Structural crossover of complex networks: bridging degree correlation and fractality

We investigate the relationship between long-range degree correlations and fractality in scale-free random networks. By analyzing degree correlations in scale-free random networks at the percolation threshold, at which a giant component emerges, we derive the ball volume ${\tilde{\nu }}_k(l)$, defined as the average number of nodes within distance $l$ from a root node with degree $k$. The resulting expression predicts a degree-dependent structural crossover. For $l\ll k^{1\mathrm{/}(\text{df}-1)}$, the ball volume grows as ${\tilde{\nu }}_k(l)\sim kl$, whereas for $l\gg k^{1\mathrm{/}(\text{df}-1)}$, it crosses over to the global fractal scaling ${\tilde{\nu }}_k(l)\sim l^{\text{df}}$. Here, $\text{df}$ is the fractal dimension of critical scale-free random networks. We further show that the structural crossover can be naturally understood in terms of critical branching processes and is also observed in empirical fractal networks.