nonlocal spin update rule which allows an analytic approach we show analytically that the characteristic relaxation time scale grows algebraically with the total number of nodes n as t n z from a scaling argument we also show the power law decay of the autocorrelation function c sigma t t alpha which is the probability to find the ising spins in the initial state sigma after t time steps with the state dependent nonuniversal exponent alpha it turns out that the power law scaling behavior has its origin in a quasiultrametric structure of the configuration space

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

griffiths singularities and algebraic order in the exact solution of an ising model on a fractal modular network we use an exact renormalization group transformation to study the ising model on a complex network composed of tightly knit communities nested hierarchically with the fractal scaling recently discovered in a variety of real world networks varying the ratio k j of intercommunity to intracommunity couplings we obtain an unusual phase diagram at high temperatures or small k j we have a disordered phase with a griffiths singularity in the free energy due to the presence of rare large clusters which we analyze through the yang lee zeros in the complex magnetic field plane as the temperature is lowered true long range order is not seen but there is a transition to algebraic order where pair correlations have power law decay with distance reminiscent of the xy model the transition is infinite order at small k j and becomes second order above a threshold value k j m the existence of such slowly decaying correlations is unexpected in a fat tailed scale free network where correlations longer than nearest neighbor are typically suppressed

modeling complex networks with self similar outerplanar unclustered graphs this paper introduces a family of modular self similar small world graphs with clustering zero relevant properties of this family are comparable to those of some networks associated with technological systems with a low clustering like the power grid or some electronic circuits moreover the graphs are outerplanar and it is know that many algorithms that are np complete for general graphs perform polynomial in outerplanar graphs therefore the graphs constitute a good mathematical model for these systems

standard random walks and trapping on the koch network with scale free behavior and small world effect a vast variety of real life networks display the ubiquitous presence of scale free phenomenon and small world effect both of which play a significant role in the dynamical processes running on networks although various dynamical processes have been investigated in scale free small world networks analytical research about random walks on such networks is much less in this paper we will study analytically the scaling of the mean first passage time mfpt for random walks on scale free small world networks to this end we first map the classical koch fractal to a network called koch network according to this proposed mapping we present an iterative algorithm for generating the koch network based on which we derive closed form expressions for the relevant topological features such as degree distribution clustering coefficient average path length and degree correlations the obtained solutions show that the koch network exhibits scale free behavior and small world effect then we investigate the standard random walks and trapping issue on the koch network through the recurrence relations derived from the structure of the koch network we obtain the exact scaling for the mfpt we show that in the infinite network order limit the mfpt grows linearly with the number of all nodes in the network the obtained analytical results are corroborated by direct extensive numerical calculations in addition we also determine the scaling efficiency exponents characterizing random walks on the koch network

target problem on small world networks in this work we focus on reactions on small world networks swn s disordered graphs of much recent interest we study the target problem since it allows an exact solution on regular lattices on swn s we find that the decay of the targets for which we extend the formalism to disordered lattices is again related to s n the mean number of distinct sites visited in n steps although the s n vs n dependence changes here drastically in going from regular linear chains to their swn

efficiency of trapping processes in regular and disordered networks we use a markov method to study the efficiency of trapping processes involving both a random walker and a deep trap in regular and disordered networks the efficiency is gauged by the mean absorption time average of the mean number of steps performed by the random walker before being absorbed by the trap we compute this quantity in terms of different control parameters namely the length of the walker jumps the mobility of the trap and the degree of spatial disorder of the network for a proper choice of the system size we find in all cases a nonmonotonic behavior of the efficiency in terms of the corresponding control parameter we thus arrive at the conclusion that despite the decrease of the effective system size underlying the increase of the control parameter the efficiency is reduced as a result of an increase of the escape probability of the walker once it finds itself in the interaction zone of the trap this somewhat anti intuitive effect is very robust in the sense that it is observed regardless of the specific choice of the control parameter for the case of a ring lattice results for the mean absorption time in systems of arbitrary size are given in terms of a two parameter scaling function for the case of a mobile trap we deal with both trapping via a single channel walker trap overlap and via two channels walker trap overlap and walker trap crossing thereby generalizing previous work as for the disordered case our analysis concerns small world networks for which we see several crossovers of the absorption time as a function of the control parameter and the system size the methodology used may be well suited to exploring characteristic time scales of encounter controlled phenomena in networks with a few interacting elements and the effect of geometric constraints in nanoscale systems with a very small number of particles

