arXiv subjects

Zhongzhi Zhang

Publications and source records attributed to Zhongzhi Zhang.

At least 109 records · Page 6Linked to original sources

arXiv2012

Optimal and suboptimal networks for efficient navigation measured by mean-first passage time of random walks

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 optimal networks with a minimum Kemeny constant, which grows linearly with the network size. Then, we study the Kemeny constant of a class of sparse networks that exhibit remarkable scale-free and fractal features as observed in many real-life networks, which cannot be described by complete graphs. To this end, we determine the closed-form solutions to all eigenvalues and their degeneracies of the networks. Employing these eigenvalues, we derive the exact solution to the Kemeny constant, which also behaves linearly with the network size for some particular cases of networks. We further use the eigenvalue spectra to determine the number of spanning trees in the networks under consideration, which is in complete agreement with previously reported results. Our work demonstrates that scale-free and fractal properties are favorable for efficient navigation, which could be considered when designing networks with high navigation efficiency.

cond-mat.stat-mech↗

arXiv2012

Mean first-passage time for random walks in general graphs with a deep trap

We provide an explicit formula for the global mean first-passage time (GMFPT) for random walks in a general graph with a perfect trap fixed at an arbitrary node, where GMFPT is the average of mean first-passage time to the trap over all starting nodes in the whole graph. The formula is expressed in terms of eigenvalues and eigenvectors of Laplacian matrix for the graph. We then use the formula to deduce a tight lower bound for the GMFPT in terms of only the numbers of nodes and edges, as well as the degree of the trap, which can be achieved in both complete graphs and star graphs. We show that for a large sparse graph the leading scaling for this lower bound is proportional to the system size and the reciprocal of the degree for the trap node. Particularly, we demonstrate that for a scale-free graph of size $N$ with a degree distribution $P(d)\sim d^{-γ}$ characterized by $γ$, when the trap is placed on a most connected node, the dominating scaling of the lower bound becomes $N^{1-1/γ}$, which can be reached in some scale-free graphs. Finally, we prove that the leading behavior of upper bounds for GMFPT on any graph is at most $N^{3}$ that can be reached in the bar-bell graphs. This work provides a comprehensive understanding of previous results about trapping in various special graphs with a trap located at a specific location.

cond-mat.stat-mech↗

arXiv2012

Trapping in dendrimers and regular hyperbranched polymers

Dendrimers and regular hyperbranched polymers are two classic families of macromolecules, which can be modeled by Cayley trees and Vicsek fractals, respectively. In this paper, we study the trapping problem in Cayley trees and Vicsek fractals with different underlying geometries, focusing on a particular case with a perfect trap located at the central node. For both networks, we derive the exact analytic formulas in terms of the network size for the average trapping time (ATT)---the average of node-to-trap mean first-passage time over the whole networks. The obtained closed-form solutions show that for both Cayley trees and Vicsek fractals, the ATT display quite different scalings with various system sizes, which implies that the underlying structure plays a key role on the efficiency of trapping in polymer networks. Moreover, the dissimilar scalings of ATT may allow to differentiate readily between dendrimers and hyperbranched polymers.

physics.chem-ph↗

arXiv2012

Exact eigenvalue spectrum of a class of fractal scale-free networks

The eigenvalue spectrum of the transition matrix of a network encodes important information about its structural and dynamical properties. We study the transition matrix of a family of fractal scale-free networks and analytically determine all the eigenvalues and their degeneracies. We then use these eigenvalues to evaluate the closed-form solution to the eigentime for random walks on the networks under consideration. Through the connection between the spectrum of transition matrix and the number of spanning trees, we corroborate the obtained eigenvalues and their multiplicities.

cond-mat.stat-mech↗

arXiv2012

Counting spanning trees in a small-world Farey graph

The problem of spanning trees is closely related to various interesting problems in the area of statistical physics, but determining the number of spanning trees in general networks is computationally intractable. In this paper, we perform a study on the enumeration of spanning trees in a specific small-world network with an exponential distribution of vertex degrees, which is called a Farey graph since it is associated with the famous Farey sequence. According to the particular network structure, we provide some recursive relations governing the Laplacian characteristic polynomials of a Farey graph and its subgraphs. Then, making use of these relations obtained here, we derive the exact number of spanning trees in the Farey graph, as well as an approximate numerical solution for the asymptotic growth constant characterizing the network. Finally, we compare our results with those of different types of networks previously investigated.

cond-mat.stat-mech↗

arXiv2011

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 non-trap 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.

