Empirical studies and topological implications of Fermat difference statistics for complex networks

The network science community requires robust metrics that effectively characterize key structural and functional attributes of complex networks. In this paper, we investigate the Fermat difference statistics of real-world and artificial networks. The Fermat difference statistics fundamentally governs multi-vertex interaction patterns and topological correlations in complex network architectures through key mechanisms. We provide two analytical findings to enhance our understandings for these Fermat statistics. Empirical data analysis demonstrates that the Fermat difference statistics can capture and quantify the existence of hub-and-spoke topology or the non-democracy phenomenon of networks. A network with small Fermat difference statistics may be aristocratic with few critical elements dominating the entire network. Based on solid theoretical foundations, we present a strong guarantee that the Sierpiński carpet networks exhibiting fractal features converge to the value zero for the average-case Fermat difference statistic as the network order approaches infinity. Our conclusions also lead to several conjectures regarding the networks generated by iterated function systems and circle graphs.