In this paper, we consider the random walk process on a kind of fractal (or transfractal) scale free networks, which also called as $(u,v)$ flowers, and we focus on the global first passage time (GFPT) and first return time (FRT). Here, we present method to derive exactly the probability generation function, mean and variance of the GFPT and FRT for a given hub (i.e., node with the highest degree) and then the scaling properties of the mean and the variance of the GFPT and FRT are disclosed. Our results show that, for the case of $u>1$, while the networks are fractals, the mean of the GFPT scales with the volume of the network as $\langle GMFPT_t\rangle \sim N_t^{{2}/{d_s}}$, where $\langle * \rangle$ denotes the mean of random variable $*$, $N_t$ is the volume of the network with generation $t$ and $d_s$ is the spectral dimension of the network; but, for the case of $u=1$, while the networks are nonfractals, the mean of the GFPT scales as $\langle GMFPT_t\rangle \sim N_t^{{2}/{\tilde{d}_s}}$, where $\tilde{d}_s$ is the transspectral dimension of the network, which is introduced in this paper. Results also show that, the variance of the GFPT scales as $Var(GFPT_t) \sim \langle GFPT_{t}\rangle^2$, where $Var(*)$ denotes variance of of random variable $*$; whereas the variance of the FRT scales as $Var(FRT_t) \sim \langle FRT_{t} \rangle \langle GFPT_{t}\rangle$. %$Var(GFPT_t) \sim \langle GFPT_{t}\rangle^2$ and $Var(FRT_t) \sim \langle FRT_{t} \rangle \langle GFPT_{t}\rangle$, where $Var(GFPT_t)$ and $Var(FRT_t)$ denote the variance of the GFPT and the FRT, $\langle GFPT_{t}\rangle$ and $\langle FRT_{t}\rangle$ denote the mean of the GFPT and the FRT respectively.
Fractal (or transfractal) features are common in real-life networks and are known to influence the dynamic processes taking place in the network itself. Here we consider a class of scale-free deterministic networks, called $(u,v)$-flowers, whose topological properties can be controlled by tuning the parameters $u$ and $v$; in particular, for $u>1$, they are fractals endowed with a fractal dimension $d_f$, while for $u=1$, they are transfractal endowed with a transfractal dimension $\tilde{d}_f$. In this work we investigate dynamic processes (i.e., random walks) and topological properties (i.e.…
We explore the concepts of self-similarity, dimensionality, and (multi)scaling in a new family of recursive scale-free nets that yield themselves to exact analysis through renormalization techniques. All nets in this family are self-similar and some are fractals - possessing a finite fractal dimension - while others are small world (their diameter grows logarithmically with their size) and are infinite-dimensional. We show how a useful measure of "transfinite" dimension may be defined and applied to the small world nets. Concerning multiscaling, we show how first-passage time for diffusion and…
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 <T> for five fractals. We also find an exact expression for the second moment <T^2> 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) ~ N^(4/\bar{d}), where \bar{d} is the spectral dimension.
We present general methods to exactly calculate mean-first passage quantities on self-similar networks defined recursively. In particular, we calculate the mean first-passage time and the splitting probabilities associated to a source and one or several targets; averaged quantities over a given set of sources (e.g., same-connectivity nodes) are also derived. The exact estimate of such quantities highlights the dependency of first-passage processes with respect to the source-target distance, which has recently revealed to be a key parameter to characterize transport in complex media. We explici…
In this work we consider a class of recursively-grown fractal networks $G_n(t)$, whose topology is controlled by two integer parameters $t$ and $n$. We first analyse the structural properties of $G_n(t)$ (including fractal dimension, modularity and clustering coefficient) and then we move to its transport properties. The latter are studied in terms of first-passage quantities (including the mean trapping time, the global mean first-passage time and the Kemeny's constant) and we highlight that their asymptotic behavior is controlled by network's size and diameter. Remarkably, if we tune $n$ (or…
For transport processes in geometrically restricted domains, the mean first-passage time (MFPT) admits a general scaling dependence on space parameters for diffusion, anomalous diffusion, and diffusion in disordered or fractal media. For transport in self-similar fractal structures, we obtain a new expression for the source-target distance dependence of the MFPT that exhibits both the leading power law behavior, depending on the Hausdorff and spectral dimension of the fractal, as well as small log periodic oscillations that are a clear and definitive signal of the underlying fractal structure.…
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 determi…
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 employ simulations of model proteins to study folding on rugged energy landscapes. We construct ``first-passage'' networks as the system transitions from unfolded to native states. The nodes and bonds in these networks correspond to basins and transitions between them in the energy landscape. We find power-laws between the folding time and number of nodes and bonds. We show that these scalings are determined by the fractal properties of first-passage networks. Reliable folding is possible in systems with rugged energy landscapes because first passage networks have small fractal dimension.
