Table 4 Inverse participation ratio (IPR)

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.