cond-mat.stat-mech↗

arXiv2011

Traffic Fluctuations on Weighted Networks

Traffic fluctuation has so far been studied on unweighted networks. However many real traffic systems are better represented as weighted networks, where nodes and links are assigned a weight value representing their physical properties such as capacity and delay. Here we introduce a general random diffusion (GRD) model to investigate the traffic fluctuation in weighted networks, where a random walk's choice of route is affected not only by the number of links a node has, but also by the weight of individual links. We obtain analytical solutions that characterise the relation between the average traffic and the fluctuation through nodes and links. Our analysis is supported by the results of numerical simulations. We observe that the value ranges of the average traffic and the fluctuation, through nodes or links, increase dramatically with the level of heterogeneity in link weight. This highlights the key role that link weight plays in traffic fluctuation and the necessity to study traffic fluctuation on weighted networks.

cond-mat.stat-mech↗

arXiv2011

Counting spanning trees in self-similar networks by evaluating determinants

Spanning trees are relevant to various aspects of networks. Generally, the number of spanning trees in a network can be obtained by computing a related determinant of the Laplacian matrix of the network. However, for a large generic network, evaluating the relevant determinant is computationally intractable. In this paper, we develop a fairly generic technique for computing determinants corresponding to self-similar networks, thereby providing a method to determine the numbers of spanning trees in networks exhibiting self-similarity. We describe the computation process with a family of networks, called $(x,y)$-flowers, which display rich behavior as observed in a large variety of real systems. The enumeration of spanning trees is based on the relationship between the determinants of submatrices of the Laplacian matrix corresponding to the $(x,y)$-flowers at different generations and is devoid of the direct laborious computation of determinants. Using the proposed method, we derive analytically the exact number of spanning trees in the $(x,y)$-flowers, on the basis of which we also obtain the entropies of the spanning trees in these networks. Moreover, to illustrate the universality of our technique, we apply it to some other self-similar networks with distinct degree distributions, and obtain explicit solutions to the numbers of spanning trees and their entropies. Finally, we compare our results for networks with the same average degree but different structural properties, such as degree distribution and fractal dimension, and uncover the effect of these topological features on the number of spanning trees.

cond-mat.stat-mech↗

arXiv2011

Mean first-passage time for random walks on undirected networks

In this paper, by using two different techniques we derive an explicit formula for the mean first-passage time (MFPT) between any pair of nodes on a general undirected network, which is expressed in terms of eigenvalues and eigenvectors of an associated matrix similar to the transition matrix. We then apply the formula to derive a lower bound for the MFPT to arrive at a given node with the starting point chosen from the stationary distribution over the set of nodes. We show that for a correlated scale-free network of size $N$ with a degree distribution $P(d)\sim d^{-γ}$, the scaling of the lower bound is $N^{1-1/γ}$. Also, we provide a simple derivation for an eigentime identity. Our work leads to a comprehensive understanding of recent results about random walks on complex networks, especially on scale-free networks.

cond-mat.stat-mech↗

arXiv2011

Complete spectrum of stochastic master equation for random walks on treelike fractals

We study random walks on a family of treelike regular fractals with a trap fixed on a central node. We obtain all the eigenvalues and their corresponding multiplicities for the associated stochastic master equation, with the eigenvalues being provided through an explicit recursive relation. We also evaluate the smallest eigenvalue and show that its reciprocal is approximately equal to the mean trapping time. We expect that our technique can also be adapted to other regular fractals with treelike structures.

cond-mat.stat-mech↗

arXiv2011

Role of fractal dimension in random walks on scale-free networks

Fractal dimension is central to understanding dynamical processes occurring on networks; however, the relation between fractal dimension and random walks on fractal scale-free networks has been rarely addressed, despite the fact that such networks are ubiquitous in real-life world. In this paper, we study the trapping problem on two families of networks. The first is deterministic, often called $(x,y)$-flowers; the other is random, which is a combination of $(1,3)$-flower and $(2,4)$-flower and thus called hybrid networks. The two network families display rich behavior as observed in various real systems, as well as some unique topological properties not shared by other networks. We derive analytically the average trapping time for random walks on both the $(x,y)$-flowers and the hybrid networks with an immobile trap positioned at an initial node, i.e., a hub node with the highest degree in the networks. Based on these analytical formulae, we show how the average trapping time scales with the network size. Comparing the obtained results, we further uncover that fractal dimension plays a decisive role in the behavior of average trapping time on fractal scale-free networks, i.e., the average trapping time decreases with an increasing fractal dimension.

