APPLICATIONS OF LAPLACIAN SPECTRA ON A 3-PRISM GRAPH

2014 ◽  
Vol 28 (02) ◽  
pp. 1450009 ◽  
Author(s):  
QINGYAN DING ◽  
WEIGANG SUN ◽  
FANGYUE CHEN
Keyword(s):  
Spanning Trees ◽  
First Passage Time ◽  
Laplacian Matrix ◽  
Passage Time ◽  
Network Size ◽  
Initial State ◽  
Laplacian Eigenvalues ◽  
Regular Structures ◽  
Laplacian Spectra ◽  
Prism Graph

In this paper, we calculate the Laplacian spectra of a 3-prism graph and apply them. This graph is both planar and polyhedral, and belongs to the generalized Petersen graph. Using the regular structures of this graph, we obtain the recurrent relationships for Laplacian matrix between this graph and its initial state — a triangle — and further derive the corresponding relationships for Laplacian eigenvalues between them. By these relationships, we obtain the analytical expressions for the product and the sum of the reciprocals of all nonzero Laplacian eigenvalues. Finally we apply these expressions to calculate the number of spanning trees and mean first-passage time (MFPT) and see that the scaling of MFPT with the network size N is N2, which is larger than those performed on some uniformly recursive trees.

Determining entire mean first-passage time for Cayley networks

2018 ◽  
Vol 29 (01) ◽  
pp. 1850009 ◽  
Author(s):  
Xiaoqian Wang ◽  
Meifeng Dai ◽  
Yufei Chen ◽  
Yue Zong ◽  
Yu Sun ◽  
...  
Keyword(s):  
First Passage Time ◽  
Polymer Networks ◽  
Laplacian Matrix ◽  
Constant Term ◽  
Passage Time ◽  
Mean First Passage Time ◽  
First Passage ◽  
Laplacian Spectra ◽  
Self Similar ◽  
The Relationship

In this paper, we consider the entire mean first-passage time (EMFPT) with random walks for Cayley networks. We use Laplacian spectra to calculate the EMFPT. Firstly, we calculate the constant term and monomial coefficient of characteristic polynomial. By using the Vieta theorem, we then obtain the sum of reciprocals of all nonzero eigenvalues of Laplacian matrix. Finally, we obtain the scaling of the EMFPT for Cayley networks by using the relationship between the sum of reciprocals of all nonzero eigenvalues of Laplacian matrix and the EMFPT. We expect that our method can be adapted to other types of self-similar networks, such as vicsek networks, polymer networks.

Mean first-passage time on a family of small-world treelike networks

2014 ◽  
Vol 25 (03) ◽  
pp. 1350097 ◽  
Author(s):  
Long Li ◽  
Weigang Sun ◽  
Guixiang Wang ◽  
Guanghui Xu
Keyword(s):  
Random Walks ◽  
First Passage Time ◽  
Small World ◽  
Passage Time ◽  
Network Size ◽  
Mean First Passage Time ◽  
First Passage ◽  
Initial Node ◽  
Laplacian Spectra ◽  
Two Parameters

In this paper, we obtain exact scalings of mean first-passage time (MFPT) of random walks on a family of small-world treelike networks formed by two parameters, which includes three kinds. First, we determine the MFPT for a trapping problem with an immobile trap located at the initial node, which is defined as the average of the first-passage times (FPTs) to the trap node over all possible starting nodes, and it scales linearly with network size N in large networks. We then analytically obtain the partial MFPT (PMFPT) which is the mean of FPTs from the trap node to all other nodes and show that it increases with N as N ln N. Finally we establish the global MFPT (GMFPT), which is the average of FPTs over all pairs of nodes. It also grows with N as N ln N in the large limit of N. For these three kinds of random walks, we all obtain the analytical expressions of the MFPT and they all increase with network parameters. In addition, our method for calculating the MFPT is based on the self-similar structure of the considered networks and avoids the calculations of the Laplacian spectra.

TRAPPING PROBLEM OF THE WEIGHTED SCALE-FREE TRIANGULATION NETWORKS FOR BIASED WALKS

Fractals ◽  
2019 ◽  
Vol 27 (03) ◽  
pp. 1950028 ◽  
Author(s):  
MEIFENG DAI ◽  
TINGTING JU ◽  
YUE ZONG ◽  
JIAOJIAO HE ◽  
CHUNYU SHEN ◽  
...  
Keyword(s):  
Nearest Neighbor ◽  
First Passage Time ◽  
Passage Time ◽  
Network Size ◽  
Large Network ◽  
Dominant Term ◽  
Scale Free ◽  
Trapping Problem ◽  
Standard Weight ◽  
Trapping Process

