Maximum matchings and minimum dominating sets in Apollonian networks and extended Tower of Hanoi graphs

Jin, Yujia; Li, Huan; Zhang, Zhongzhi

doi:10.1016/j.tcs.2017.08.024

Cited by 17 publications

(5 citation statements)

References 52 publications

(85 reference statements)

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“…The Sierpiński graphs S(n, k) = (V(S(n, k)), E(S(n, k))) (n ≥ 1 and k ≥ 3) were introduced by Klavžar and Milutinović [38] as a two-parametric generalization of the Tower of Hanoi graph [60], [61]. They are defined on the vertex set comprising of all n-tuples of integers 1, 2, · · · , k, that is, V(S(n, k)) = {1, 2, · · · , k} n .…”

Section: B Sierpiński Graphsmentioning

confidence: 99%

Consensus in Self-Similar Hierarchical Graphs and Sierpiński Graphs: Convergence Speed, Delay Robustness, and Coherence

Zhang

et al. 2019

IEEE Trans. Cybern.

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The hierarchical graphs and Sierpiński graphs are constructed iteratively, which have the same number of vertices and edges at any iteration, but exhibit quite different structural properties: the hierarchical graphs are nonfractal and small-world, while the Sierpiński graphs are fractal and ''large-world.'' Both graphs have found broad applications. In this paper, we study consensus problems in hierarchical graphs and Sierpiński graphs, focusing on three important quantities of consensus problems, that is, convergence speed, delay robustness, and coherence for first-order (and second-order) dynamics, which are, respectively, determined by algebraic connectivity, maximum eigenvalue, and sum of reciprocal (and square of reciprocal) of each nonzero eigenvalue of Laplacian matrix. For both graphs, based on the explicit recursive relation of eigenvalues at two successive iterations, we evaluate the second smallest eigenvalue, as well as the largest eigenvalue, and obtain the closed-form solutions to the sum of reciprocals (and square of reciprocals) of all nonzero eigenvalues. We also compare our obtained results for consensus problems on both graphs and show that they differ in all quantities concerned, which is due to the marked difference of their topological structures.

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Section: B Sierpiński Graphsmentioning

confidence: 99%

Consensus in Self-Similar Hierarchical Graphs and Sierpiński Graphs: Convergence Speed, Delay Robustness, and Coherence

Zhang

et al. 2019

IEEE Trans. Cybern.

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“…At present, many network operations have been used to design and construct complex network models and to study the topological and dynamic properties on them, including: q−subdivision, planar triangulation, Cartesian product, hierarchical product, corona product, Kronecker product, etc. [Wood, 2005;Jin et al, 2017;Imrich & Klavzar, 2000;Barriere et al, 2016;Sharma et al, 2017;Mahdian & Xu, 2007]. However, the network iteration methods in the above articles all use nodes as the basic unit, and the iterative model with local node sets as the unit has never been studied.…”

Section: Introductionmentioning

confidence: 99%

Spectral Analysis and its applications for a class of scale-free network based on the weighted m-clique annex operation

Zhang¹,

Alsaedi²,

Wu³

2022

Preprint

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The spectrum of network is an important tool to study the function and dynamic properties of network, and graph operation and product is an effective mechanism to construct a specific local and global topological structure. In this study, a class of weighted m−clique annex operation τ r m (•) controlled by scale factor m and weight factor r is defined, through which an iterative weighted network model G t with small-world and scale-free properties is constructed. In particular, when the number of iterations t tends to infinity, the network has transfinite fractal property. Then, through the iterative features of the network structure, the iterative relationship of the eigenvalues of the normalized Laplacian matrix corresponding to the network is studied. Accordingly, some applications of the spectrum of the network, including the Kenemy constant, Multiplicative Degree-Kirchhoff index and the number of weighted spanning trees, are further given. In addition, we also study the effect of the two factors controlling network operation on the structure and function of the iterative weighted network G t , so that the network operation can better simulate the real network and have more application potential in the field of artificial network.

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“…These striking structural patterns have great impact on other structural and dynamical properties of networks. For example, scale-free topology strongly affects structural characteristics (e.g., perfect matchings [39] and minimum dominating sets [40]) and dynamical processes (e.g., epidemic spreading [41], game [42], controllability [43]) on scale-free networks.…”

Section: Introductionmentioning

confidence: 99%

Scale-Free Loopy Structure is Resistant to Noise in Consensus Dynamics in Complex Networks

Zhang

Patterson

2020

IEEE Trans. Cybern.

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The vast majority of real-world networks are scalefree, loopy, and sparse, with a power-law degree distribution and a constant average degree. In this paper, we study first-order consensus dynamics in binary scale-free networks, where vertices are subject to white noise. We focus on the coherence of networks characterized in terms of the H2-norm, which quantifies how closely agents track the consensus value. We first provide a lower bound of coherence of a network in terms of its average degree, which is independent of the network order. We then study the coherence of some sparse, scale-free real-world networks, which approaches a constant. We also study numerically the coherence of Barabási-Albert networks and high-dimensional random Apollonian networks, which also converges to a constant when the networks grow. Finally, based on the connection of coherence and the Kirchhoff index, we study analytically the coherence of two deterministically-growing sparse networks and obtain the exact expressions, which tend to small constants. Our results indicate that the effect of noise on the consensus dynamics in power-law networks is negligible. We argue that scale-free topology, together with loopy structure, is responsible for the strong robustness with respect to noisy consensus dynamics in power-law networks.

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Maximum matchings and minimum dominating sets in Apollonian networks and extended Tower of Hanoi graphs

Cited by 17 publications

References 52 publications

Consensus in Self-Similar Hierarchical Graphs and Sierpiński Graphs: Convergence Speed, Delay Robustness, and Coherence

Consensus in Self-Similar Hierarchical Graphs and Sierpiński Graphs: Convergence Speed, Delay Robustness, and Coherence

Spectral Analysis and its applications for a class of scale-free network based on the weighted m-clique annex operation

Scale-Free Loopy Structure is Resistant to Noise in Consensus Dynamics in Complex Networks

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