cond-mat.stat-mech↗

arXiv2011

Random walks in small-world exponential treelike networks

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 the order of $N \ln N$. After that, we compute the precise global mean first-passage time among all pairs of nodes and find that it also varies in the order of $N \ln N$ in the large limit of $N$. After obtaining the relevant quantities, we compare them with each other and related our results to the efficiency for information transmission by regarding the walker as an information messenger. Finally, we compare our results with those previously reported for other trees with different structural properties (e.g., degree distribution), such as the standard fractal trees and the scale-free small-world trees, and show that the shortest path between a pair of nodes in a tree is responsible for the scaling of FPT between the two nodes.

cond-mat.stat-mech↗

arXiv2011

Diffusion-annihilation proecesses in weighted scale-free networks with identical degree sequence

The studies based on $A+A \rightarrow \emptyset$ and $A+B\rightarrow \emptyset$ diffusion-annihilation processes have so far been studied on weighted uncorrelated scale-free networks and fractal scale-free networks. In the previous reports, it is widely accepted that the segregation of particles in the processes is introduced by the fractal structure. In this paper, we study these processes on a family of weighted scale-free networks with identical degree sequence. We find that the depletion zone and segregation are essentially caused by the disassortative mixing, namely, high-degree nodes tend to connect with low-degree nodes. Their influence on the processes is governed by the correlation between the weight and degree. Our finding suggests both the weight and degree distribution don't suffice to characterize the diffusion-annihilation processes on weighted scale-free networks.

cond-mat.dis-nn↗

arXiv2011

Scaling of mean first-passage time as efficiency measure of nodes sending information on scale-free Koch networks

A lot of previous work showed that the sectional mean first-passage time (SMFPT), i.e., the average of mean first-passage time (MFPT) for random walks to a given hub node (node with maximum degree) averaged over all starting points in scale-free small-world networks exhibits a sublinear or linear dependence on network order $N$ (number of nodes), which indicates that hub nodes are very efficient in receiving information if one looks upon the random walker as an information messenger. Thus far, the efficiency of a hub node sending information on scale-free small-world networks has not been addressed yet. In this paper, we study random walks on the class of Koch networks with scale-free behavior and small-world effect. We derive some basic properties for random walks on the Koch network family, based on which we calculate analytically the partial mean first-passage time (PMFPT) defined as the average of MFPTs from a hub node to all other nodes, excluding the hub itself. The obtained closed-form expression displays that in large networks the PMFPT grows with network order as $N \ln N$, which is larger than the linear scaling of SMFPT to the hub from other nodes. On the other hand, we also address the case with the information sender distributed uniformly among the Koch networks, and derive analytically the entire mean first-passage time (EMFPT), namely, the average of MFPTs between all couples of nodes, the leading scaling of which is identical to that of PMFPT. From the obtained results, we present that although hub nodes are more efficient for receiving information than other nodes, they display a qualitatively similar speed for sending information as non-hub nodes. Moreover, we show that the location of information sender has little effect on the transmission efficiency. The present findings are helpful for better understanding random walks performed on scale-free small-world networks.

cond-mat.stat-mech↗

arXiv2010

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.

cond-mat.stat-mech↗

arXiv2010

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 $P(k)\sim k^{-γ}$, 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.

cond-mat.stat-mech↗

arXiv2010

Mapping Koch curves into scale-free small-world networks

The class of Koch fractals is one of the most interesting families of fractals, and the study of complex networks is a central issue in the scientific community. In this paper, inspired by the famous Koch fractals, we propose a mapping technique converting Koch fractals into a family of deterministic networks, called Koch networks. This novel class of networks incorporates some key properties characterizing a majority of real-life networked systems---a power-law distribution with exponent in the range between 2 and 3, a high clustering coefficient, small diameter and average path length, and degree correlations. Besides, we enumerate the exact numbers of spanning trees, spanning forests, and connected spanning subgraphs in the networks. All these features are obtained exactly according to the proposed generation algorithm of the networks considered. The network representation approach could be used to investigate the complexity of some real-world systems from the perspective of complex networks.

cond-mat.stat-mech↗

arXiv2010

Enumeration of spanning trees in a pseudofractal scale-free web

Spanning trees are an important quantity characterizing the reliability of a network, however, explicitly determining the number of spanning trees in networks is a theoretical challenge. In this paper, we study the number of spanning trees in a small-world scale-free network and obtain the exact expressions. We find that the entropy of spanning trees in the studied network is less than 1, which is in sharp contrast to previous result for the regular lattice with the same average degree, the entropy of which is higher than 1. Thus, the number of spanning trees in the scale-free network is much less than that of the corresponding regular lattice. We present that this difference lies in disparate structure of the two networks. Since scale-free networks are more robust than regular networks under random attack, our result can lead to the counterintuitive conclusion that a network with more spanning trees may be relatively unreliable.

cond-mat.stat-mech↗