B. Kahng

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.

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 between ness centrality displays a power-law distribution with an exponent eta, which is robust, and use it to classify the scale-free networks. We have observed two universality classes with eta approximately equal 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.

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. Supporting information for this article is available on the WWW under http://www.wiley-vch.de/contents/jc_2089/2008/adma200702024_s.pdf or from the author. Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.

Scale-free (SF) networks exhibiting a power-law degree distribution can be grouped into the assortative, dissortative, and neutral networks according to the behavior of the degree-degree correlation coefficient. Here we investigate the betweenness centrality (BC) correlation for each type of SF networks. While the BC-BC correlation coefficients behave similarly to the degree-degree correlation coefficients for the dissortative and neutral networks, the BC correlation is nontrivial for the assortative ones found mainly in social networks. The mean BC of neighbors of a vertex with BC g(i) is almost independent of g(i), implying that each person is surrounded by almost the same influential environments of people no matter how influential the person may be.

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 $\ensuremath{\tau}$. Applying the theory of the multiplicative branching process, we obtain the exponent $\ensuremath{\tau}$ and the dynamic exponent $z$ as a function of the degree exponent $\ensuremath{\gamma}$ of SF networks as $\ensuremath{\tau}=\ensuremath{\gamma}/(\ensuremath{\gamma}\ensuremath{-}1)$ and $z=(\ensuremath{\gamma}\ensuremath{-}1)/(\ensuremath{\gamma}\ensuremath{-}2)$ in the range $2&lt;\ensuremath{\gamma}&lt;3$ and the mean-field values $\ensuremath{\tau}=1.5$ and $z=2.0$ for $\ensuremath{\gamma}&gt;3$, with a logarithmic correction at $\ensuremath{\gamma}=3$. 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.