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Register a new userBetweenness Centrality of Fractal and Non-Fractal Scale-Free Model Networks and Tests on Real Networks
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We study the betweenness centrality of fractal and non-fractal scale-free network models as well as real networks. We show that the correlation between degree and betweenness centrality $C$ of nodes is much weaker in fractal network models compared to non-fractal models. We also show that nodes of both fractal and non-fractal scale-free networks have power law betweenness centrality distribution $P(C)sim C^{-delta}$. We find that for non-fractal scale-free networks $delta = 2$, and for fractal scale-free networks $delta = 2-1/d_{B}$, where $d_{B}$ is the dimension of the fractal network. We support these results by explicit calculations on four real networks: pharmaceutical firms (N=6776), yeast (N=1458), WWW (N=2526), and a sample of Internet network at AS level (N=20566), where $N$ is the number of nodes in the largest connected component of a network. We also study the crossover phenomenon from fractal to non-fractal networks upon adding random edges to a fractal network. We show that the crossover length $ell^{*}$, separating fractal and non-fractal regimes, scales with dimension $d_{B}$ of the network as $p^{-1/d_{B}}$, where $p$ is the density of random edges added to the network. We find that the correlation between degree and betweenness centrality increases with $p$.
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Self-similarity is a property of fractal structures, a concept introduced by Mandelbrot and one of the fundamental mathematical results of the 20th century. The importance of fractal geometry stems from the fact that these structures were recognized in numerous examples in Nature, from the coexistence of liquid/gas at the critical point of evaporation of water, to snowflakes, to the tortuous coastline of the Norwegian fjords, to the behavior of many complex systems such as economic data, or the complex patterns of human agglomeration. Here we review the recent advances in self-similarity of complex networks and its relation to transport, diffusion, percolations and other topological properties such us degree distribution, modularity, and degree-degree correlations.
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In this paper, we study traffic dynamics in scale-free networks in which packets are generated with non-homogeneously selected sources and destinations, and forwarded based on the local routing strategy. We consider two situations of packet generation: (i) packets are more likely generated at high-degree nodes; (ii) packets are more likely generated at low-degree nodes. Similarly, we consider two situations of packet destination: (a) packets are more likely to go to high-degree nodes; (b) packets are more likely to go to low-degree nodes. Our simulations show that the network capacity and the optimal value of $alpha$ corresponding to the maximum network capacity greatly depend on the configuration of packets sources and destinations. In particular, the capacity is greatly enhanced when most packets travel from low-degree nodes to high-degree nodes.
We investigate the accumulated wealth distribution by adopting evolutionary games taking place on scale-free networks. The system self-organizes to a critical Pareto distribution (1897) of wealth $P(m)sim m^{-(v+1)}$ with $1.6 < v <2.0$ (which is in agreement with that of U.S. or Japan). Particularly, the agents personal wealth is proportional to its number of contacts (connectivity), and this leads to the phenomenon that the rich gets richer and the poor gets relatively poorer, which is consistent with the Matthew Effect present in society, economy, science and so on. Though our model is simple, it provides a good representation of cooperation and profit accumulation behavior in economy, and it combines the network theory with econophysics.
Using data on the Berlin public transport network, the present study extends previous observations of fractality within public transport routes by showing that also the distribution of inter-station distances along routes displays non-trivial power law behaviour. This indicates that the routes may in part also be described as Levy-flights. The latter property may result from the fact that the routes are planned to adapt to fluctuating demand densities throughout the served area. We also relate this to optimization properties of Levy flights.
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