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:<math> \|\vec{n}\|_1=\|n_1\|+\cdots +\|n_d\|, </math>
the transition occurs at <math>\textstyle \alpha = d</math>. The definition of dimension using the complex network zeta function satisfies properties like monotonicity (a subset has a lower or the same dimension as its containing set), stability (a union of sets has the maximum dimension of the component sets forming the union) and Lipschitz invariance <ref name=falconer>K. Falconer., Fractal Geometry: Mathematical Foundations and Applications, Wiley, second edition, 2003</ref>, provided the operations involved change the distances between nodes only by finite amounts as the graph size <math>\textstyle N</math> goes to <math>\textstyle \infty</math>.
==Values for discrete regular lattices==
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