Complex network zeta function: Difference between revisions

Revision as of 00:44, 15 September 2008

Revision as of 00:56, 15 September 2008

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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>K.-I. Goh, G. Salvi, B. Kahng and D. Kim, Phys. Rev.

Lett. 96, 018701 (2006).</ref>. Here we describe the definition based on the '''complex network zeta function'''<ref name="Shankerb">{{cite journal|author=Shanker, O. Shanker|year=2007|title=Graph Zeta Function and Dimension of Complex Network|journal=Modern Physics Letters B|volume= 21(11)|pages=639–644|doi=10.1142/S0217984907013146}}</ref>. This generalises the definition based on the scaling property of the volume with distance<ref name="Shankera">{{cite journal|author=Shanker, O. Shanker|year=2007|title=Defining Dimension of a Complex Network |journal=Modern Physics Letters B|volume= 21(6)|pages=321–326|doi=10.1142/S0217984907012773}}</ref>. The best definition depends on the application.

==Definition==

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:<math> S_{d+1}(r) = 2+S_{d}(r)+2\sum^{r-1}_{i=1}S_{d}(i). </math>

From combinatorics the surface function for a regular lattice can be written<ref name="Shankerd">{{cite journal|author=O. Shanker|year=2008|title=Sharp dimension transition in a shortcut model |journal=J. Phys. A: Math. Theor. |volume= 41|pages=285001|doi=10.1088/1751-8113/41/28/285001}}</ref> as

:<math> S_{d}(r) = \sum^{d-1}_{i=0}(-1)^{i}2^{d-i}{d \choose i} { d+r-i-1 \choose d-i-1 }. </math>