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 networks compared to non-fractal networks. We 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 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 $p$. We test the new 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.
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机译：我们研究了分形和非分形无标度网络模型以及实际网络的中间性。我们表明，与非分形网络相比，分形网络中节点的度和中间中心度$ C $之间的相关性要弱得多。我们证明了分形和非分形无标度网络的节点都具有幂律中间性分布$ P（C）\ sim C ^ {-\ delta} $。我们发现，对于非分形无标度网络$ \ delta = 2 $，对于分形无标度网络$ \ delta = 2-1 / d_ {B} $，其中$ d_ {B} $是分形网络。我们还研究了向分形网络添加随机边缘后从分形网络到非分形网络的交叉现象。我们证明了交叉长度$ \ ell ^ {*} $，分离了分形和非分形状态，并以网络的维度$ d_ {B} $缩放为$ p ^ {-1 / d_ {B}} $，其中$ p $是添加到网络的随机边的密度。我们发现，程度和中间性之间的相关性随$ p $的增加而增加。我们通过在四个实际网络上的显式计算来测试新结果：制药公司（N = 6776），酵母菌（N = 1458），WWW（N = 2526）和AS级Internet网络样本（N = 20566），其中，$ N $是网络中最大连接组件中的节点数。
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