random walks on deterministic scale free networks exact results we study the random walk problem on a class of deterministic scale free networks displaying a degree sequence for hubs scaling as a power law with an exponent gamma log 3 log 2 we find exact results concerning different first passage phenomena and in particular we calculate the probability of first return to the main hub these results allow to derive the exact analytic expression for the mean time to first reach the main hub whose leading behavior is given by tau approximately v1 1 gamma where v denotes the size of the structure and the mean is over a set of starting points distributed uniformly over all the other sites of the graph interestingly the process turns out to be particularly efficient we also discuss the thermodynamic limit of the structure and some local topological properties

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 p variation of partial sum processes and the empirical process the p variation of a function f is the supremum of the sums of the p th powers of absolute increments of f over nonoverlapping intervals let f be a continuous probability distribution function dudley has shown that the p variation of the empirical process is bounded in probability as n to infty if and only if p 2 and for 1 leq p leq 2 the p variation of the empirical process is at least n 1 p 2 and is at most of the order n 1 p 2 log log n p 2 in probability in this paper we prove that the exact order of the 2 variation of the empirical process is log log n in probability and for 1 leq p 2 the p variation of the empirical process is of exact order n 1 p 2 in expectation and almost surely let s j x 1 x 2 dots x j then the p variation of the partial sum process for x 1 x 2 dots x n is defined as that of f on 0 n where f t s j for j 1 t leq j j 1 2 dots n bretagnolle has shown that the expectation of the p variation for independent centered random variables x i with bounded p th moments is of order n for 1 leq p 2 we prove that for p 2 the 2 variation of the partial sum process of i i d centered nonconstant random variables with finite 2 delta moment for some delta 0 is of exact order n log log n in probability

trapping in scale free networks with hierarchical organization of modularity a wide variety of real life networks share two remarkable generic topological properties scale free behavior and modular organization and it is natural and important to study how these two features affect the dynamical processes taking place on such networks in this paper we investigate a simple stochastic process trapping problem a random walk with a perfect trap fixed at a given location performed on a family of hierarchical networks that exhibit simultaneously striking scale free and modular structure we focus on a particular case with the immobile trap positioned at the hub node having the largest degree using a method based on generating functions we determine explicitly the mean first passage time mfpt for the trapping problem which is the mean of the node to trap first passage time over the entire network the exact expression for the mfpt is calculated through the recurrence relations derived from the special construction of the hierarchical networks the obtained rigorous formula corroborated by extensive direct numerical calculations exhibits that the mfpt grows algebraically with the network order concretely the mfpt increases as a power law function of the number of nodes with the exponent much less than 1 we demonstrate that the hierarchical networks under consideration have more efficient structure for transport by diffusion in contrast with other analytically soluble media including some previously studied scale free networks we argue that the scale free and modular topologies are responsible for the high efficiency of the trapping process on the hierarchical networks

exact scaling properties of a hierarchical network model we report on the exact results for the degree k the diameter d the clustering coefficient c and the betweenness centrality b of a hierarchical network model with a replication factor m such quantities are calculated exactly with the help of recursion relations using the results we show that i the degree distribution follows a power law p k approximately k gamma with gamma 1 ln m ln m 1 ii the diameter grows logarithmically as d approximately ln n with the number of nodes n iii the clustering coefficient of each node is inversely proportional to its degree c proportional variant 1 k and the average clustering coefficient is nonzero in the infinite n limit and iv the betweenness centrality distribution follows a power law p b approximately b 2 we discuss a classification scheme of scale free networks into the universality class with the clustering property and the betweenness centrality distribution

