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. Many complex systems can be modeled as networks, i.e., as a set of connections (edges) linking discrete elements (nodes) [1] [2] [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 networks, in which the degree distribution follows a powerlaw, [4] [5] [6] [7] [8] [9] [10] . In particular, scale-freeness has been shown to have important implications in the thermodynamic limit. For studying the properties of scalefree networks, several generative models have been proposed [1] [2] [3] [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 distributions 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 property, which is regularly exploited in data storage and algorithms, also many examples of dense networks are known [13] [14] [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 powerlaw degree distribution must have γ < 0. Calling such networks scale-free is at best dubious because they would not exhibit the characteristic properties commonly associated 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 degree distribution can be considered as a two-step procedure. 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 distribution 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 graphical, 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 impossible 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 Erdős-Gallai theorem, stated here as reformulated in [17] using recurrence relations:  where L k and R k are given by the recurrence relations and R 0 = N − 1 (4)
In general, online social networks are sparse REF , meaning that the total amount of potential links to be created is much greater than the links that are actually created.
All scale-free networks are sparse
Phys. Rev. Lett. 107, 178701 (2011)
Medicine Physics Computer Science