We find that scale-free random networks are excellently modeled by a deterministic graph. This graph has a discrete degree distribution (degree is the number of connections of a vertex) which is characterized by a power-law with exponent γ = 1 + ln 3/ ln 2. Properties of this simple structure are surprisingly close to those of growing random scale-free networks with γ in the most interesting region, between 2 and 3. We succeed to find exactly and numerically with high precision all main characteristics of the graph. In particular, we obtain the exact shortest-path-length distribution. For the large network (ln N ≫ 1) the distribution tends to a Gaussian of width ∼ √ ln N centered at ℓ ∼ ln N . We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent 2 + γ. 05.10.-a, 87.18.Sn The essence of the modern situation in network science is the change-over from study of classical random graphs with Poisson degree distributions [1,2] to exploration of complex networks with fat-tailed degree distributions [3] [4] [5] [6] [7] [8] . The prominent particular case of such nets are networks with power-law degree distributions (scalefree networks) [3] . While growing, such nets actually selforganize into scale-free structures. These networks play a great role in Nature [4] [5] [6] . The Internet, the WWW, and many basic biological networks belong to this class. Fat tails of the degree distributions produce a number of intriguing effects [9] [10] [11] [12] [13] [14] . Such networks are widespread, but very little is still known even about their basic properties [15, 16] . Most of real growing scale-free networks have γ exponent of the degree distribution P (k) ∼ k −γ in the range (2, 3), but this case turned to be the most difficult and unexplored. In particular, no exact results for the average shortest-path length ℓ are known in this situation. The only known exact shortest-path length distributions were obtained for the simplest equilibrium networks [17, 18] . Notice that if γ ≤ 3, standard estimates of ℓ [19] are inapplicable to equilibrium networks with uncorrelated vertices. Correlations in growing networks are inevitable, and the results are even less encouraging. The generic property of these networks, which makes their analytical study so hard, is a complex structure of their adjacency matrices. Scale-free random networks naturally have a continuous degree distribution spectrum, but it has recently been shown that discrete degree distributions of some deterministic graphs also have a power-law decay [20] . The leading idea of the present work is the following. If it is so hard to get exact or precise results for scale-free random networks, especially, with 2 < γ < 3, let us (i) construct a simple deterministic scale-free graph with such γ, (ii) obtain exact (analytical) and precise (numerical) answers for main structural and topological characteristics of the graph, (iii) compare its properties with known characteristics of corresponding scale-free growing random networks, and (iv) if these properties coincide, make new predictions for scale-free growing random networks. We actually model scale-free networks with 2 < γ < 3 by the deterministic graph, whose structure can be described completely. Here we present results of this program. We succeed to find a number of exact characteristics of the scale-free deterministic graph, which are still unknown for random scale-free networks. The structural properties of deterministic and random scale-free growing networks proved to be surprisingly close to each other, so that our results can be reasonably applied to random growing nets. Pseudofractal graph.-The growth starts from a single edge connecting two vertices at t = −1 (see Fig. 1 ). At each time step, to every edge of the graph, a new vertex is added, which is attached to both the end vertices of the edge. Then, at t = 0, we have a triangle of edges connecting a triple of vertices, at t = 1, the graph consists of 6 vertices connected by 9 edges, and so on. The total number of vertices at "time" t is N t = 3(3 t + 1)/2, and the total number of edges is L t = 3 t+1 , so that the average degree is k t = 2L t /N t = 4/(1 + 3 −t ). This simple rule produces a complex growing network which is certainly not a fractal [21] . Indeed, at any step, the entire graph can be set inside of a unit triangle. This means that the structure has no any fixed finite fractal dimension. On the other hand, one can depict the graph in another way, namely, like in Fig. 1 where the graph is surrounded by a long chain of edges. The length of this "perimeter" is P t = 3 · 2 t edges, whence N t ∼ L t ∼ P ln 3/ ln 2 t . We can, however, fold the graph into a more compact structure with a different border, so that ln 3/ ln 2 is not a fractal dimension of the structure but only some characteristic value. We failed to introduce a well defined spectrum of fractal dimensions (between ln 3/ ln 2 and ∞), hence the network cannot be called a multifractal. Thus, this graph is not a fractal but only parody of it, and we call it, for brevity, pseud-1
According to REF , some deterministic scale-free networks have a clustering coefficient distribution C(q) ∼ q −1 , where q is the degree.
Pseudofractal scale-free web
Physical review. E, Statistical, nonlinear, and soft matter physics
Physical review. E, Statistical, nonlinear, and soft matter physics
Medicine Physics