Multifractal analysis of structural perturbation time series in hyperbolic growing networks

Quantifying structural perturbations during network growth is crucial for understanding its dynamic mechanisms, yet fine-grained measurement of such local changes remains challenging. To address this, this study introduces two perturbation metrics, D1(t) and D2(t), to characterize the local topological and geometric responses triggered by each newly added node in the hyperbolic Popularity-Similarity Optimization model. Using multifractal detrended fluctuation analysis, we extract multiscale features including the generalized Hurst exponent h(q), the spectrum width Δα, and the spectral asymmetry Rα to describe the inhomogeneity, memory effects, and scaling behavior of the perturbation dynamics. Both {D1(t)} and {D2(t)} exhibit pronounced parameter-dependent multifractal characteristics, revealing an asymmetric coupling mechanism between node connectivity changes and the evolution of the underlying geometry. To further corroborate this mechanism, we analyze a real-world Autonomous System network and find that its perturbation sequences likewise display strong multifractal properties. The framework not only deepens the theoretical understanding of how geometric constraints shape network dynamics, but also provides new analytical tools for assessing node influence and system resilience. Its principles and methods are expected to be applied to biological, social, and technological networks, among other fields.