We propose a paradigm for modeling real-world scale-free networks, where the integration of new nodes is driven by the combined attractiveness of degree and betweenness centralities, the competition of which (expressed by a parameter $0\le p\le 1$) shapes the structure of the evolving network. We reveal the ability to seamlessly explore a vast landscape of scale-free networks, unlocking a class of complex networks that we call a stars-with-filament structure. Remarkably, the average degree $\overline{k}$ of these networks grows like $lnt$ to some power, where $t$ is time and the average shortest path length grows logarithmically with the system size for intermediate $p$ values, offering fresh insights into the structural dynamics of scale-free systems. Our approach is backed by a robust mean-field theory, which nicely captures the dynamics of $\overline{k}$. We further unveil a rich, $p$-dependent phase diagram, encompassing 47 real-world scale-free networks, shedding light on previously hidden patterns. This work opens avenues for understanding the universal properties of complex networks.

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• Complex networks
• Scale free & inhomogeneous networks
• Critical exponents
• Data analysis
• Mean field theory