We study the first passage time processes of anomalous diffusion on self similar curves in two dimensions. The scaling properties of the mean square displacement and mean first passage time of the ballistic motion, fractional Brownian motion and subordinated walk on different fractal curves (loop erased random walk, harmonic explorer and percolation front) are derived. We also define natural parametrized subordinated Schramm Loewner evolution (NS-SLE) as a mathematical tool that can model diffusion on fractal curves. The scaling properties of the mean square displacement and mean first passage…
The regular hyperbranched polymers (RHPs), also known as Vicsek fractals, are an important family of hyperbranched structures which have attracted a wide spread attention during the past several years. In this paper, we study the first-passage properties for random walks on the RHPs. Firstly, we propose a way to label all the different nodes of the RHPs and derive exact formulas to calculate the mean first-passage time (MFPT) between any two nodes and the mean trapping time (MTT) for any trap node. Then, we compare the trapping efficiency between any two nodes of the RHPs by using the MTT as t…
The first return time (FRT) is the time it takes a random walker to first return to its original site, and the global first passage time (GFPT) is the first passage time for a random walker to move from a randomly selected site to a given site. We find that in finite networks the variance of FRT, Var(FRT), can be expressed Var(FRT)~$=2\langle$FRT$ \rangle \langle $GFPT$ \rangle -\langle $FRT$ \rangle^2-\langle $FRT$ \rangle$, where $\langle \cdot \rangle$ is the mean of the random variable. Therefore a method of calculating the variance of FRT on general finite networks is presented. We then c…
A human pathogen, \textit{Neisseria gonorrhoeae} (NG), moves on surfaces by attaching and retracting polymeric structures called Type IV pili. The \textit{tug-of-war} between the pili results in a two-dimensional stochastic motion called \textit{twitching motility}. In this paper, with the help of real time NG trajectories, we develop coarse-grained models for their description. The \textit{fractal properties} of these trajectories are determined and their influence on \textit{first passage time} and formation of bacterial microcolonies is studied. Our main observations are as follows: (i) NG …
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 fo…
In the 1980s an important goal of the emergent field of fractals was to determine the relationships between their physical and geometrical properties. The fractal-Einstein and Alexander-Orbach laws, which interrelate electrical, diffusive and fractal properties, are two key theories of this type. Here we settle a long standing controversy about their exactness by showing that the properties of a class of fractal trees violate both laws. A new formula is derived which unifies the two classical results by proving that if one holds, then so must the other, and resolves a puzzling discrepancy in t…
We propose a model for anomalous transport in inhomogeneous environments, such as fractured rocks, in which particles move only along pre-existing self-similar curves (cracks). The stochastic Loewner equation is used to efficiently generate such curves with tunable fractal dimension $d_f$. We numerically compute the probability of first passage (in length or time) from one point on the edge of the semi-infinite plane to any point on the semi-circle of radius $R$. The scaled probability distributions have a variance which increases with $d_f$, a non-monotonic skewness, and tails that decay fast…
We study the mean first passage time of a one-dimensional random walker with step sizes decaying exponentially in discrete time. That is step sizes go like $λ^{n}$ with $λ\leq1$ . We also present, for pedagogical purposes, a continuum system with a diffusion constant decaying exponentially in continuous time. Qualitatively both systems are alike in their global properties. However, the discrete case shows very rich mathematical structure, depending on the value of the shrinking parameter, such as self-repetitive and fractal-like structure for the first passage characteristics. The results we p…
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 for a proper choice of highly connected target, is realized under the simple condition t…
It is known that the heterogeneity of scale-free networks helps enhancing the efficiency of trapping processes performed on them. In this paper, we show that transport efficiency is much lower in a fractal scale-free network than in non-fractal networks. To this end, we examine a simple random walk with a fixed trap at a given position on a fractal scale-free network. We calculate analytically the mean first-passage time (MFPT) as a measure of the efficiency for the trapping process, and obtain a closed-form expression for MFPT, which agrees with direct numerical calculations. We find that, in…