order statistics of rosenstock s trapping problem in disordered media the distribution of times t j n elapsed until the first j independent random walkers from a set of n ensuremath gg 1 all starting from the same site are trapped by a quenched configuration of traps randomly placed on a disordered lattice is investigated in doing so the cumulants of the distribution of the territory explored by n independent random walkers s n t and the probability ensuremath phi n t that no particle of an initial set of n is trapped by time t are considered simulation results for the two dimensional incipient percolation aggregate show that the ratio between the nth cumulant and the nth moment of s n t is for large n i very large in comparison with the same ratio in euclidean media and ii almost constant the first property implies that in contrast with euclidean media approximations of the order higher than the standard zeroth order rosenstock approximation are required to provide a reasonable description of the trapping order statistics fortunately the second property which has a geometric origin can be exploited to build these higher order rosenstock approximations simulation results for the two dimensional incipient percolation aggregate confirm the predictions of our approach

exactly solvable scale free network model we study a deterministic scale free network recently proposed by barabasi ravasz and vicsek we find that there are two types of nodes the hub and rim nodes which form a bipartite structure of the network we first derive the exact numbers p k of nodes with degree k for the hub and rim nodes in each generation of the network respectively using this we obtain the exact exponents of the distribution function p k of nodes with k degree in the asymptotic limit of k infinity we show that the degree distribution for the hub nodes exhibits the scale free nature p k proportional to k gamma with gamma ln 3 ln 2 1 584 962 while the degree distribution for the rim nodes is given by p k proportional to e gamma k with gamma ln 3 2 0 405 465 second we analytically calculate the second order average degree of nodes d third we numerically as well as analytically calculate the spectra of the adjacency matrix a for representing topology of the network we also analytically obtain the exact number of degeneracies at each eigenvalue in the network the density of states i e the distribution function of eigenvalues exhibits the fractal nature with respect to the degeneracy fourth we study the mathematical structure of the determinant of the eigenequation for the adjacency matrix fifth we study hidden symmetry zero modes and its index theorem in the deterministic scale free network finally we study the nature of the maximum eigenvalue in the spectrum of the deterministic scale free network we will prove several theorems for it using some mathematical theorems thus we show that most of all important quantities in the network theory can be analytically obtained in the deterministic scale free network model of barabasi ravasz and vicsek therefore we may call this network model the exactly solvable scale free network

zero constant formula for first passage observables in bounded domains in this letter we develop an analytical approach which provides an explicit determination of mean first passage times mfpts for random walks in bounded domains for a wide class of transport processes in particular we derive for the first time explicit expressions of mfpts for emblematic models of transport in complex media such as diffusion on deterministic and random fractals this approach relies on a scale invariance hypothesis and a large volume expansion of the mfpt which actually proves to be very accurate even for small system sizes as shown by numerical simulations this explicit determination of mfpts can be straightforwardly generalized to further useful first passage observables such as occupation times and splitting probabilities

effective target arrangement in a deterministic scale free graph we study the random walk problem on a deterministic scale free network in the presence of a set of static identical targets due to the strong inhomogeneity of the underlying structure the mean first passage time mfpt meant as a measure of transport efficiency is expected to depend sensitively on the position of targets we consider several spatial arrangements for targets and we calculate mainly rigorously the related mfpt where the average is taken over all possible starting points and over all possible paths for all the cases studied the mfpt asymptotically scales like ensuremath sim n ensuremath theta being n the volume of the substrate and ensuremath theta ranging from 1 ensuremath text log text 2 text log text 3 for central target s to 1 for a single peripheral target

