We introduce Multivariate Multiscale Graph-based Dispersion Entropy (mvDEG), a novel, computationally efficient method for analyzing multivariate time series data in graph and complex network frameworks, and demonstrate its application in real-world data. mvDEG effectively combines temporal dynamics with topological relationships, offering enhanced analysis compared to traditional nonlinear entropy methods. Its efficacy is established through testing on synthetic signals, such as uncorrelated and correlated noise, showcasing its adeptness in discerning various levels of dependency and complexity. The robustness of mvDEG is further validated with real-world datasets, effectively differentiating various two-phase flow regimes and capturing distinct dynamics in weather data analysis. An important advancement of mvDEG is its computational efficiency. Our optimized algorithm displays a computational time that grows linearly with the number of vertices or nodes, in contrast to the exponential growth observed in classical methods. This efficiency is achieved through refined matrix power calculations that exploit matrix and Kronecker product properties, making our method faster than the state of the art. The significant acceleration in computational time positions mvDEG as a transformative tool for extensive and real-time applications, setting a new benchmark in the analysis of time series recorded at distributed locations and opening avenues for innovative applications.

翻译：我们提出了一种新颖且计算高效的多变量多尺度图基离散熵方法（mvDEG），用于在图与复杂网络框架下分析多变量时间序列数据，并展示了其在真实世界数据中的应用。mvDEG有效融合了时间动态与拓扑关系，相较于传统非线性熵方法提供了增强的分析能力。通过针对合成信号（如非相关和相关噪声）的测试，验证了其在区分不同依赖性与复杂性水平方面的能力。mvDEG的鲁棒性进一步通过真实世界数据集得到验证，成功区分了多种两相流态并捕捉了天气数据分析中的独特动态特征。mvDEG的重要进步在于其计算效率。我们的优化算法显示计算时间随顶点或节点数量线性增长，而传统方法则呈现指数增长。这一效率得益于利用矩阵与克罗内克积性质改进的矩阵幂计算，使该方法快于当前最优技术。计算时间的显著加速使mvDEG成为大规模及实时应用的变革性工具，为分布式位置记录的时间序列分析树立了新基准，并为创新应用开辟了道路。