Loops of any size and Hamilton cycles in random scale-free networks

Bianconi, Ginestra; Marsili, Matteo

doi:10.1088/1742-5468/2005/06/p06005

Cited by 74 publications

(127 citation statements)

References 28 publications

(51 reference statements)

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“…The clustering coefficient is the ratio of number of loops of size 3 [16,17] to the number of triples of connected nodes, which is i k i (k i − 1). So using Eq.…”

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confidence: 99%

Scale-free networks with an exponent less than two

2006

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We study scale free simple graphs with an exponent of the degree distribution γ less than two. Generically one expects such extremely skewed networks -which occur very frequently in systems of virtually or logically connected units -to have different properties than those of scale free networks with γ > 2: The number of links grows faster than the number of nodes and they naturally posses the small world property, because the diameter increases by the logarithm of the size of the network and the clustering coefficient is finite. We discuss a simple prototype model of such networks, inspired by real world phenomena, which exhibits these properties and allows for a detailed analytical investigation. [7]. Indeed in each of these systems nodes -web pages or actors -are linked -by hyperlinks or collaboration in the same movie -to a number k of other nodes, which is called the degree of the node [27], and which obeys a power law distribution P (k) ∼ k −γ . In many cases (table I) the exponent γ of such a distribution is larger than two which its occurrence has been related to some interaction mechanismsuch as preferential attachment [3] -in simplified models.Scale-free networks with an exponent γ < 2 have received less attention, despite of their widespread appearance (table I), in the peer-to-peer Gnutella network [28] [8, 9], outgoing E-mails network [10], traffic in networks [11], co-authorship network in high energy physics [12] and in the network of dependency among software packages [13,14].The aim of this letter is to show that simple graphs with γ < 2 have markedly different properties than simple graphs with γ > 2. We shall do this first on the basis of general arguments and then using a prototype model motivated by the above mentioned real networks. This model reproduces all the discussed generic properties. Furthermore we show that its generalization to a weighted network exhibits non-trivial statistical properties.

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“…The clustering coefficient is the ratio of number of loops of size 3 [16,17] to the number of triples of connected nodes, which is i k i (k i − 1). So using Eq.…”

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confidence: 99%

Scale-free networks with an exponent less than two

2006

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“…For example, ER graphs with finite average connectivity have a finite number of finite loops [4,7]. On the contrary scale-free graphs have a number of finite loops which increases with the number N of vertices, provided that γ ≤ 3 [8,9]. The abundance of some subgraphs of small size -the so-called motifs -in biological networks has been shown to be related to important functional properties selected by evolution [10,11,12].…”

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confidence: 99%

Emergence of large cliques in random scale-free networks

Bianconi

Marsili

2006

Europhys. Lett.

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In a network cliques are fully connected subgraphs that reveal which are the tight communities present in it. Cliques of size c > 3 are present in random Erdös and Renyi graphs only in the limit of diverging average connectivity. Starting from the finding that real scale free graphs have large cliques, we study the clique number in uncorrelated scale-free networks finding both upper and lower bounds. Interesting we find that in scale-free networks large cliques appear also when the average degree is finite, i.e. even for networks with power-law degree distribution exponents ! ! (2, 3). Moreover as long as ! < 3 scale-free networks have a maximal clique which diverges with the system size

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“…Analysis of correlations in complex networks have been in fact mainly confined to the degree of adjacent nodes [3,4,11], or to the presence of loops and cliques [15][16][17] which are a manifestation of correlations. Particular attention has been paid to the "configuration model" [1,3], i.e., the "uncorrelated" network model"…”

Section: Massimoostilli@gmailcommentioning

confidence: 99%

“…It is well known that, in the thermodynamic limit, in the configuration model the presence of loops of finite length is negligible, provided the exponent γ of the associated power law degree-distribution P (k) ∼ k −γ is sufficiently large [3,15]. In turn, this has led perhaps to the erroneous conclusion that, in any synthetic or real network, fluctuations are always negligible for sufficiently large γ.…”

Section: Massimoostilli@gmailcommentioning

confidence: 99%

Fluctuations of motifs and non–self-averaging in complex networks: A self- vs. non–self-averaging phase transition scenario

Ostilli¹

2014

EPL

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-Complex networks have been mostly characterized from the point of view of the degree distribution of their nodes and a few other motifs (or modules), with a special attention to triangles and cliques. The most exotic phenomena have been observed when the exponent γ of the associated power law degree-distribution is sufficiently small. In particular, a zero percolation threshold takes place for γ < 3, and an anomalous critical behavior sets in for γ < 5. In this Letter we prove that in sparse scale-free networks characterized by a cut-off scaling with the sistem size N , relative fluctuations are actually never negligible: given a motif Γ, we analyze the relative fluctuations RΓ of the associated density of Γ, and we show that there exists an interval in γ, [γ1, γ2], where RΓ does not go to zero in the thermodynamic limit, where γ1 ≈ kmin and γ2 ≈ 2kmax, kmin and kmax being the smallest and the largest degree of Γ, respectively. Remarkably, in (γ1, γ2) RΓ diverges, implying the instability of Γ to small perturbations.

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Loops of any size and Hamilton cycles in random scale-free networks

Cited by 74 publications

References 28 publications

Scale-free networks with an exponent less than two

Scale-free networks with an exponent less than two

Emergence of large cliques in random scale-free networks

Fluctuations of motifs and non–self-averaging in complex networks: A self- vs. non–self-averaging phase transition scenario

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