AFRACT: Autocorrelation-Aware Fractal Dimension for Complex Networks

Estimating the fractal dimension Df of a complex network is fundamental to understanding its self-similar structure, yet existing methods based on boxcovering are sensitive to hub concentration and provide no way to incorporate node-property information. We introduce AFRACT (Autocorrelation-aware Fractal dimension for Complex neTworks), a ball-mass scaling algorithm that addresses both limitations. The core idea is simple: instead of counting nodes within a growing ball, AFRACT weights each node by a function of its property value and its spatial autocorrelation with the centre, capturing how local order decays with topological distance. We establish a complete axiomatic framework for the weighting function, prove that property-weighted and unweighted estimates converge to the same Df asymptotically (power-law preservation theorem), and show that the weight matrix is circulant on vertex-transitive graphs, enabling an exact FFT-based computation that achieves a 471× speedup over the direct method. On four deterministic networks with analytically known Df (Sierpi«ski gasket and carpet, 2D and 3D lattices, N up to 3375), AFRACT achieves R2 ≥ 0.998. We derive a universal nite-size correction law Df ≈ ˆDf + cG/N1/Df confirmed across three graph classes with R2 > 0.99. Against Compact Box Burning on five diverse networks, AFRACT achieves higher goodness-of-fit, 4.7× better noise robustness, and avoids the hub-induced overestimation that causes CBB to return ˆDf = 3.98 on BarabásiAlbert networks.