Full eigenvalues of the Markov matrix for scale-free polymer networks

Zhang, Zhongzhi; Guo, Xiaoye; Lin, Yandan

doi:10.1103/physreve.90.022816

Cited by 18 publications

(20 citation statements)

References 55 publications

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“…Thus, for the studied scale-free network, the random target access time grows linearly with the number of nodes, which is the minimal scaling for random walks on graphs [59] and is in sharp contrast to those previously obtained for other networks [28][29][30][33][34][35]59], where the random target access timeF scales with the network size N asF ∼ N θ (θ > 1) or F ∼ N ln N .…”

Section: B Random Target Access Timementioning

confidence: 51%

“…More specifically, for T g , the number of different eigenvalues N g (λ) is 2g + 1, which has a logarithmic dependence on the network size N g , that is, N g (λ) ∼ ln N g . This should be compared with that for other previously studied deterministic networks, including Sierpinski gaskets [30], fractal trees [34], and polymer networks [35], for which the number of different eigenvalues behaves linearly with the network size.…”

Section: Spectral Densitymentioning

confidence: 98%

“…Thus, we have obtained exact simple expressions for all eigenvalues of the transition matrix P g . We note that analytical solutions to eigenvalues of transition matrix for other deterministic networks are [28][29][30][33][34][35] invariably of the forms having radicals or trigonometric functions.…”

Section: B Explicit Expressions For All Eigenvaluesmentioning

confidence: 99%

“…In the context of the transition matrix, a conscientious effort has been devoted to determine the eigenvalues for some classic fractals [28][29][30][31] and treelike networks [32][33][34][35]. However, these previously studied networks cannot simultaneously mimic some common and distinctive properties of real-world systems [36][37][38], including scale-free behavior [39] and the small-world effect [40] characterized by a small average distance and a high clustering coefficient.…”

Section: Introductionmentioning

confidence: 99%

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

2015

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The eigenvalues of the transition matrix for random walks on a network play a significant role in the structural and dynamical aspects of the network. Nevertheless, it is still not well understood how the eigenvalues behave in small-world and scale-free networks, which describe a large variety of real systems. In this paper, we study the eigenvalues for the transition matrix of a network that is simultaneously scale-free, small-world, and clustered. We derive explicit simple expressions for all eigenvalues and their multiplicities, with the spectral density exhibiting a power-law form. We then apply the obtained eigenvalues to determine the mixing time and random target access time for random walks, both of which exhibit unusual behaviors compared with those for other networks, signaling discernible effects of topologies on spectral features. Finally, we use the eigenvalues to count spanning trees in the network.

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Section: B Random Target Access Timementioning

confidence: 51%

Section: Spectral Densitymentioning

confidence: 98%

Section: B Explicit Expressions For All Eigenvaluesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2015

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“…Asymptotic results related to the circular law were obtained for Erdös-Renyi graphs with mean connectivity diverging in the thermodynamic limit [20]. For some recent related results concerning spectra of graph Laplacians, we refer to [21][22][23].…”

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confidence: 99%

Spectra of random stochastic matrices and relaxation in complex systems

Kühn

2015

EPL

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We compute spectra of large stochastic matrices W , defined on sparse random graphs, where edges (i, j) of the graph are given positive random weights Wij > 0 in such a fashion that column sums are normalized to one. We compute spectra of such matrices both in the thermodynamic limit, and for single large instances. The structure of the graphs and the distribution of the non-zero edge weights Wij are largely arbitrary, as long as the mean vertex degree remains finite in the thermodynamic limit and the Wij satisfy a detailed balance condition. Knowing the spectra of stochastic matrices is tantamount to knowing the complete spectrum of relaxation times of stochastic processes described by them, so our results should have many interesting applications for the description of relaxation in complex systems. Our approach allows to disentangle contributions to the spectral density related to extended and localized states, respectively, allowing to differentiate between time-scales associated with transport processes and those associated with the dynamics of local rearrangements. There are numerous processes, both natural and artificial, which can be understood in terms of random walks on complex networks [1][2][3], including the spread of diseases in social networks [4,5], the transmission of information in communication networks (e.g.[6]), search algorithms [7,8], the out-of-equilibrium dynamics of glassy systems at low temperatures as described in terms of hopping between long-lived states in state space [9][10][11], the dynamics of major conformational changes in macro-molecules [12], or cell-signalling through protein-protein interaction networks [13], to name but a few. For reviews that cover several of these topics, see e.g. [14][15][16].The purpose of the present letter is to use random matrix theory to contribute to the understanding of systems of this type. We compute spectra of transition matrices for discrete Markov chains describing stochastic dynamics in complex systems. We construct these in terms of sparse random graphs in such a way that an edge (i, j) in a graph corresponds to a possible transition j → i, with the edge weight W ij > 0 quantifying the associated transitions probability, requiring i W ij = 1 for all j. We are interested in the limit, where the number N of possible states becomes large, with the average number of possible transitions at each state remaining finite in the thermodynamic limit (N → ∞).Given a time-dependent probability vector p(t) = (p i (t)), we have an evolution equation of the formThe condition W ij ≥ 0 for all (i, j) and the column sum constraint together entail that the spectrum of W is contained in the unit disc of the complex plane, σ(W ) ⊆ {z; |z| ≤ 1}. If W satisfies a detailed balance condition with an equilibrium distribution, p i = p eq i , such that W ij p j = W ji p i for all pairs (i, j), then W can be symmetrized by a similarity transformation -Our main interest here is the relation between eigenvalues of W and relaxation times of the Markov chain it describes...

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