Network functional complexity

Differs from other methods of assessing entropy because it is predicated in space, not time.

Introduced by Zamora-Lopez2016:

The functional complexity $C$ of the network is the difference between the observed distribution $p(r_{ij})$ and the uniform distribution. If $p(r_{ij})$ is estimated in $m$ bins, the uniform distribution is ${\bar{p}}_{\mu }=\frac{1}{m}$ for all bins $\mu =1,2,\dots ,m$

Functional complexity is therefore quantified as the sum of the differences of the two distributions over the bins:

$$\begin{array}{c}C=1-\frac{1}{C_m}\sum _{\mu =1}^m|p_{\mu }(r_{ij})-\frac{1}{m}|\end{array}$$

where $C_m=2\frac{m-1}{m}$ is the normalisation factor representing the extreme case where $p(r_{ij})$ is a Dirac-delta function ${\delta }_m$. This occurs when all $r_{ij}$ fall in the same bin, where all nodes are mutually independent or globally synchronised.

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