Incompatibility networks as models of scale-free small-world graphs

Zhang, Zhongzhi; Zhou, Shuigeng; Zou, Tao; Chen, Lichao; Guan, Jihong

doi:10.1140/epjb/e2007-00344-7

Cited by 35 publications

(35 citation statements)

References 49 publications

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“…The fact that there is no degree correlation in RSN is compared with that of its deterministic variant, which shows a negative degree correlation [11]. We guess that the reason behind this difference between RSN and its deterministic counterpart may stem from a biased choice of "active triangles".…”

Section: Degree Correlationsmentioning

confidence: 99%

“…The analytical approaches are completely deferent from those applied to the deterministic Sierpinski network [11].…”

Section: Structural Characteristicsmentioning

confidence: 99%

“…From this famous fractal we have defined the Sierpinski network [11] as illustrated in figure 1(b). The translation from the fractal to network generation is quite straightforward.…”

Section: Brief Introduction To Sierpinski Networkmentioning

confidence: 99%

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Random Sierpinski network with scale-free small-world and modular structure

Zhang

Zhou

et al. 2008

Eur. Phys. J. B

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Abstract. In this paper, we define a stochastic Sierpinski gasket, on the basis of which we construct a network called random Sierpinski network (RSN). We investigate analytically or numerically the statistical characteristics of RSN. The obtained results reveal that the properties of RSN is particularly rich, it is simultaneously scale-free, small-world, uncorrelated, modular, and maximal planar. All obtained analytical predictions are successfully contrasted with extensive numerical simulations. Our network representation method could be applied to study the complexity of some real systems in biological and information fields.

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Section: Degree Correlationsmentioning

confidence: 99%

“…The analytical approaches are completely deferent from those applied to the deterministic Sierpinski network [11].…”

Section: Structural Characteristicsmentioning

confidence: 99%

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Random Sierpinski network with scale-free small-world and modular structure

Zhang

Zhou

et al. 2008

Eur. Phys. J. B

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“…To mimic simultaneously the two prominent characteristics, Barabási, Ravasz and Vicsek proposed a deterministic model [24], hereafter called BRV model, which is the progenitor of deterministic models for complex networks and has led to an increasing number of theoretical investigations on deterministic networks that are an interesting class of networks and have been proved to be a useful tool [25,26,27,28,29,30,31,32,33,34,35,36]. Many structural and dynamical properties of the BRV model have been studied in much detail, including degree distribution [24], spectra of adjacency matrix [37], random walks [38], to name but a few.…”

Section: Introductionmentioning

confidence: 99%

Average distance in a hierarchical scale-free network: an exact solution

et al. 2009

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Abstract. Various real systems simultaneously exhibit scale-free and hierarchical structure. In this paper, we study analytically average distance in a deterministic scale-free network with hierarchical organization. Using a recursive method based on the network construction, we determine explicitly the average distance, obtaining an exact expression for it, which is confirmed by extensive numerical calculations. The obtained rigorous solution shows that the average distance grows logarithmically with the network order (number of nodes in the network). We exhibit the similarity and dissimilarity in average distance between the network under consideration and some previously studied networks, including random networks and other deterministic networks. On the basis of the comparison, we argue that the logarithmic scaling of average distance with network order could be a generic feature of deterministic scalefree networks.

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“…We call them deterministic uniform recursive trees (DURTs), since they are deterministic versions of URT. This kind of deterministic models have received considerable attention from the scientific communities and have turned out to be a useful tool [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51]. Although uniform recursive tree is well understood [26,28,29,30,31,52,53], relatively less is known about the structural and other nature of the DURTs [33].…”

Section: Introductionmentioning

confidence: 99%

Structural and spectral properties of a family of deterministic recursive trees: rigorous solutions

Qi¹,

Zhang²,

Ding³

et al. 2009

J. Phys. A: Math. Theor.

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Abstract. As one of the most significant models, the uniform recursive tree (URT) has found many applications in a variety of fields. In this paper, we study rigorously the structural features and spectral properties of the adjacency matrix for a family of deterministic uniform recursive trees (DURTs) that are deterministic versions of URT. Firstly, from the perspective of complex networks, we investigate analytically the main structural characteristics of DURTs, and obtain the accurate solutions for these properties, which include degree distribution, average path length, distribution of node betweenness, and degree correlations. Then we determine the complete eigenvalues and their corresponding eigenvectors of the adjacency matrix for DURTs. Our research may shed light in better understanding of the features for URT. Also, the analytical methods used here is capable of extending to many other deterministic networks, making the precise computation of their properties (especially the full spectrum characteristics) possible.

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Incompatibility networks as models of scale-free small-world graphs

Cited by 35 publications

References 49 publications

Random Sierpinski network with scale-free small-world and modular structure

Random Sierpinski network with scale-free small-world and modular structure

Average distance in a hierarchical scale-free network: an exact solution

Structural and spectral properties of a family of deterministic recursive trees: rigorous solutions

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