Classification of scale-free networks

Abstract

While the emergence of a power-law degree distribution in complex networks is intriguing, the degree exponent is not universal. Here we show that the betweenness centrality displays a powerlaw distribution with an exponent , which is robust, and use it to classify the scale-free networks. We have observed two universality classes with Ϸ 2.2(1) and 2.0, respectively. Real-world networks for the former are the protein-interaction networks, the metabolic networks for eukaryotes and bacteria, and the coauthorship network, and those for the latter one are the Internet, the World Wide Web, and the metabolic networks for Archaea. Distinct features of the mass-distance relation, generic topology of geodesics, and resilience under attack of the two classes are identified. Various model networks also belong to either of the two classes, while their degree exponents are tunable.E mergence of a power law in the degree distribution P D (k) ϳ k Ϫ␥ in complex networks is an interesting self-organized phenomenon in complex systems (1-3). Here, the degree k means the number of edges incident upon a given vertex. Such a network is called scale-free (SF; ref. 4). Real-world networks that are SF include the author-collaboration network (5) in social systems, the protein-interaction network (PIN; ref. 6), and the metabolic network (7) in biological systems, and the Internet (8) and World Wide Web (WWW; refs. 9 and 10) in communication systems. The power-law behavior means that most vertices are connected sparsely, while a few vertices are connected intensively to many others and play an important role in functionality. While the emergence of such a SF behavior in degree distribution itself is surprising, the degree exponent ␥ is not universal and depends on the detail of network structure. As listed in Table 1, numerical values of the exponent ␥ for various systems are diverse, but most of them are in the range of 2 Ͻ ␥ Յ 3. From the viewpoint of theoretical physics, it would be interesting to search a universal quantity associated with SF networks.Recently a physical quantity called ''load'' was introduced as a candidate for the universal quantity in SF networks. It quantifies the load of a vertex in the transport of data packet along the shortest pathways in SF networks (11). It was shown that the load distribution exhibits a power law, P L (ᐉ) ϳ ᐉ Ϫ␦ , and the exponent ␦ is robust as ␦ Ϸ 2.2 for diverse SF networks with various degree exponents in the range of 2 Ͻ ␥ Յ3. Since the universal behavior of the load exponent was obtained empirically, fundamental questions such as how the load exponent is robust in association with network topology or the possibility of any other universal classes existing have not been explored yet. In this paper, we address those issues in detail.While the load is a dynamic quantity, it is closely related to a static quantity, the ''betweenness centrality'' (BC), commonly used in sociology to quantify how influential a given person in a society is (12). To be specific, BC is defined as follows. Let us...

Search and Congestion in Complex Networks

“…Load, independently of search, has been analyzed in different classes of networks [28][29][30][31]. The load, as introduced in these works, is equivalent to the betweenness as it has been defined in social networks [32,28].…”

“…On the other side, the load distribution of small-world networks shows a combined behavior of two Poisson-type decays. In subsequent work, the authors in [31] suggested that real-world networks should be classified in two different universality classes, according to the exponent of the power-law distribution of loads. Finally, the distribution of loads was analytically computed for scale-free trees in [30].…”

Reliable anycast and unicast routing: protection against attacks

“…For the purpose of working path computations, we propose to use the metric based on the so called betweenness centrality parameter (BC) defined in [10] and shown in (2), since it provides us with good estimates on the centrality of node n, and therefore on its vulnerability to attacks.…”

Evolution of the Internet Map and Load Distribution

“…(2) [17]. The Internet AS map is no exception and the load exponent δ of the power law is estimated to be approximately δ ≈ 2.0 [17,18].…”

“…(2) [17]. The Internet AS map is no exception and the load exponent δ of the power law is estimated to be approximately δ ≈ 2.0 [17,18]. The power-law load distribution means that a few ASes should handle an extraordinarily large amount of load while most others should do only a little.…”