In this paper, we study the trapping problem in the weighted scale-free triangulation networks with the growth factor [Formula: see text] and the weight factor [Formula: see text]. We propose two biased walks, one is standard weight-dependent walk only including the nearest-neighbor jumps, the other is mixed weight-dependent walk including both the nearest-neighbor and the next-nearest-neighbor jumps. For the weighted scale-free triangulation networks, we derive the exact analytic formulas of the average trapping time (ATT), the average of node-to-trap mean first-passage time over the whole networks, which measures the efficiency of the trapping process. The obtained results display that for two biased walks, in the large network, the ATT grows as a power-law function of the network size [Formula: see text] with the exponent, represented by [Formula: see text] when [Formula: see text]. Especially when the case of [Formula: see text] and [Formula: see text], the ATT grows linear with the network size [Formula: see text]. That is the smaller the value of [Formula: see text], the more efficient the trapping process is. Furthermore, comparing the standard weight-dependent walk with mixed weight-dependent walk, the obtained results show that although the next-nearest-neighbor jumps have no main effect on the trapping process, they can modify the coefficient of the dominant term for the ATT. The smaller the value of probability parameter [Formula: see text], the more efficient the trapping process for the mixed weight-dependent walk is.

FIRST-ORDER NETWORK COHERENCE AND EIGENTIME IDENTITY ON THE WEIGHTED CAYLEY NETWORKS

Fractals ◽  
2017 ◽  
Vol 25 (05) ◽  
pp. 1750049 ◽  
Author(s):  
MEIFENG DAI ◽  
XIAOQIAN WANG ◽  
YUE ZONG ◽  
JIAHUI ZOU ◽  
YUFEI CHEN ◽  
...  
Keyword(s):  
First Passage Time ◽  
Analytical Formula ◽  
Passage Time ◽  
Small Degree ◽  
Network Size ◽  
Mean First Passage Time ◽  
Weight Factor ◽  
First Passage ◽  
First Order ◽  
Definition Of

In this paper, we first study the first-order network coherence, characterized by the entire mean first-passage time (EMFPT) for weight-dependent walk, on the weighted Cayley networks with the weight factor. The analytical formula of the EMFPT is obtained by the definition of the EMFPT. The obtained results show that the scalings of first-order coherence with network size obey four laws along with the range of the weight factor. Then, we study eigentime identity quantifying as the sum of reciprocals of all nonzero normalized Laplacian eigenvalues on the weighted Cayley networks with the weight factor. We show that all their eigenvalues can be obtained by calculating the roots of several small-degree polynomials defined recursively. The obtained results show that the scalings of the eigentime identity on the weighted Cayley networks obey two laws along with the range of the weight factor.

MEAN FIRST PASSAGE TIME OF RANDOM WALKS ON THE GENERALIZED PSEUDOFRACTAL WEB

2013 ◽  
Vol 27 (10) ◽  
pp. 1350070 ◽  
Author(s):  
LONG LI ◽  
WEIGANG SUN ◽  
JING CHEN ◽  
GUIXIANG WANG
Keyword(s):  
Random Walks ◽  
First Passage Time ◽  
Passage Time ◽  
Scaling Exponent ◽  
Mean First Passage Time ◽  
Initial State ◽  
First Passage ◽  
Subsequent Step ◽  
Power Law Function ◽  
Exact Scaling

In this paper, we study the scaling for mean first passage time (MFPT) of random walks on the generalized pseudofractal web (GPFW) with a trap, where an initial state is transformed from a triangle to a r-polygon and every existing edge gives birth to finite nodes in the subsequent step. We then obtain an analytical expression and an exact scaling for the MFPT, which shows that the MFPT grows as a power-law function in the large limit of network order. In addition, we determine the exponent of scaling efficiency characterizing the random walks, with the exponent less than 1. The scaling exponent of the MFPT is same for the initial state of the web being a polygon with finite nodes. This method could be applied to other fractal networks.

scholarly journals On the Simulation of a Special Class of Time-Inhomogeneous Diffusion Processes

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 818
Author(s):  
Virginia Giorno ◽  
Amelia G. Nobile
Keyword(s):  
Probability Density ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Transition Probability ◽  
Passage Time ◽  
Probability Density Functions ◽  
Periodic Functions ◽  
Initial State ◽  
Transition Probability Density ◽  
Inhomogeneous Diffusion

General methods to simulate probability density functions and first passage time densities are provided for time-inhomogeneous stochastic diffusion processes obtained via a composition of two Gauss–Markov processes conditioned on the same initial state. Many diffusion processes with time-dependent infinitesimal drift and infinitesimal variance are included in the considered class. For these processes, the transition probability density function is explicitly determined. Moreover, simulation procedures are applied to the diffusion processes obtained starting from Wiener and Ornstein–Uhlenbeck processes. Specific examples in which the infinitesimal moments include periodic functions are discussed.

CONVERGENCE RATE AND GLOBAL MEAN WEIGHTED FIRST-PASSAGE TIME IN A 1D CHAIN NETWORK WITH A WEIGHTED ADDING REVERSE EDGE

Fractals ◽  
2020 ◽  
Vol 28 (05) ◽  
pp. 2050078
Author(s):  
TINGTING CHEN ◽  
MEIFENG DAI ◽  
FANG HUANG ◽  
SHILIN FENG
Keyword(s):  
Convergence Rate ◽  
Positive Relationship ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Laplacian Eigenvalues ◽  
Characteristic Determinant ◽  
1D Chain ◽  
Analytic Expression ◽  
Global Mean

In this paper, a 1D chain network with a reverse weighted edge is introduced. We focus on studying the relationships including the convergence rate and the length, the convergence rate and weight of adding reverse edge relationships. Laplacian characteristic determinant is calculated and subsequently, the sum of the reciprocals of all nonzero Laplacian eigenvalues is obtained. Hence, the analytic expression of global mean weighted first-passage time (GMWFPT) can be deduced. The obtained results show that there exists a linearly positive relationship between GMWFPT and the weight [Formula: see text].