scaling theory of transport in complex biological networks transport is an important function in many network systems and understanding its behavior on biological social and technological networks is crucial for a wide range of applications however it is a property that is not well understood in these systems probably because of the lack of a general theoretical framework here based on the finding that renormalization can be applied to bionetworks we develop a scaling theory of transport in self similar networks we demonstrate the networks invariance under length scale renormalization and we show that the problem of transport can be characterized in terms of a set of critical exponents the scaling theory allows us to determine the influence of the modular structure on transport in metabolic and protein interaction networks we also generalize our theory by presenting and verifying scaling arguments for the dependence of transport on microscopic features such as the degree of the nodes and the distance between them using transport concepts such as diffusion and resistance we exploit this invariance and we are able to explain based on the topology of the network recent experimental results on the broad flow distribution in metabolic networks

geometric fractal growth model for scale free networks we introduce a deterministic model for scale free networks whose degree distribution follows a power law with the exponent ensuremath gamma at each time step each vertex generates its offspring whose number is proportional to the degree of that vertex with proportionality constant m ensuremath 1 mg1 we consider the two cases first each offspring is connected to its parent vertex only forming a tree structure second it is connected to both its parent and grandparent vertices forming a loop structure we find that both models exhibit power law behaviors in their degree distributions with the exponent ensuremath gamma 1 mathrm ln 2m ensuremath 1 mathrm ln m thus by tuning m the degree exponent can be adjusted in the range 2l ensuremath gamma l3 we also solve analytically a mean shortest path distance d between two vertices for the tree structure showing the small world behavior that is d ensuremath sim mathrm ln n mathrm ln overline k where n is system size and overline k is the mean degree finally we consider the case that the number of offspring is the same for all vertices and find that the degree distribution exhibits an exponential decay behavior

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

influences of degree inhomogeneity on average path length and random walks in disassortative scale free networks various real life networks exhibit degree correlations and heterogeneous structure with the latter being characterized by power law degree distribution p k k where the degree exponent describes the extent of heterogeneity in this paper we study analytically the average path length apl of and random walks rws on a family of deterministic networks recursive scale free trees rsfts with negative degree correlations and various 2 1 ln 3 ln 2 with an aim to explore the impacts of structure heterogeneity on the apl and rws we show that the degree exponent has no effect on the apl d of rsfts in the full range of d behaves as a logarithmic scaling with the number of network nodes n i e d ln n which is in sharp contrast to the well known double logarithmic scaling d ln ln n previously obtained for uncorrelated scale free networks with 2 3 in addition we present that some scaling efficiency exponents of random walks are reliant on the degree exponent

close or connected distance and connectivity effects on transport in networks laboratoire de physique th eorique de la mati ere condens ee umr cnrs upmc universit e pierre et marie curie 4 place jussieu 75255 paris cedex dated february 25 2011 we develop an analytical approach which provides the dependence of the mean rst passage time mfpt for random walks on complex networks both on the target connectivity and on the source target distance our approach puts forward two strongly di erent behaviors depending on the type compact or non compact of the random walk in the case of non compact exploration we showthat the mfpt scales linearly with the inverse connectivity of the target and is largely independentof the starting point on the contrary in the compact case the mfpt is controlled by the source target distance and we nd that unexpectedly the target connectivity becomes irrelevant for remotetargets

random walks on complex trees we study the properties of random walks on complex trees we observe that the absence of loops is reflected in physical observables showing large differences with respect to their looped counterparts first both the vertex discovery rate and the mean topological displacement from the origin present a considerable slowing down in the tree case second the mean first passage time mfpt displays a logarithmic degree dependence in contrast to the inverse degree shape exhibited in looped networks this deviation can be ascribed to the dominance of source target topological distance in trees to show this we study the distance dependence of a symmetrized mfpt and derive its logarithmic profile obtaining good agreement with simulation results these unique properties shed light on the recently reported anomalies observed in diffusive dynamical systems on trees

pseudofractal scale free web we find that scale free random networks are excellently modeled by simple deterministic graphs our graph has a discrete degree distribution degree is the number of connections of a vertex which is characterized by a power law with exponent gamma 1 ln 3 ln 2 properties of this compact structure are surprisingly close to those of growing random scale free networks with gamma in the most interesting region between 2 and 3 we succeed to find exactly and numerically with high precision all main characteristics of the graph in particular we obtain the exact shortest path length distribution for a large network ln n 1 the distribution tends to a gaussian of width approximately sqrt ln n centered at l approximately ln n we show that the eigenvalue spectrum of the adjacency matrix of the graph has a power law tail with exponent 2 gamma

