Optimizing self-organized topology of recurrence-based complex networks

Networks and graphs have emerged as powerful tools to model and analyze nonlinear dynamical systems. By constructing an adjacency matrix from recurrence networks, it is possible to capture critical structural and geometric information about the underlying dynamics of a time series. However, randomization of data often raises concerns about the potential loss of deterministic relationships. Here, in using the spring-electrical-force model, we demonstrate that by optimizing the distances between randomized points through minimizing an entropy-related energy measure, the deterministic structure of the original system is not destroyed. This process allows us to approximate the time series shape and correct the phase, effectively reconstructing the initial invariant set and attracting dynamics of the system. Our approach highlights the importance of adjacency matrices derived from recurrence plots, which preserve crucial information about the nonlinear dynamics. By using recurrence plots and the entropy of diagonal line lengths and leveraging the Kullback-Leibler divergence as a relative entropic measure, we fine-tune the parameters and initial conditions for recurrence plots, ensuring an optimal representation of the system's dynamics. Through the integration of network geometry and energy minimization, we show that data-driven graphs can self-organize to retain and regenerate the fundamental features of the time series, including its phase space structures. This study underscores the robustness of recurrence networks as a tool for analyzing nonlinear systems and demonstrates that randomization, when guided by informed optimization, does not erase deterministic relationships, opening new avenues for reconstructing dynamical systems from observational data.