Explicit determination of mean first-passage time for random walks on deterministic uniform recursive trees

Heterogeneous Mean First-Passage Time Scaling in Fractal Media

The mean first passage time (MFPT) of random walks is a key quantity characterizing dynamic processes on disordered media. In a random fractal embedded in the Euclidean space, the MFPT is known to obey the power law scaling with the distance between a source and a target site with a universal exponent. We find that the scaling law for the MFPT is not determined solely by the distance between a source and a target but also by their locations. The role of a site in the first passage processes is quantified by the random walk centrality. It turns out that the site of highest random walk centrality, dubbed as a hub, intervenes in first passage processes. We show that the MFPT from a departure site to a target site is determined by a competition between direct paths and indirect paths detouring via the hub. Consequently, the MFPT displays a crossover scaling between a short distance regime, where direct paths are dominant, and a long distance regime, where indirect paths are dominant. The two regimes are characterized by power laws with different scaling exponents. The crossover scaling behavior is confirmed by extensive numerical calculations of the MFPTs on the critical percolation cluster in two dimensional square lattices.

Stochastic growth tree networks with an identical fractal dimension: Construction and mean hitting time for random walks

There is little attention paid to stochastic tree networks in comparison with the corresponding deterministic analogs in the current study of fractal trees. In this paper, we propose a principled framework for producing a family of stochastic growth tree networks _T possessing fractal characteristic, where t represents the time step and parameter m is the number of vertices newly created for each existing vertex at generation. To this end, we introduce two types of generative ways, i.e., Edge-Operation and Edge-Vertex-Operation. More interestingly, the resulting stochastic trees turn out to have an identical fractal dimension _d = ln ⁡ 2 ( m + 1 ) / ln ⁡ 2 regardless of the introduction of randomness in the growth process. At the same time, we also study many other structural parameters including diameter and degree distribution. In both extreme cases, our tree networks are deterministic and follow multiple-point degree distribution and power-law degree distribution, respectively. Additionally, we consider random walks on stochastic growth tree networks _T and derive an expectation estimation for mean hitting time ⟨ _H ⟩ in an effective combinatorial manner instead of commonly used spectral methods. The result shows that on average, the scaling of mean hitting time ⟨ _H ⟩ obeys ⟨ _H ⟩ = | _T ^|, where | _T | represents vertex number and exponent λ is equivalent to 1 + ln ⁡ 2 / ln ⁡ 2 ( m + 1 ). In the meantime, we conduct extensive experimental simulations and observe that empirical analysis is in strong agreement with theoretical results.

Mean first-passage time for random walks on random growth tree networks

As opposed to most previous works focusing on random walks on deterministic tree networks, in this paper, we pay more attention on random growth tree networks. Specifically, we propose two distinct types of random growth tree networks T(1;t,p) and T(2;t,p) where t represents time step and p is a probability parameter (0≤p≤1). Tree T(1;t,p) is iteratively built in a fractal manner; however, T(2;t,p) is generated using a nonfractal operation. Then we study random walks on tree network T(i;t,p) (i=1,2) and derive the analytical solution to mean first-passage time 〈F_{T(i;t,p)}〉. The results suggest that two growth ways have remarkably different influence on parameters 〈F_{T(i;t,p)}〉. More precisely, the fractal-growth manner makes topological structure of tree network more loose than the nonfractal one and thus increases drastically mean first-passage time in the large graph limit. Finally, we extensively conduct experimental simulations, and the results demonstrate that computer simulations are in strong agreement with the theoretical analysis.

Exact results for the first-passage properties in a class of fractal networks

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 Kemeny's constant), and we highlight that their asymptotic behavior is controlled by the network's size and diameter. Remarkably, if we tune n (or, analogously, t) while keeping the network size fixed, as n increases (t decreases) the network gets more and more clustered and modular while its diameter is reduced, implying, ultimately, a better transport performance. The connection between this class of networks and models for polymer architectures is also discussed.

Anomalous behavior of trapping in extended dendrimers with a perfect trap

Compact and extended dendrimers are two important classes of dendritic polymers. The impact of the underlying structure of compact dendrimers on dynamical processes has been much studied, yet the relation between the dynamical and structural properties of extended dendrimers remains not well understood. In this paper, we study the trapping problem in extended dendrimers with generation-dependent segment lengths, which is different from that of compact dendrimers where the length of the linear segments is fixed. We first consider a particular case that the deep trap is located at the central node, and derive an exact formula for the average trapping time (ATT) defined as the average of the source-to-trap mean first passage time over all starting points. Then, using the obtained result we deduce a closed-form expression for the ATT to an arbitrary trap node, based on which we further obtain an explicit solution to the ATT corresponding to the trapping issue with the trap uniformly distributed in the polymer systems. We show that the trap location has a substantial influence on the trapping efficiency measured by the ATT, which increases with the shortest distance from the trap to the central node, a phenomenon similar to that for compact dendrimers. In contrast to this resemblance, the leading terms of ATTs for the three trapping problems differ drastically between extended and compact dendrimers, with the trapping processes in the extended dendrimers being less efficient than in compact dendrimers.

