For reversible random networks, we exhibit a relationship between the almost sure spectral dimension and the Euclidean growth exponent, which is the smallest asymptotic rate of volume growth over all embeddings of the network into a Hilbert space. Using metric embedding theory, it is then shown that the Euclidean growth exponent coincides with the metric growth exponent. This simplifies and generalizes a powerful tool for bounding the spectral dimension in random networks.

中文翻译：

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谱维数、欧几里德嵌入和度量增长指数

对于可逆随机网络，我们展示了几乎确定的谱维数和欧几里得增长指数之间的关系，欧几里得增长指数是网络到希尔伯特空间的所有嵌入的最小体积增长渐近率。利用度量嵌入理论，证明欧几里得增长指数与度量增长指数一致。这简化并概括了用于限制随机网络中频谱维度的强大工具。