The universal properties of the geomagnetic/solar indices critically describe their time series’ dynamics. Here, we investigate the complexity behavior of the time series of the geomagnetic and solar activity indices Kp, DST, Ap, F10.7, \(\hbox {Lyman}_{\alpha }\) , AE, AL, AU, pc, and R. To do this, we apply multiscale entropy (MSE) analysis, detrended fluctuation analysis (DFA), rescaled range (R/S) test, and visibility graph (VG) to the time series of indices recorded daily from January 1st, 1996 to January 1st, 2020. MSE analysis reveals that AE exhibits the lowest entropy (minimal complexity), followed by F10.7, \(\hbox {Lyman}_{\alpha }\) , and R. Ap shows intermediate complexity, while AL, Dst, AU, and pc maintain higher entropy values. The range of Hurst exponents between 0.5 and 1 extracted from DFA and R/S analyses confirms the long-term memory of geomagnetic indices. In contrast, solar activity indices exhibit Hurst exponents consistent with pink noise or non-stationarity. VG analysis identifies AU, Dst, pc, and Kp as possessing “small-world” characteristics—corroborating strong long-range correlations. Synthesized results indicate that the indices Kp, Dst, AL, AU, and pc exhibit stronger evidence of long-term correlated memory, while the solar indices, along with the geomagnetic index AE, are comparatively suitable for studying transient phenomena.
