Scaling law of cliques and holes in chaotic systems

We investigate the algebraic topology of strange attractors in chaotic systems. By mapping strange attractors into networks, the box-covering algorithm is employed to observe the high-order structures at different length scales on the networks. Notably, for a great variety of chaotic systems, we find that the numbers of cliques and holes exhibit distinct scaling laws as the networks expand. Furthermore, we observe that this interesting scaling law behaviors are also established in real-world chaotic systems. Our work unveils the function of cliques and holes in shaping chaotic attractors.