Spectrum of walk matrix for Koch network and its application

Various structural and dynamical properties of a network are encoded in the eigenvalues of walk matrix describing random walks on the network. In this paper, we study the spectra of walk matrix of the Koch network, which displays the prominent scale-free and small-world features. Utilizing the parti...

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Bibliographic Details
Published in:The Journal of chemical physics 2015-06, Vol.142 (22), p.224106-224106
Main Authors: Xie, Pinchen, Lin, Yuan, Zhang, Zhongzhi
Format: Article
Language:English
Subjects:
Access time
Coding
Eigenvalues
Graph theory
Physics
Random walk
Random walk theory
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Summary:Various structural and dynamical properties of a network are encoded in the eigenvalues of walk matrix describing random walks on the network. In this paper, we study the spectra of walk matrix of the Koch network, which displays the prominent scale-free and small-world features. Utilizing the particular architecture of the network, we obtain all the eigenvalues and their corresponding multiplicities. Based on the link between the eigenvalues of walk matrix and random target access time defined as the expected time for a walker going from an arbitrary node to another one selected randomly according to the steady-state distribution, we then derive an explicit solution to the random target access time for random walks on the Koch network. Finally, we corroborate our computation for the eigenvalues by enumerating spanning trees in the Koch network, using the connection governing eigenvalues and spanning trees, where a spanning tree of a network is a subgraph of the network, that is, a tree containing all the nodes.
ISSN:0021-9606
1089-7690
DOI:10.1063/1.4922265