Multiplex graphs, characterised by their layered structure, exhibit informative interdependencies within layers that are crucial for understanding complex network dynamics. Quantifying the interaction and shared information among these layers is challenging due to the non-Euclidean structure of graphs. Our paper introduces a comprehensive theory of multivariate information measures for multiplex graphs. We introduce graphon mutual information for pairs of graphs and expand this to graphon interaction information for three or more graphs, including their conditional variants. We then define graphon total correlation and graphon dual total correlation, along with their conditional forms, and introduce graphon $O-$information. We discuss and quantify the concepts of synergy and redundancy in graphs for the first time, introduce consistent nonparametric estimators for these multivariate graphon information--theoretic measures, and provide their convergence rates. We also conduct a simulation study to illustrate our theoretical findings and demonstrate the relationship between the introduced measures, multiplex graph structure, and higher--order interdependecies. Real-world applications further show the utility of our estimators in revealing shared information and dependence structures in real-world multiplex graphs. This work not only answers fundamental questions about information sharing across multiple graphs but also sets the stage for advanced pattern analysis in complex networks.

翻译：多重图以其分层结构为特征，在层内展现出丰富的信息互依性，这对于理解复杂网络动态至关重要。由于图的非欧几里得结构，量化这些层之间的相互作用与共享信息具有挑战性。本文提出了一个针对多重图的多元信息测度的完备理论。我们引入了用于成对图的图互信息，并将其扩展至用于三个或更多图的图交互信息，包括它们的条件变体。接着，我们定义了图总相关性与图对偶总相关性及其条件形式，并引入了图$O-$信息。我们首次讨论并量化了图中的协同与冗余概念，为这些多元图信息论测度引入了一致的非参数估计量，并给出了它们的收敛速率。我们还进行了一项模拟研究以阐明理论发现，并展示了所引入的测度、多重图结构以及高阶互依性之间的关系。实际应用进一步证明了我们的估计量在揭示现实世界多重图中共享信息与依赖结构方面的效用。这项工作不仅回答了关于跨多个图信息共享的基本问题，也为复杂网络中的高级模式分析奠定了基础。