Eigenvalues for the transition matrix of a small-world scale-free network: Explicit expressions and applications

Zhang, Zhongzhi; Lin, Yandan; Guo, Xiaoye

doi:10.1103/physreve.91.062808

Cited by 20 publications

(9 citation statements)

References 57 publications

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“…✷ The two particular eigenvalues 1 and 3 2 of L n are called exceptional eigenvalues for the family of matrices {L n }, for which the spectra shows self-similarity characteristics [1,34]. Theorem 3.5 confirms a previous result on the spectrum of the scalefree pseudofractal graph, obtained with other related methods by one of the authors [39].…”

Section: Definition 22supporting

confidence: 82%

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

Xie

Zhang

Comellas

2016

Applied Mathematics and Computation

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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.Postprint (author's final draft

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Section: Definition 22supporting

confidence: 82%

“…Finally, the values for Kemeny's constant and the number of spanning trees of a scale-free pseudofractal graph, which is another particular case of n-triangulation graph, were obtained by one of the authors in [39]. We find the same expressions by using N 0 = E 0 = 3 in Eq.…”

Section: Spanning Trees Theorem 43supporting

confidence: 60%

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

Xie

Zhang

Comellas

2016

Applied Mathematics and Computation

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“…The first-passage problem on a (1, 2)-flower has been extensively investigated in the past few years [14,[42][43][44][45]; the mean of the FPT to a given hub is also obtained for some special cases, such as (u = 2, v = 2) and (u = 1, v = 3) [13,17]. However, for generic u and v, the exact expressions for FPT, FRT, and GFPT at finite size N t are still unknown and will be obtained in this paper.…”

Section: Network Modelmentioning

confidence: 99%

Scaling laws for diffusion on (trans)fractal scale-free networks

Peng,

Agliari

2019

Preprint

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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 df . In this work we investigate dynamic processes (i.e., random walks) and topological properties (i.e., the Laplacian spectrum) and we show that, under proper conditions, the same scalings (ruled by the related dimensions), emerge for both fractal and transfractal.

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“…First passage time (FPT), which is the time it takes a random walker to reach a given site for the first time, and first return time (FRT), which is the time it takes a random walker to return to the starting site for the first time, are two important quantities inTherefore, the PSFW has attracted lots of attentions in the past several years and much effort has been devoted to the study of its properties, such as degree distribution, degree correlation, clustering coefficient [47,50], diameter [50], average path length [49], the number of spanning trees [51], and eigenvalues [52]. As for random walks on the PSFW, the MFRT for any node v is 2m/d v ; Zhang and etc [53] obtained the recursive relation of the MFPT from any starting node to the hub (i.e., node with highest degree) and then gained the mean trap time to the hub by averaging the MFPTs over all the possible starting nodes.…”

Section: Introductionmentioning

confidence: 99%

Volatilities analysis of first-passage time and first-return time on a small-world scale-free network

Peng¹

2016

J. Stat. Mech.

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Abstract.In this paper, we study random walks on a small-world scale-free network, also called as pseudofractal scale-free web (PSFW), and analyze the volatilities of first passage time (FPT) and first return time (FRT) by using the variance and the reduced moment as the measures. Note that the FRT and FPT are deeply affected by the starting or target site. We don't intend to enumerate all the possible cases and analyze them. We only study the volatilities of FRT for a given hub (i.e., node with highest degree) and the volatilities of the global FPT (GFPT) to a given hub, which is the average of the FPTs for arriving at a given hub from any possible starting site selected randomly according to the equilibrium distribution of the Markov chain. Firstly, we calculate exactly the probability generating function of the GFPT and FRT based on the self-similar structure of the PSFW. Then, we calculate the probability distribution, the mean, the variance and reduced moment of the GFPT and FRT by using the generating functions as a tool. Results show that: the reduced moment of FRT grows with the increasing of the network order N and tends to infinity while N → ∞; but for the reduced moments of GFPT, it is almost a constant(≈ 1.1605) for large N . Therefore, on the PSFW of large size, the FRT has huge fluctuations and the estimate provided by MFRT is unreliable, whereas the fluctuations of the GFPT is much smaller and the estimate provided by its mean is more reliable. The method we propose can also be used to analyze the volatilities of FPT and FRT on other networks with selfsimilar structure, such as (u, v) flowers and recursive scale-free trees.Volatilities of FPT and FRT on a small-world scale-free network 2 IntroductionFirst passage time (FPT), which is the time it takes a random walker to reach a given site for the first time, and first return time (FRT), which is the time it takes a random walker to return to the starting site for the first time, are two important quantities in the random walk literature [1][2][3][4]. The importance lies in the fact that many physical processes are controlled by first passage events [5][6][7][8][9][10][11][12], and that FRT can model the time intervals between two successive extreme events [13][14][15][16][17], such as traffic jams in roads, the floods, the droughts, and power blackouts in electrical power grid, etc [18][19][20]. Both FPT and FRT are random variables which can not be determined exactly and researchers can only try to find suitable quantities to estimate them. A first step consists in the analysis of the mean of the two random variables, the mean first-passage time (MFPT) and the mean first return time (MFRT). For the MFRT, it can be calculated from the stationary distribution directly. That is to say, the MFRT of node v is 2m/d v on any finite connected network, where m is the total numbers of edges of the network and d v is the degree of node v [21,22]. For the MFPT, general formula to calculate the MPFT between any two nodes in any finite networks was presented [...

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Eigenvalues for the transition matrix of a small-world scale-free network: Explicit expressions and applications

Cited by 20 publications

References 57 publications

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

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

Scaling laws for diffusion on (trans)fractal scale-free networks

Volatilities analysis of first-passage time and first-return time on a small-world scale-free network

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