Quantifying the relations (e.g., similarity) between complex networks paves the way for studying the latent information shared across networks. However, fundamental relation metrics are not well-defined between networks. As a compromise, prevalent techniques measure network relations in data-driven manners, which are inapplicable to analytic derivations in physics. To resolve this issue, we present a theory for obtaining an optimal characterization of network topological properties. We show that a network can be fully represented by a Gaussian variable defined by a function of the Laplacian, which simultaneously satisfies network-topology-dependent smoothness and maximum entropy properties. Based on it, we can analytically measure diverse relations between complex networks. As illustrations, we define encoding (e.g., information divergence and mutual information), decoding (e.g., Fisher information), and causality (e.g., Granger causality and conditional mutual information) between networks. We validate our framework on representative networks (e.g., random networks, protein structures, and chemical compounds) to demonstrate that a series of science and engineering challenges (e.g., network evolution, embedding, and query) can be tackled from a new perspective. An implementation of our theory is released as a multi-platform toolbox.

翻译：量化复杂网络之间的关系（如相似性）为研究网络间共享的潜在信息铺平了道路。然而，网络之间缺乏定义明确的基础关系度量。作为一种折衷方案，现有技术通常以数据驱动的方式测量网络关系，这无法应用于物理学中的解析推导。为解决这一问题，我们提出了一种理论，用于获得网络拓扑性质的最优表征。我们证明，网络可以通过由拉普拉斯算子函数定义的高斯变量完全表示，该变量同时满足依赖网络拓扑的光滑性与最大熵性质。基于此，我们能够解析地测量复杂网络之间的多种关系。作为示例，我们定义了网络间的编码（如信息散度与互信息）、解码（如Fisher信息）与因果关系（如Granger因果关系与条件互信息）。我们在代表性网络（如随机网络、蛋白质结构与化合物）上验证了该框架，表明一系列科学与工程挑战（如网络演化、嵌入与查询）可从新角度得以解决。本理论的实现已作为跨平台工具箱发布。