We aim to study the relationship between the fractal dimension and the Lin-Lu-Yau Ricci curvature in complex networks in this paper. We analyzed two types of networks with fractal properties, and compared the total edge Lin-Lu-Yau Ricci curvature with the fractal dimension of the complex networks from a global point of view. Except for a special case of one type of fractal network (i.e., a tree graph), we observed a quantitative relationship with the diameter of the network, thereby linking two important properties of complex networks. We found that the sum of the Lin-Lu-Yau Ricci curvatures of tree graphs is constantly 2. These properties provide a new perspective for analyzing complex networks and help us better understand their characteristics.
This paper investigates the effect of the variable-order Riemann-Liouville fractional integral operator on the box dimension of graphs of continuous functions. By constructing a counterexample, we show that even when the integrand is a constant function, the graph of the resulting integral has box dimension strictly greater than 1 if the order function exhibits fractal characteristics, revealing an essential difference between variable-order and constant-order fractional operators. On the other hand, we provide a sufficient condition ensuring that the output graph has box dimension equal to 1: if the order function is Lipschitz continuous with a positive lower bound and the integrand satisfies an appropriate H¨older continuity condition, then the resulting integral belongs to the class of absolutely continuous functions, and hence its graph has box dimension 1. Numerical simulations illustrate the conclusion of the counterexample.
The Black-Scholes model plays a significant role in the analysis of financial derivatives and option pricing. However, the classical framework may not fully capture the memory and hereditary effects observed in financial markets. This study investigates a time-fractional Black-Scholes equation involving the Caputo fractional derivative and develops a semi-analytical approach for its solution. The proposed scheme transforms the governing fractional differential equation into a recursive algebraic formulation without requiring temporal or spatial discretization, linearization, or restrictive assumptions. The effectiveness of the proposed approach is demonstrated through several illustrative examples, and the resulting solutions are compared with the corresponding classical solutions. Furthermore, the numerical results and graphical analyses demonstrate the accuracy and effciency of the proposed method, while the rapid convergence of the resulting series confirms its consistency and computational effectiveness.
Research on fractional-order Hermite-Hadamard (H-H) inequalities has predominantly relied on single convexity assumptions, limiting its adaptability to generalized convexity contexts such as h-preinvex functions. This paper refines the framework of fractional-order H-H inequalities for preinvex functions by leveraging the generalized Riemann-Liouville fractional integral operator. Specifically, a novel class of H-H-type inequalities is constructed and rigorously proved for h-preinvex functions. These results not only generalize and improve classical findings in existing literature but also reveal intrinsic connections between the inequalities. The validity of our conclusions is demonstrated through concrete examples and numerical visualizations, offering new insights for generalized convexity theory and fractional integral inequalities.
The present paper puts forward a time-delayed fractional-order phytoplankton-zooplankton system and systematically investigates its fractal dynamics, stability, bifurcation mechanism, and control strategy. Firstly, the existence, uniqueness, positivity and boundedness of solutions are derived. The equilibrium’s stability is analyzed, and the critical condition for delay-induced Hopf bifurcation is obtained. Subsequently, a nonlinear time-delayed feedback controller is introduced with the objective of suppressing unstable oscillations and bifurcation behaviors. The discussion encompasses the stability and bifurcation of the controlled system, demonstrating the efficacy of the controller in expanding the stable region and constraining fractal-like oscillatory patterns. Finally, the results of the theoretical model are verified by means of numerical simulations. This study contributes to the fractal and fractional dynamics of ecological models and provides a theoretical basis for the stability regulation of plankton systems.
