From: Network spectra for drug-target identification in complex diseases: new guns against old foes
The distribution of eigenvectors components can be used to obtain non-random system dependent information (Fig. 4). The inverse participation ratio (IPR) has long been employed to analyze localization properties of the eigenvectors (Haake and Zyczkowski 1990). For \(E_{l}^{k}\) denoting lth component of kth eigenvector E^{k}, the IPR of an eigenvector can be defined as |
\( I^{k} = \frac { \sum _{l=1}^{N} \left [E_{l}^{k}\right ]^{4}}{ \left (\sum _{l=1}^{N} \left [E_{l}^{k}\right ]^{2}\right)^{2}} \text {(3)} \) |
which shows two limiting values: (i) a vector with identical components \(E_{l}^{k} \equiv 1/\sqrt {N}\) has I^{k}=1/N, whereas (ii) a vector, with one component \(E_{1}^{k}=1\) and the remainders zero, has I^{k}=1. Thus, the IPR quantifies the reciprocal of the number of eigenvector components that contribute significantly. |
Further, the average IPR in order to measure an overall localization of the network is calculated as (Jalan and et al. 2011), |
\( \langle IPR \rangle = \frac { \sum _{k=1}^{N} \left [I^{k}\right ]}{N} \text {(4)} \) |
Note that IPR defined as above, separates out the top contributing nodes by keeping the threshold as 1/IPR. These Top Contributing Nodes (TCNs) are further found to have important role in the underlying system. |