Network representations often cannot fully account for the structural richness of complex systems spanning multiple levels of organisation. Recently proposed high-order information-theoretic signals are well-suited to capture synergistic phenomena that transcend pairwise interactions; however, the exponential-growth of their cardinality severely hinders their applicability. In this work, we combine methods from harmonic analysis and combinatorial topology to construct efficient representations of high-order information-theoretic signals. The core of our method is the diagonalisation of a discrete version of the Laplace–de Rham operator, that geometrically encodes structural properties of the system. We capitalise on these ideas by developing a complete workflow for the construction of hyperharmonic representations of high-order signals, which is applicable to a wide range of scenarios.

网络表示通常不能完全解释跨越多个组织层次的复杂系统的结构丰富性。最近提出的高阶信息理论信号非常适合捕捉超越成对相互作用的协同现象；然而，它们基数的指数增长严重阻碍了它们的适用性。在这项工作中，我们结合谐波分析和组合拓扑的方法来构建高阶信息论信号的有效表示。我们方法的核心是 Laplace-de Rham 算子的离散版本的对角化，它对系统的结构特性进行几何编码。我们通过开发用于构建高阶信号的超谐波表示的完整工作流程来利用这些想法，