Degree distribution and patches of the image horizontal visibility graph mapped from two-dimensional Thue–Morse words

Abstract

In this paper, we aim to study the image horizontal visibility graph of order $n$ (IHVG${}_n$) mapped from the 2D Thue–Morse words which also help us to understand their limit, i.e., 2D Thue–Morse sequence. Firstly, we map the series to complex networks by using the image horizontal visibility algorithm of order 4. Subsequently, using the iterative method and the construction of sequences, we obtain the exact analytic expressions of the degree distribution of the 2D Thue–Morse IHVG₄. Furthermore, we also investigate 3-order patches of the 2D Thue–Morse IHVG₄ and find that the patch profile converges to a constant. Finally, numerical simulations are used to verify that theoretical values of the degree distribution and patch profile are in agreement with the simulated values.