Exact calculations of first-passage properties on the pseudofractal scale-free web

In this paper, we consider discrete time random walks on the pseudofractal scale-free web (PSFW) and we study analytically the related first passage properties. First, we classify the nodes of the PSFW into different levels and propose a method to derive the generation function of the first passage probability from an arbitrary starting node to the absorbing domain, which is located at one or more nodes of low-level (i.e., nodes with large degree). Then, we calculate exactly the first passage probability, the survival probability, the mean, and the variance of first passage time by using the generating functions as a tool. Finally, for some illustrative examples corresponding to given choices of starting node and absorbing domain, we derive exact and explicit results for such first passage properties. The method we propose can as well address the cases where the absorbing domain is located at one or more nodes of high-level on the PSFW, and it can also be used to calculate the first passage properties on other networks with self-similar structure, such as (u, v) flowers and recursive scale-free trees.

Random walks on non-homogenous weighted Koch networks

In this paper, we introduce new models of non-homogenous weighted Koch networks on real traffic systems depending on the three scaling factors r1,r2,r3∈(0,1). Inspired by the definition of the average weighted shortest path (AWSP), we define the average weighted receiving time (AWRT). Assuming that the walker, at each step, starting from its current node, moves uniformly to any of its neighbors, we show that in large network, the AWRT grows as power-law function of the network order with the exponent, represented by θ(r1,r2,r3)=log4(1+r1+r2+r3). Moreover, the AWSP, in the infinite network order limit, only depends on the sum of scaling factors r1,r2,r3.

Random walks in weighted networks with a perfect trap: an application of Laplacian spectra

Trapping processes constitute a primary problem of random walks, which characterize various other dynamical processes taking place on networks. Most previous works focused on the case of binary networks, while there is much less related research about weighted networks. In this paper, we propose a general framework for the trapping problem on a weighted network with a perfect trap fixed at an arbitrary node. By utilizing the spectral graph theory, we provide an exact formula for mean first-passage time (MFPT) from one node to another, based on which we deduce an explicit expression for average trapping time (ATT) in terms of the eigenvalues and eigenvectors of the Laplacian matrix associated with the weighted graph, where ATT is the average of MFPTs to the trap over all source nodes. We then further derive a sharp lower bound for the ATT in terms of only the local information of the trap node, which can be obtained in some graphs. Moreover, we deduce the ATT when the trap is distributed uniformly in the whole network. Our results show that network weights play a significant role in the trapping process. To apply our framework, we use the obtained formulas to study random walks on two specific networks: trapping in weighted uncorrelated networks with a deep trap, the weights of which are characterized by a parameter, and Lévy random walks in a connected binary network with a trap distributed uniformly, which can be looked on as random walks on a weighted network. For weighted uncorrelated networks we show that the ATT to any target node depends on the weight parameter, that is, the ATT to any node can change drastically by modifying the parameter, a phenomenon that is in contrast to that for trapping in binary networks. For Lévy random walks in any connected network, by using their equivalence to random walks on a weighted complete network, we obtain the optimal exponent characterizing Lévy random walks, which have the minimal average of ATTs taken over all target nodes.

Impact of degree heterogeneity on the behavior of trapping in Koch networks

Previous work shows that the mean first-passage time (MFPT) for random walks to a given hub node (node with maximum degree) in uncorrelated random scale-free networks is closely related to the exponent γ of power-law degree distribution P(k) ∼ k(-γ), which describes the extent of heterogeneity of scale-free network structure. However, extensive empirical research indicates that real networked systems also display ubiquitous degree correlations. In this paper, we address the trapping issue on the Koch networks, which is a special random walk with one trap fixed at a hub node. The Koch networks are power-law with the characteristic exponent γ in the range between 2 and 3, they are either assortative or disassortative. We calculate exactly the MFPT that is the average of first-passage time from all other nodes to the trap. The obtained explicit solution shows that in large networks the MFPT varies lineally with node number N, which is obviously independent of γ and is sharp contrast to the scaling behavior of MFPT observed for uncorrelated random scale-free networks, where γ influences qualitatively the MFPT of trapping problem.

Spectral dimensions of hierarchical scale-free networks with weighted shortcuts

Spectral dimensions have been widely used to understand transport properties on regular and fractal lattices. However, they have received little attention with regard to complex networks such as scale-free and small-world networks. Here, we study the spectral dimension and the return-to-origin probability of random walks on hierarchical scale-free networks, which can be either fractal or nonfractal depending on the weight of the shortcuts. Applying the renormalization-group (RG) approach to a Gaussian model, we obtain the exact spectral dimension. While the spectral dimension varies between 1 and 2 for the fractal case, it remains at 2, independent of the variation in the network structure, for the nonfractal case. The crossover behavior between the two cases is studied by carrying out the RG flow analysis. The analytical results are confirmed by simulation results and their implications for the architecture of complex systems are discussed.

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.

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.

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.