We study a problem of data packet transport in scale-free networks whose degree distribution follows a power law with the exponent gamma. Load, or "betweenness centrality," of a vertex is the accumulated total number of data packets passing through that vertex when every pair of vertices sends and receives a data packet along the shortest path connecting the pair. It is found that the load distribution follows a power law with the exponent delta approximately 2.2(1), insensitive to different values of gamma in the range, 2 < gamma < or = 3, and different mean degrees, which is valid for both undirected and directed cases. Thus, we conjecture that the load exponent is a universal quantity to characterize scale-free networks.

We investigate the avalanche dynamics of the Bak-Tang-Wiesenfeld sandpile model on scale-free (SF) networks, where the threshold height of each node is distributed heterogeneously, given as its own degree. We find that the avalanche size distribution follows a power law with an exponent tau. Applying the theory of the multiplicative branching process, we obtain the exponent tau and the dynamic exponent z as a function of the degree exponent gamma of SF networks as tau=gamma divided by (gamma-1) and z=(gamma-1) divided by (gamma-2) in the range 23, with a logarithmic correction at gamma=3. The analytic solution supports our numerical simulation results. We also consider the case of a uniform threshold, finding that the two exponents reduce to the mean-field ones.

The random circuit breaker network model is proposed for unipolar resistance switching behavior. This model describes reversible dynamic processes involving two quasi‐metastable states. The formation and rupture of conducting channels (see figure) in the polycrystalline TiO2 thin films may be analyzed by the self organized avalanche process in the random circuit breaker network model.

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...