Structural crossover of complex networks: bridging degree correlation and fractality

Abstract

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 $l\ll k^{1/(\df-1)}$, the ball volume grows as ${\tilde{\nu }}_k(l)\sim kl$, whereas for $l\gg k^{1/(\df-1)}$, it crosses over to the global fractal scaling $\tildeν_k(l)\sim l^{\df}$. Here, $\df$ 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.

arXiv comment9 pages, 6 figures