arXiv subjects

Zhongzhi Zhang

Publications and source records attributed to Zhongzhi Zhang.

At least 91 records · Page 5Linked to original sources

arXiv2015

Farey Graphs as Models for Complex Networks

Farey sequences of irreducible fractions between 0 and 1 can be related to graph constructions known as Farey graphs. These graphs were first introduced by Matula and Kornerup in 1979 and further studied by Colbourn in 1982 and they have many interesting properties: they are minimally 3-colorable, uniquely Hamiltonian, maximally outerplanar and perfect. In this paper we introduce a simple generation method for a Farey graph family, and we study analytically relevant topological properties: order, size, degree distribution and correlation, clustering, transitivity, diameter and average distance. We show that the graphs are a good model for networks associated with some complex systems.

cond-mat.stat-mech↗

arXiv2015

The number and degree distribution of spanning trees in the Tower of Hanoi graph

The number of spanning trees of a graph is an important invariant related to topological and dynamic properties of the graph, such as its reliability, communication aspects, synchronization, and so on. However, the practical enumeration of spanning trees and the study of their properties remain a challenge, particularly for large networks. In this paper, we study the num- ber and degree distribution of the spanning trees in the Hanoi graph. We first establish recursion relations between the number of spanning trees and other spanning subgraphs of the Hanoi graph, from which we find an exact analytical expression for the number of spanning trees of the n-disc Hanoi graph. This result allows the calculation of the spanning tree entropy which is then compared with those for other graphs with the same average degree. Then, we introduce a vertex labeling which allows to find, for each vertex of the graph, its degree distribution among all possible spanning trees.

cs.DM↗

arXiv2015

The normalized Laplacian spectrum of subdivisions of a graph

Determining and analyzing the spectra of graphs is an important and exciting research topic in theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to random walks. In this paper, we give the spectra of the normalized Laplacian of iterated subdivisions of simple connected graphs. As an example of application of these results we find the exact values of their multiplicative degree-Kirchhoff index, Kemeny's constant and number of spanning trees.

math.CO↗

arXiv2015

Mixed random walks with a trap in scale-free networks including nearest-neighbor and next-nearest-neighbor jumps

Random walks including non-nearest-neighbor jumps appear in many real situations such as the diffusion of adatoms and have found numerous applications including PageRank search algorithm, however, related theoretical results are much less for this dynamical process. In this paper, we present a study of mixed random walks in a family of fractal scale-free networks, where both nearest-neighbor and next-nearest-neighbor jumps are included. We focus on trapping problem in the network family, which is a particular case of random walks with a perfect trap fixed at the central high-degree node. We derive analytical expressions for the average trapping time (ATT), a quantitative indicator measuring the efficiency of the trapping process, by using two different methods, the results of which are consistent with each other. Furthermore, we analytically determine all the eigenvalues and their multiplicities for the fundamental matrix characterizing the dynamical process. Our results show that although next-nearest-neighbor jumps have no effect on the leading sacling of the trapping efficiency, they can strongly affect the prefactor of ATT, providing insight into better understanding of random-walk process in complex systems.

physics.chem-ph↗

arXiv2015

On the spectrum of the normalized Laplacian of iterated triangulations of graphs

The eigenvalues of the normalized Laplacian of a graph provide information on its topological and structural characteristics and also on some relevant dynamical aspects, specifically in relation to random walks. In this paper we determine the spectra of the normalized Laplacian of iterated triangulations of a generic simple connected graph. As an application, we also find closed-forms for their multiplicative degree-Kirchhoff index, Kemeny's constant and number of spanning trees.

math.SP↗

arXiv2015

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.

cond-mat.soft↗

arXiv2014

Mean first-passage time for maximal-entropy random walks in complex networks

We perform an in-depth study for mean first-passage time (MFPT)---a primary quantity for random walks with numerous applications---of maximal-entropy random walks (MERW) performed in complex networks. For MERW in a general network, we derive an explicit expression of MFPT in terms of the eigenvalues and eigenvectors of the adjacency matrix associated with the network. For MERW in uncorrelated networks, we also provide a theoretical formula of MFPT at the mean-field level, based on which we further evaluate the dominant scalings of MFPT to different targets for MERW in uncorrelated scale-free networks, and compare the results with those corresponding to traditional unbiased random walks (TURW). We show that the MFPT to a hub node is much lower for MERW than for TURW. However, when the destination is a node with the least degree or a uniformly chosen node, the MFPT is higher for MERW than for TURW. Since MFPT to a uniformly chosen node measures real efficiency of search in networks, our work provides insight into general searching process in complex networks.

cond-mat.stat-mech↗

arXiv2014

Maximal entropy random walk improves efficiency of trapping in dendrimers

We use maximal entropy random walk (MERW) to study the trapping problem in dendrimers modeled by Cayley trees with a deep trap fixed at the central node. We derive an explicit expression for the mean first passage time from any node to the trap, as well as an exact formula for the average trapping time (ATT), which is the average of the source-to-trap mean first passage time over all non-trap starting nodes. Based on the obtained closed-form solution for ATT, we further deduce an upper bound for the leading behavior of ATT, which is the fourth power of $\ln N$, where $N$ is the system size. This upper bound is much smaller than the ATT of trapping depicted by unbiased random walk in Cayley trees, the leading scaling of which is a linear function of $N$. These results show that MERW can substantially enhance the efficiency of trapping performed in dendrimers.