For a random walk on a network, the mean first-passage time from a node $i$ to another node $j$ chosen stochastically according to the equilibrium distribution of Markov chain representing the random walk is called Kemeny constant, which is closely related to the navigability on the network. Thus, the configuration of a network that provides optimal or suboptimal navigation efficiency is a question of interest. It has been proved that complete graphs have the exact minimum Kemeny constant over all graphs. In this paper, by using another method we first prove that complete graphs are the optima…
We investigate the multifractals of the normalized first passage time on one-dimensional small-world network with both reflecting and absorbing barriers. The multifractals is estimated from the distribution of the normalized first passage time charactrized by the random walk on the small-world network with three fractions of edges rewired randomly. Particularly, our estimate is the fractal dimension D_0 = 0.917, 0.926, 0.930 for lattice points L = 80 and a randomly rewired fraction p = 0.2. The numerical result is found to disappear multifractal properties in the regime p> p_c, where p_c is th…
Subdiffusive motion of tracer particles in complex crowded environments, such as biological cells, has been shown to be widepsread. This deviation from brownian motion is usually characterized by a sublinear time dependence of the mean square displacement (MSD). However, subdiffusive behavior can stem from different microscopic scenarios, which can not be identified solely by the MSD data. In this paper we present a theoretical framework which permits to calculate analytically first-passage observables (mean first-passage times, splitting probabilities and occupation times distributions) in di…
How long does it take a random walker to reach a given target point? This quantity, known as a first passage time (FPT), has led to a growing number of theoretical investigations over the last decade1. The importance of FPTs originates from the crucial role played by first encounter properties in various real situations, including transport in disordered media, neuron firing dynamics, spreading of diseases or target search processes. Most methods to determine the FPT properties in confining domains have been limited to effective 1D geometries, or for space dimensions larger than one only to ho…
We derive a general exact formula for the mean first passage time (MFPT) from a fixed point inside a planar domain to an escape region on its boundary. The underlying mixed Dirichlet-Neumann boundary value problem is conformally mapped onto the unit disk, solved exactly, and mapped back. The resulting formula for the MFPT is valid for an arbitrary space-dependent diffusion coefficient, while the leading logarithmic term is explicit, simple, and remarkably universal. In contrast to earlier works, we show that the natural small parameter of the problem is the harmonic measure of the escape regio…
Explicit determination of the mean first-passage time (MFPT) for trapping problem on complex media is a theoretical challenge. In this paper, we study random walks on the Apollonian network with a trap fixed at a given hub node (i.e. node with the highest degree), which are simultaneously scale-free and small-world. We obtain the precise analytic expression for the MFPT that is confirmed by direct numerical calculations. In the large system size limit, the MFPT approximately grows as a power-law function of the number of nodes, with the exponent much less than 1, which is significantly differe…
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…
In this paper, we investigate random walks in a family of small-world trees having an exponential degree distribution. First, we address a trapping problem, that is, a particular case of random walks with an immobile trap located at the initial node. We obtain the exact mean trapping time defined as the average of first-passage time (FPT) from all nodes to the trap, which scales linearly with the network order $N$ in large networks. Then, we determine analytically the mean sending time, which is the mean of the FPTs from the initial node to all other nodes, and show that it grows with $N$ in t…
We study numerically the geometrical properties of minimally weighted paths that appear in the negative-weight percolation (NWP) model on two-dimensional lattices assuming a combination of periodic and free boundary conditions (BCs). Each realization of the disorder consists of a random fraction 1-rho of bonds with unit strength and a fraction rho of bond strengths drawn from a Gaussian distribution with zero mean and unit width. For each such sample, the path is forced to span the lattice along the direction with the free BCs. The path and a set of negatively weighted loops form a ground stat…
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 networ…
