On The Scale Dependence and Spacetime Dimension of the Internet with Causal Sets

TL;DR

This paper introduces a statistical measure to analyze the effective spatial dimension of the Internet across multiple scales, revealing a complex, multi-scale structure with dimensions ranging from approximately 1.66 to 7.7.

Contribution

It applies a novel scale-dependent dimension measure to Internet data, uncovering its multi-scale, hierarchical structure and providing insights into its hidden dimensions.

Findings

Internet has a multi-scale 'foamy' hierarchy.

Effective dimension varies from about 1.66 to 7.7.

Long-range order emerges at certain scales.

Abstract

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.