Measuring the dimension of partially embedded networks

TL;DR

This paper introduces a new network dimension measurement method that uses local link information without requiring global embedding data, applicable to various real-world networks and revealing complex scaling behaviors.

Contribution

It proposes a novel dimension measure based on local link data, generalizing spectral dimension, and demonstrates its effectiveness on synthetic and real networks.

Findings

The new measure works with networks where previous methods fail, such as with P(r) ~ 1/r link length distribution.

It reveals two distinct scaling regimes in network dimension.

Link length distribution alone does not determine network dimensionality.

Abstract

Scaling phenomena have been intensively studied during the past decade in the context of complex networks. As part of these works, recently novel methods have appeared to measure the dimension of abstract and spatially embedded networks. In this paper we propose a new dimension measurement method for networks, which does not require global knowledge on the embedding of the nodes, instead it exploits link-wise information (link lengths, link delays or other physical quantities). Our method can be regarded as a generalization of the spectral dimension, that grasps the network's large-scale structure through local observations made by a random walker while traversing the links. We apply the presented method to synthetic and real-world networks, including road maps, the Internet infrastructure and the Gowalla geosocial network. We analyze the theoretically and empirically designated case when the length distribution of the links has the form P(r) ~ 1/r. We show that while previous dimension concepts are not applicable in this case, the new dimension measure still exhibits scaling with two distinct scaling regimes. Our observations suggest that the link length distribution is not sufficient in itself to entirely control the dimensionality of complex networks, and we show that the proposed measure provides information that complements other known measures.