Dynamics of complex networks through the use of p-adic differential equations linked to fractional pseudo-differential operators

Abstract The article examines how p -adic analysis is applied to modeling dynamics in complex systems, particularly in hierarchical networks, using ultrametric spaces. These spaces, defined by ultrametric distances, provide a framework for transition networks where transition probabilities between system states depend on both distances and local potentials. Mathematical models are developed, including master equations that describe how probabilities evolve over time, highlighting the influence of hierarchical distances on dynamics. The article also explores applications in areas such as physics, quantum systems, and hierarchical neural networks. Furthermore, it investigates how p -adic fractional differential equations can describe complex phenomena with non-local interactions, using a fractional pseudo-differential operator combined with a potential. An explicit solution is provided using the Fourier transform, detailing how the operator and potential affect dynamics. Specific examples illustrate the dispersion of the solution and the interplay between diffusion and amplification. The results include graphical representations that depict the temporal evolution of solutions, showing how initial conditions can lead to either dissipation or concentration of the distribution. Finally, a spectral analysis of the solution to the p -adic differential equation is performed, decomposing it into eigenfunctions and eigenvalues of the operator. This approach helps understand, for instance, the long-term behavior of tumor cell diffusion, emphasizing how the characteristics of the operator and potential can stabilize or accelerate the growth of the cancer front, depending on their properties.