hierarchical regular small world networks two new networks are introduced that resemble small world properties these networks are recursively constructed but retain a fixed regular degree they possess a unique one dimensional lattice backbone overlaid by a hierarchical sequence of long distance links mixing real space and small world features both networks one 3 regular and the other 4 regular lead to distinct behaviors as revealed by renormalization group studies the 3 regular network is planar has a diameter growing as with system size n and leads to super diffusion with an exact anomalous exponent dw 1 306 but possesses only a trivial fixed point tc 0 for the ising ferromagnet in turn the 4 regular network is non planar has a diameter growing as exhibits ballistic diffusion dw 1 and a non trivial ferromagnetic transition tc 0 it suggests that the 3 regular network is still quite geometric while the 4 regular network qualifies as a true small world with mean field properties as an engineering application we discuss synchronization of processors on these networks

first passage properties of the erdos renyi random graph we study the mean time for a random walk to traverse between two arbitrary sites of the erd s renyi random graph we develop an effective medium approximation that predicts that the mean first passage time between pairs of nodes as well as all moments of this first passage time are insensitive to the fraction p of occupied links this prediction qualitatively agrees with numerical simulations away from the percolation threshold near the percolation threshold the statistically meaningful quantity is the mean transit rate namely the inverse of the first passage time this rate varies non monotonically with p near the percolation transition much of this behaviour can be understood by simple heuristic arguments

planar unclustered scale free graphs as models for technological and biological networks many real life networks present an average path length logarithmic with the number of nodes and a degree distribution which follows a power law often these networks have also a modular and self similar structure and in some cases usually associated with topological restrictions their clustering is low and they are almost planar in this paper we introduce a family of graphs which share all these properties and are defined by two parameters as their construction is deterministic we obtain exact analytic expressions for relevant properties of the graphs including the degree distribution degree correlation diameter and average distance as a function of the two defining parameters thus the graphs are useful to model some complex networks in particular several families of technological and biological networks and in the design of new practical communication algorithms in relation to their dynamical processes they can also help understanding the underlying mechanisms that have produced their particular structure

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

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

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

inverted berezinskii kosterlitz thouless singularity and high temperature algebraic order in an ising model on a scale free hierarchical lattice small world network we have obtained exact results for the ising model on a hierarchical lattice incorporating three key features characterizing many real world networks a scale free degree distribution a high clustering coefficient and the small world effect by varying the probability p of long range bonds the entire spectrum from an unclustered non small world network to a highly clustered small world system is studied using the self similar structure of the network we obtain analytic expressions for the degree distribution p k and clustering coefficient c for all p as well as the average path length ensuremath ell for p 0 and 1 the ferromagnetic ising model on this network is studied through an exact renormalization group transformation of the quenched bond probability distribution using up to 562 phantom rule 0 2em 0ex 500 renormalized probability bins to represent the distribution for pl0 494 we find power law critical behavior of the magnetization and susceptibility with critical exponents continuously varying with p and exponential decay of correlations away from t c for p ensuremath ge 0 494 in fact where the network exhibits small world character the critical behavior radically changes we find a highly unusual phase transition namely an inverted berezinskii kosterlitz thouless singularity between a low temperature phase with nonzero magnetization and finite correlation length and a high temperature phase with zero magnetization and infinite correlation length with power law decay of correlations throughout the phase approaching t c from below the magnetization and the susceptibility respectively exhibit the singularities of mathrm exp ensuremath c sqrt t c ensuremath t and mathrm exp d sqrt t c ensuremath t with c and d positive constants with long range bond strengths decaying with distance we see a phase transition with power law critical singularities for all p and evaluate an unusually narrow critical region and important corrections to power law behavior that depend on the exponent characterizing the decay of long range interactions