Exploring multifractal geometry in directed complex networks

Complex networks are fundamental for modelling and understanding a wide range of real-world applications or non-trivial systems. Understanding their multifractal geometry has become relevant in recent years. Various algorithms have been developed either to refine existing methods or to adapt them to different network topologies. However, multifractal analysis has never been applied to directed networks. In this work, we present a novel multifractal box-covering method for directed networks, referred as MF-BCd. To assess its applicability, state-of-the-art undirected networks have been used to generate the directed ones. Specifically, (u,v)-flower, the generalized minimal model, scale-free and random networks are used as weakly connected networks. Krapivsky and small-world networks are used to generate the strongly connected ones. To evaluate the impact of the degree of directionality, MF-BCd has been evaluated with different percentages of directed edges. Results show that Krapivsky networks exhibit a clear multifractal structure, with a ΔD(q) approximately 0.9, confirming the method’s effectiveness. In contrast, small-world networks display limited multifractality, with ΔD(q) less than 0.16, decreasing as the rewiring probability p becomes smaller. Moreover, adding directed shortcuts in small-world networks increases multifractality compared to the non-directed case. The weakly connected networks studied exhibit multifractality, which diminishes as the proportion of directed links increases. Higher directionality also increases the number of boxes required to cover the network and leads to a loss of multifractality, approaching monofractal behaviour. This loss is particularly pronounced in scale-free and random networks, whereas ring-based networks such as (u,v)-flower and generalized minimal model preserve multifractality up to certain thresholds. The proposed MF-BCd method is further validated using two real-world directed networks, enabling a consistent multifractal analysis and distinguishing different multifractal behaviour.