A new spectral connectivity measure considering the effect of network scale

Motivated by the need to distinguish connectivity from robustness and to enable comparability across networks of varying scales, this work introduces a new measure termed the spectral connectivity extent $(\hat {\mu })$ . $\hat {\mu }$ is defined as the average of the second and third smallest Laplacian eigenvalues, normalized by network size and rescaled to the interval [0, 100]. Higher values of $\hat {\mu }$ correspond to stronger network connectivity. Several networks are analyzed to evaluate the proposed measure. By analyzing a real-world network, it is found that when the network undergoes a critical change, the relative variation of $\hat {\mu }$ and the global efficiency $(E_{\text {glob}})$ are 16.485 and 0.037, respectively, whereas under an unimportant change, the relative variations of $\hat {\mu }$ and $E_{\text {glob}}$ are 0.2 and 0.162, respectively. These findings indicate that, compared with existing measures, $\hat {\mu }$ more effectively captures significant structural changes within the network. Since $\hat {\mu }$ emphasizes the efficiency of interactions among nodes, it has potential applications in the design of communication and transportation networks, the analysis of disease spreading, and the assessment of infrastructure resilience.