基于因果集的互联网的尺度依赖和时空维数
On The Scale Dependence and Spacetime Dimension of the Internet with Causal Sets
摘要：
维度的统计度量用于计算互联网和其他图的有效平均空间维度，基于来自起点集合的类型化边（链接）。该方法被应用到CAIDA的ITDK数据的互联网。使用低维超立方体将不同尺度下的有效维数校准到传统的欧氏维数。互联网时空有一个“泡沫”的多尺度包容层次结构，交错的语义类型。有一个紧急规模的设备节点频谱中的近似长距离秩序，但这是不明显的AS级，其中有有限的距离遏制。因此，统计维数是一个局部变化的度量，它是尺度依赖的，这给了卡鲁扎-克莱因理论中隐藏的尺度依赖维数一个直观的类比。互联网的特征外部尺寸在1.66 +- 0.00和6.12 +- 0.00之间，最大内部尺寸上升到7.7。
英文摘要：
A statistical measure of dimension is used to compute the effective average space dimension for the Internet and other graphs, based on typed edges (links) from an ensemble of starting points. The method is applied to CAIDA's ITDK data for the Internet. The effective dimension at different scales is calibrated to the conventional Euclidean dimension using low dimensional hypercubes. Internet spacetime has a 'foamy' multi-scale containment hierarchy, with interleaving semantic types. There is an emergent scale for approximate long range order in the device node spectrum, but this is not evident at the AS level, where there is finite distance containment. Statistical dimension is thus a locally varying measure, which is scale-dependent, giving an visual analogy for the hidden scale-dependent dimensions of Kaluza-Klein theories. The characteristic exterior dimensions of the Internet lie between 1.66 +- 0.00 and 6.12 +- 0.00, and maximal interior dimensions rise to 7.7.
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