cond-mat.stat-mech↗

arXiv2014

The Number of Spanning Trees in Apollonian Networks

In this paper we find an exact analytical expression for the number of spanning trees in Apollonian networks. This parameter can be related to significant topological and dynamic properties of the networks, including percolation, epidemic spreading, synchronization, and random walks. As Apollonian networks constitute an interesting family of maximal planar graphs which are simultaneously small-world, scale-free, Euclidean and space filling and highly clustered, the study of their spanning trees is of particular relevance. Our results allow also the calculation of the spanning tree entropy of Apollonian networks, which then we compare with those of other graphs with the same average degree.

math.CO↗

arXiv2014

The Number of Spanning Trees of an Infinite Family of Outerplanar, Small-World and Self-Similar Graphs

In this paper we give an exact analytical expression for the number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs. This number is an important graph invariant related to different topological and dynamic properties of the graph, such as its reliability, synchronization capability and diffusion properties. The calculation of the number of spanning trees is a demanding and difficult task, in particular for large graphs, and thus there is much interest in obtaining closed expressions for relevant infinite graph families. We have also calculated the spanning tree entropy of the graphs which we have compared with those for graphs with the same average degree.

math.CO↗

arXiv2013

Random walks in unweighted and weighted modular scale-free networks with a perfect trap

Designing optimal structure favorable to diffusion and effectively controlling the trapping process are crucial in the study of trapping problem---random walks with a single trap. In this paper, we study the trapping problem occurring on unweighted and weighted networks, respectively. The networks under consideration display the striking scale-free, small-world, and modular properties, as observed in diverse real-world systems. For binary networks, we concentrate on three cases of trapping problems with the trap located at a peripheral node, a neighbor of the root with the least connectivity, and a farthest node, respectively. For weighted networks with edge weights controlled by a parameter, we also study three trapping problems, in which the trap is placed separately at the root, a neighbor of the root with the least degree, and a farthest node. For all the trapping problems, we obtain the analytical formulas for the average trapping time (ATT) measuring the efficiency of the trapping process, as well as the leading scaling of ATT. We show that for all the trapping problems in the binary networks with a trap located at different nodes, the dominating scalings of ATT reach the possible minimum scalings, implying that the networks have optimal structure that is advantageous to efficient trapping. Furthermore, we show that for trapping in the weighted networks, the ATT is controlled by the weight parameter, through modifying which, the ATT can behave superlinealy, linearly, sublinearly, or logarithmically with the system size. This work could help improving the design of systems with efficient trapping process and offers new insight into control of trapping in complex systems.

cond-mat.stat-mech↗

arXiv2013

Controlling the efficiency of trapping in treelike fractals

Efficiently controlling the trapping process, especially the trapping efficiency, is central in the study of trap problem in complex systems, since it is a fundamental mechanism for diverse other dynamic processes. Thus, it is of theoretical and practical significance to study the control technique for trapping problem. In this paper, we study the trapping problem in a family of proposed directed fractals with a deep trap at a central node. The directed fractals are a generalization of previous undirected fractals by introducing the directed edge weights dominated by a parameter. We characterize all the eigenvalues and their degeneracies for an associated matrix governing the trapping process. The eigenvalues are provided through an exact recursive relation deduced from the self-similar structure of the fractals. We also obtain the expressions for the smallest eigenvalue and the mean first-passage time (MFPT) as a measure of trapping efficiency, which is the expected time for the walker to first visit the trap. The MFPT is evaluated according to the proved fact that it is approximately equal to reciprocal of the smallest eigenvalue. We show that the MFPT is controlled by the weight parameter, by modifying which, the MFPT can scale superlinealy, linearly, or sublinearly with the system size. Thus, this work paves a way to delicately controlling the trapping process in the fractals.

cond-mat.stat-mech↗

arXiv2013

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

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.

cond-mat.stat-mech↗

arXiv2013

Eigenvalues of normalized Laplacian matrices of fractal trees and dendrimers: Analytical results and applications

The eigenvalues of the normalized Laplacian matrix of a network plays an important role in its structural and dynamical aspects associated with the network. In this paper, we study the spectra and their applications of normalized Laplacian matrices of a family of fractal trees and dendrimers modeled by Cayley trees, both of which are built in an iterative way. For the fractal trees, we apply the spectral decimation approach to determine analytically all the eigenvalues and their corresponding multiplicities, with the eigenvalues provided by a recursive relation governing the eigenvalues of networks at two successive generations. For Cayley trees, we show that all their eigenvalues can be obtained by computing the roots of several small-degree polynomials defined recursively. By using the relation between normalized Laplacian spectra and eigentime identity, we derive the explicit solution to the eigentime identity for random walks on the two treelike networks, the leading scalings of which follow quite different behaviors. In addition, we corroborate the obtained eigenvalues and their degeneracies through the link between them and the number of spanning trees.

