We propose a generalized degree-biased random walk model (GDBRW) for scale-free networks, where transition probabilities inversely depend on source and target node degrees via tunable exponents $\alpha$ and $\beta$. We derive equilibrium probability distributions using a continuum approach and simulate exploration times across diverse network densities. The GDBRW model significantly boosts exploration efficiency in sparse networks, outperforming the traditional popularity-driven random walk. Through a detailed analysis of fallbacks—localized oscillations where the walker immediately returns to the previously visited node—we demonstrate that our model effectively suppresses hub-leaf trapping motifs. This symmetry-driven suppression of fallbacks explains the improved coverage efficiency and establishes GDBRW as a robust and efficient exploration strategy for scale-free networks, particularly in low-connectivity regimes.

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• Network diffusion
• Network optimization