Time-Dependent Fractional Dynamics of Anomalous Diffusion Process on Complex Networks

Fractional equations are usually applied to represent the anomalous diffusion process in network system. As an important system parameter, the role of time in the dynamical diffusion process are not fully considered. This paper introduced time-dependent fractional approach to depict the dynamical diffusion process in networks. For a general network, the general fractional derivative order in the diffusion equation are replaced with a time function and obtained the corresponding time-dependent Laplacian matrix of the network. Random walks are generated by obtained matrix of networks with different time functions, and simulated the anomalous diffusion process on a network with fractal tree structure. Furthermore, the average fractional return probability are calculated with different time functions. The results shows the efficiency of the network exploration are significant improved with time-depemdemt order. Therefore, the proposed time-dependent fractional diffusion approach provided an important tool to analyze the time-related process on network systems.