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Different definitions have been given for the dimension of a [[complex network]] or [[graph theory|graph]]. For example, [[Metric dimension (graph theory)|metric dimension]] is defined in terms of the resolving set for a graph. Dimension has also been [[Fractal dimension on networks|defined]] based on the [[Minkowski–Bouligand dimension|box covering method]] applied to graphs.<ref name=goh>{{cite journal |
==Definition==
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:<math> \langle k \rangle = \lim_{\alpha \rightarrow \infty} \zeta_G ( \alpha ). </math>
The need for taking an average over all nodes can be avoided by using the concept of supremum over nodes, which makes the concept much easier to apply for formally infinite graphs.<ref name="ShankerTCS">{{cite journal|author=O. Shanker|year=2010|title=Complex Network Dimension and Path Counts|journal=Theoretical Computer Science|volume= 411|pages=2454–2458|doi=10.1016/j.tcs.2010.02.013|issue=26–28|doi-access=free}}</ref> The definition can be expressed as a weighted sum over the node distances. This gives the Dirichlet series relation
:<math> \zeta_G ( \alpha ) = \sum_r S(r)/r^{\alpha}. </math>
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