physics.chem-ph↗

arXiv2013

Laplacian spectra of recursive treelike small-world polymer networks: Analytical solutions and applications

A central issue in the study of polymer physics is to understand the relation between the geometrical properties of macromolecules and various dynamics, most of which are encoded in the Laplacian spectra of a related graph describing the macrostructural structure. In this paper, we introduce a family of treelike polymer networks with a parameter, which has the same size as the Vicsek fractals modeling regular hyperbranched polymers. We study some relevant properties of the networks and show that they have an exponentially decaying degree distribution and exhibit the small-world behavior. We then study the Laplacian eigenvalues and their corresponding eigenvectors of the networks under consideration, with both quantities being determined through the recursive relations deduced from the network structure. Using the obtained recursive relations we can find all the eigenvalues and eigenvectors for the networks with any size. Finally, as some applications, we use the eigenvalues to study analytically or semi-analytically three dynamical processes occurring in the networks, including random walks, relaxation dynamics in the framework of generalized Gaussian structure, as well as the fluorescence depolarization under quasiresonant energy transfer. Moreover, we compare the results with those corresponding to Vicsek fractals, and show that the dynamics differ greatly for the two network families, which thus enables us to distinguish between them.

cond-mat.stat-mech↗

arXiv2013

Influence of trap location on the efficiency of trapping in dendrimers and regular hyperbranched polymers

In this paper, we study the trapping problem in two representative polymer networks, Cayley trees and Vicsek fractals, which separately model dendrimers and regular hyperbranched polymers. Our goal is to explore the impact of trap location on the efficiency of trapping in these two important polymer systems, with the efficiency being measured by the average trapping time (ATT) that is the average of source-to-trap mean first-passage time over every staring point in the whole networks. For Cayley trees, we derive an exact analytic formula for the ATT to an arbitrary trap node, based on which we further obtain the explicit expression of ATT for the case that the trap is uniformly distributed. For Vicsek fractals, we provide the closed-form solution for ATT to a peripheral node farthest from the central node, as well as the numerical solutions for the case when the trap is placed on other nodes. Moreover, we derive the exact formula for the ATT corresponding to the trapping problem when the trap has an uniform distribution over all nodes. Our results show that the influence of trap location on the trapping efficiency is completely different for the two polymer networks. In Cayley trees, the leading scaling of ATT increases with the shortest distance between the trap and the central node, implying that trap's position has an essential impact on the trapping efficiency; while in Vicsek fractals, the effect of location of the trap is negligible, since the dominant behavior of ATT is identical, respective of the location where the trap is placed. We also present that for all cases of trapping problems being studied, the trapping process is more efficient in Cayley trees than in Vicsek fractals. We demonstrate that all differences related to trapping in the two polymer systems are rooted in their underlying topological structures.

cond-mat.stat-mech↗

arXiv2013

Optimal scale-free network with a minimum scaling of transport efficiency for random walks with a perfect trap

Average trapping time (ATT) is central in the trapping problem since it is a key indicator characterizing the efficiency of the problem. Previous research has provided the scaling of a lower bound of the ATT for random walks in general networks with a deep trap. However, it is still not well understood in which networks this minimal scaling can be reached. Particularly, explicit quantitative results for ATT in such networks, even in a specific network, are lacking, in spite that such networks shed light on the design for optimal networks with the highest trapping efficiency. In this paper, we study the trapping problem taking place on a hierarchical scale-free network with a perfect trap. We focus on four representative cases with the immobile trap located at the root, a peripheral node, a neighbor of the root with a single connectivity, and a farthest node from the root, respectively. For all the four cases, we obtain the closed-form formulas for the ATT, as well as its leading scalings. We show that for all the four cases of trapping problems, the dominating scalings of ATT can reach the predicted minimum scalings. This work deepens the understanding of behavior of trapping in scale-free networks, and is helpful for designing networks with the most efficient transport process.

cond-mat.stat-mech↗

arXiv2012

Random walks on weighted networks

Random walks constitute a fundamental mechanism for a large set of dynamics taking place on networks. In this article, we study random walks on weighted networks with an arbitrary degree distribution, where the weight of an edge between two nodes has a tunable parameter. By using the spectral graph theory, we derive analytical expressions for the stationary distribution, mean first-passage time (MFPT), average trapping time (ATT), and lower bound of the ATT, which is defined as the average MFPT to a given node over every starting point chosen from the stationary distribution. All these results depend on the weight parameter, indicating a significant role of network weights on random walks. For the case of uncorrelated networks, we provide explicit formulas for the stationary distribution as well as ATT. Particularly, for uncorrelated scale-free networks, when the target is placed on a node with the highest degree, we show that ATT can display various scalings of network size, depending also on the same parameter. Our findings could pave a way to delicately controlling random-walk dynamics on complex networks.

cond-mat.stat-mech↗