Complex network zeta function: Difference between revisions

Revision as of 04:00, 16 December 2007

Revision as of 19:48, 20 December 2007

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where <math>\textstyle N</math> is the graph size, measured by the number of nodes. When <math>\textstyle \alpha</math> is zero all nodes contribute equally to the sum in Eq. . This means that <math>\textstyle \zeta_{G}(0)</math> is <math>\textstyle N-1</math>, and it diverges when <math>\textstyle N \rightarrow \infty</math>. When the exponent <math>\textstyle \alpha</math> tends to infinity, the sum in Eq. gets contributions only from the nearest neighbours of a node. The other terms tend to zero. Thus, <math>\textstyle \zeta_G ( \alpha )</math> tends to the average degree <math>\textstyle <k></math> for the graph as <math>\textstyle \alpha \rightarrow \infty</math>.

:<math> <\langle k> \rangle = \lim_{\alpha \rightarrow \infty} \zeta_G ( \alpha ). </math>

The definition Eq. can be expressed as a weighted sum over the node distances. This gives the Dirichlet series relation