What is the dimension of a network? Here, we view it as the smallest dimension of Euclidean space into which nodes can be embedded so that pairwise distances accurately reflect the connectivity structure. We show that a recently proposed and extremely efficient algorithm for data clouds, based on computing first and second nearest neighbour distances, can be used as the basis of an approach for estimating the dimension of a network with weighted edges. We also show how the algorithm can be extended to unweighted networks when combined with spectral embedding. We illustrate the advantages of this technique over the widely-used approach of characterising dimension by visually searching for a suitable gap in the spectrum of the Laplacian.

翻译：网络维度是什么？在此，我们将其视为节点可嵌入的欧几里得空间的最小维度，使得成对距离能够准确反映网络的连通结构。我们证明，一种近期提出的、基于计算第一和第二近邻距离的高效数据云算法，可被用作估算带权重边网络维度的基础方法。我们还展示了如何将该算法与谱嵌入相结合，推广至无权重网络。我们通过实例说明了该技术相较于广泛使用的、通过目视搜索拉普拉斯算子谱中适当间隙来表征维度的方法所具有的优势。