Evolving pseudofractal networks

Zhang, Zhongzhi; Zhou, Shuigeng; Chen, Lichao

doi:10.1140/epjb/e2007-00229-9

Cited by 51 publications

(45 citation statements)

References 52 publications

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“…For example, it has been suggested that for stochastic scale-free networks with degree distribution exponent γ < 3 and network order N, their average distance d(N) behaves as a double logarithmic scaling with N: d(N) ∼ ln ln N [7,6], which is in sharp contrast to the logarithmic scaling obtained for the BRV model addressed here, in despite of the fact that the latter has a degree distribution exponent γ = 1 + ln 3 ln 2 less than 3. Actually, this logarithmic scaling of average distance with network order has also been shown in other deterministic scale-free networks with γ < 3 [19,25,28,36,41,42]. Thus, deterministic scale-free networks present an obvious difference from their stochastic scale-free counterparts in the aspect of structural property of average distance.…”

Section: Analytical Resulstssupporting

confidence: 62%

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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Section: Analytical Resulstssupporting

confidence: 62%

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

et al. 2009

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“…It is also interesting to stress that this linear scaling is in contrast to the sub-linear scaling of mean trapping time obtained for the two-dimensional Apollonian network [60] and the pseudofractal scale-free web [67], in spite of the fact that they have a similar topological structure [57,58,59,62,63,64] as that of the Koch network. The reason for this disparity is worth studying in the future.…”

Section: B Analytical Solution For Mean Trapping Timementioning

confidence: 74%

“…Equation (39) also reveals that δ ij is a constant 1. Notice that the linear scaling of the average traverse time with network order has been previously obtained by numerical simulations for the Apollonian networks [37] and the pseudofractal scale-free web [41], both of which have been well studied [52,57,58,59,60,61,62,63,64,65,66,67].…”

Section: First-passage Time For All Nodesmentioning

confidence: 74%

Standard random walks and trapping on the Koch network with scale-free behavior and small-world effect

et al. 2009

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A vast variety of real-life networks display the ubiquitous presence of scale-free phenomenon and small-world effect, both of which play a significant role in the dynamical processes running on networks. Although various dynamical processes have been investigated in scale-free small-world networks, analytical research about random walks on such networks is much less. In this paper, we will study analytically the scaling of the mean first-passage time (MFPT) for random walks on scale-free small-world networks. To this end, we first map the classical Koch fractal to a network, called Koch network. According to this proposed mapping, we present an iterative algorithm for generating the Koch network, based on which we derive closed-form expressions for the relevant topological features, such as degree distribution, clustering coefficient, average path length, and degree correlations. The obtained solutions show that the Koch network exhibits scale-free behavior and small-world effect. Then, we investigate the standard random walks and trapping issue on the Koch network. Through the recurrence relations derived from the structure of the Koch network, we obtain the exact scaling for the MFPT. We show that in the infinite network order limit, the MFPT grows linearly with the number of all nodes in the network. The obtained analytical results are corroborated by direct extensive numerical calculations. In addition, we also determine the scaling efficiency exponents characterizing random walks on the Koch network.

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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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Evolving pseudofractal networks

Cited by 51 publications

References 52 publications

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

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

Standard random walks and trapping on the Koch network with scale-free behavior and small-world effect

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

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