All scale-free networks are sparse

📝 Abstract

We study the realizability of scale free-networks with a given degree sequence, showing that the fraction of realizable sequences undergoes two first-order transitions at the values 0 and 2 of the power-law exponent. We substantiate this finding by analytical reasoning and by a numerical method, proposed here, based on extreme value arguments, which can be applied to any given degree distribution. Our results reveal a fundamental reason why large scale-free networks without constraints on minimum and maximum degree must be sparse.

💡 Analysis

We study the realizability of scale free-networks with a given degree sequence, showing that the fraction of realizable sequences undergoes two first-order transitions at the values 0 and 2 of the power-law exponent. We substantiate this finding by analytical reasoning and by a numerical method, proposed here, based on extreme value arguments, which can be applied to any given degree distribution. Our results reveal a fundamental reason why large scale-free networks without constraints on minimum and maximum degree must be sparse.

📄 Content

arXiv:1106.5150v2 [physics.soc-ph] 26 Aug 2011 All scale-free networks are sparse Charo I. Del Genio,1 Thilo Gross,1 and Kevin E. Bassler2, 3 1Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Straße 38, 01187 Dresden, Germany 2Department of Physics, 617 Science and Research 1, University of Houston, Houston, Texas 77204-5005, USA 3Texas Center for Superconductivity, 202 Houston Science Center, University of Houston, Houston, Texas 77204-5002, USA (Dated: September 10, 2018) We study the realizability of scale free-networks with a given degree sequence, showing that the fraction of realizable sequences undergoes two first-order transitions at the values 0 and 2 of the power-law exponent. We substantiate this finding by analytical reasoning and by a numerical method, proposed here, based on extreme value arguments, which can be applied to any given degree distribution. Our results reveal a fundamental reason why large scale-free networks without constraints on minimum and maximum degree must be sparse. PACS numbers: 89.75.Hc 89.75.-k 02.10.Ox 89.65.-s Many complex systems can be modeled as networks, i.e., as a set of connections (edges) linking discrete ele- ments (nodes) [1–3]. A characteristic of a network that affects many physical properties is its degree distribution P (k), the probability of finding a node with k edges. Considerable attention has been paid to scale-free net- works, in which the degree distribution follows a power- law, P (k) ∼k−γ [4–10]. In particular, scale-freeness has been shown to have important implications in the ther- modynamic limit. For studying the properties of scale- free networks, several generative models have been pro- posed [1–4]. However, no models creating networks with γ < 2 have been found [11, 12], and γ < 2 is observed only in networks that are relatively small or in which the power-law behavior has some cutoff[9]. In this Letter, we explain the absence of large networks that exhibit a power-law with 0 < γ < 2 in the tail of the distribution. Specifically, we show that fundamental constraints exist that prevent the realization of any such network. It is well known that the mean degree of scale-free dis- tributions with exponents γ less than 2 diverges in the thermodynamic limit, i.e., when the number of nodes N →∞[2]. Scale-free networks with γ < 2 would therefore be called dense networks, whereas networks γ > 2 are sparse. While sparseness is a common prop- erty, which is regularly exploited in data storage and algorithms, also many examples of dense networks are known [13–15]. It is thus reasonable to ask why there are no examples of dense scale-free networks. We answer this question by showing that dense networks with a power- law degree distribution must have γ < 0. Calling such networks scale-free is at best dubious because they would not exhibit the characteristic properties commonly asso- ciated with scale-freeness for N →∞. The absence of dense scale-free networks is explained by a discontinuous transition in the realizability of such networks. Below, we show numerically, analytically, and by a hybrid method proposed here, that the probability of finding a scale free-network with a given γ is 0 for 0 < γ < 2. We emphasize that these results are not contingent on a specific generative model, but arise directly from fundamental mathematical constraints. The generation of scale-free networks with a given de- gree distribution can be considered as a two-step proce- dure. First, one creates a number of nodes and assigns to each node a number of connection “stubs” drawn from the degree distribution. The realization of the degree dis- tribution that is thus created is called degree sequence. Second, one connects the stubs such that every stub on a given node links to a stub on a different node, without forming self-loops or double links. However, not every degree sequence can be realized in a network. Sequences that admit realizations as simple graphs are called graph- ical, and their realizability property is commonly referred to as graphicality [18]. Graphicality fails trivially if the number of stubs is odd, as one needs two stubs to form every link, or if the degree of any node is equal to or greater than the number of nodes, as it would be impos- sible to connect all its stubs to different nodes. Below we do not consider sequences for which graphicality is such trivially violated, but note that further conditions must be met for a sequence to be graphical [16, 17]. The main result used for testing the graphicality of a degree sequence is the Erd˝os-Gallai theorem, stated here as reformulated in [17] using recurrence relations: Theorem 1. Let D = {d0, d1, . . . , dN−1} be a non- increasing degree sequence on N nodes. Define xk = min {i : di < k + 1} and k⋆= min {i : xi < i + 1}. Then, D is graphical if and only if PN−1 i=0 di is even, and Lk ⩽Rk ∀0 ⩽k < N −1 , (1) where Lk and Rk are given by the recurrence relations L0 = d0 (2) Lk = Lk−1 + dk (3) and R0

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