Scale-free networks with an exponent less than two

Abstract

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.

Fundamental Structural Constraint of Random Scale-Free Networks

“…[14] This also agrees with the upper cutoff found by Seyed-allaei et al [16], which is required for scale-free networks with γ < 2 generated using a node-fitness mechanism [17]. We also numerically found that the cutoff coefficient c affects the realizability of degree sequences for the special case of the linear cutoff α = 1, which has been overlooked.…”

Redes Sociais e Complexas: um modelo computacional para a investigação da pós-graduação Brasileira em Ensino de Física

“…Por meio da metodologia referenciada no tópico anterior, foi possível a construção da rede semântica apresentada na Figura 1: [1] apresentaram γ ≤ 2,2 e γ < 2, na pesquisa de Seyed-allaei et al [10]. Há uma preferência na rede por determinados vértices (hubs), conforme Quadro 2:…”

Emergence of Scale-free Spike Flow Graphs in Recurrent Neural Networks

“…These approximations are rather rough, since the tail of the distribution is scattered thus biasing the outcome exponents, but one can say with a high degree of confidence that γ in < γ out and that γ in < 2. Scale-free networks with exponents within range of (1, 2) were investigated in [17]. It is worth mentioning that in all instances of the simulation the output spike flow network had only one connected component, which was not Spectral density for the analyzed network (after dropping edge directions) and for the corresponding Erdős-Rényi random graph with the same edge density (exhibiting the usual semicircular law) for comparison.…”