Centrality and universality in scale-free networks

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≤p≤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 k[over ¯] 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 k[over ¯]. 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.