CONSENSUS PROBLEMS IN WEIGHTED HIERARCHICAL GRAPHS

Fractals ◽  
2019 ◽  
Vol 27 (06) ◽  
pp. 1950086
Author(s):  
XIAOQIAN WANG ◽  
HUILING XU ◽  
CHANGBING MA
Keyword(s):  
Dynamical Evolution ◽  
Laplacian Matrix ◽  
Convergence Speed ◽  
Weight Factor ◽  
Laplacian Eigenvalues ◽  
First Order ◽  
Laplacian Spectra ◽  
Weighted Laplacian ◽  
Structural Growth ◽  
Recursive Equations

We present the hierarchical graph for the growth of weighted networks in which the structural growth is coupled with the edges’ weight dynamical evolution. We investigate consensus problems of the graph from weighted Laplacian spectra perspective, focusing on three important quantities of consensus problems, convergence speed, delay robustness, and first-order coherence, which are determined by the second smallest eigenvalue, largest eigenvalue, and sum of reciprocals of each nonzero eigenvalue of weighted Laplacian matrix, respectively. Unlike previous enquiries, we want to emphasize the importance of weight factor in the study of coherence problems. In what follows, we attempt to study that the weighted Laplacian eigenvalues of the weighted hierarchical graphs, which are determined through analytic recursive equations. We find in our study that the value of convergence speed and delay robustness in weighted hierarchical graphs increases as weight factor increases and the value of first-order coherence decreases as weight factor increases. Moreover, as is expected, weight factor affects the performance of consensus behavior and can be regarded as a leverage in the problem of consensus problems. This paper puts forward the proposal and the countermeasure for stability optimization of networks from the perspective of weight factor for future researchers.

Study on Applications of Laplacian Spectra for a Network

2013 ◽  
Vol 753-755 ◽  
pp. 2859-2862
Author(s):  
Hai Tang Wang
Keyword(s):  
Recurrence Relations ◽  
Spanning Trees ◽  
Small World ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Scale Free ◽  
Structure And Dynamics ◽  
Laplacian Spectra ◽  
Number Of Spanning Trees ◽  
Biological Field

Systems composing of dynamical units are ubiquitous in nature, ranging from physical to technological, and to biological field. These systems can be naturally described by networks, knowledge of its Laplacian eigenvalues is central to understanding its structure and dynamics for a network. In this paper, we study the Laplacian spectra of a family with scale-free and small-world properties. Based on the obtained recurrence relations, we determine explicitly the product of all nonzero Laplacian eigenvalues, as well as the sum of the reciprocals of these eigenvalues. Then, using these results, we further evaluate the number of spanning trees, Kirchhoff index.

The entire mean weighted first-passage time on infinite families of weighted tree networks

2017 ◽  
Vol 31 (07) ◽  
pp. 1750049 ◽  
Author(s):  
Yanqiu Sun ◽  
Meifeng Dai ◽  
Shuxiang Shao ◽  
Weiyi Su
Keyword(s):  
First Passage Time ◽  
Passage Time ◽  
Theory And Practice ◽  
Characteristic Polynomials ◽  
Weighted Tree ◽  
First Passage ◽  
Tree Networks ◽  
Laplacian Eigenvalues ◽  
Weighted Networks ◽  
Simplified Calculation

We propose the entire mean weighted first-passage time (EMWFPT) for the first time in the literature. The EMWFPT is obtained by the sum of the reciprocals of all nonzero Laplacian eigenvalues on weighted networks. Simplified calculation of EMWFPT is the key quantity in the study of infinite families of weighted tree networks, since the weighted complex systems have become a fundamental mechanism for diverse dynamic processes. We base on the relationships between characteristic polynomials at different generations of their Laplacian matrix and Laplacian eigenvalues to compute EMWFPT. This technique of simplified calculation of EMWFPT is significant both in theory and practice. In this paper, firstly, we introduce infinite families of weighted tree networks with recursive properties. Then, we use the sum of the reciprocals of all nonzero Laplacian eigenvalues to calculate EMWFPT, which is equal to the average of MWFPTs over all pairs of nodes on infinite families of weighted networks. In order to compute EMWFPT, we try to obtain the analytical expressions for the sum of the reciprocals of all nonzero Laplacian eigenvalues. The key step here is to calculate the constant terms and the coefficients of first-order terms of characteristic polynomials. Finally, we obtain analytically the closed-form solutions to EMWFPT on the weighted tree networks and show that the leading term of EMWFPT grows superlinearly with the network size.

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