Signed graphs are an emergent way of representing data in a variety of contexts where antagonistic interactions exist. These include data from biological, ecological, and social systems. Here we propose the concept of communicability for signed graphs and explore in depth its mathematical properties. We also prove that the communicability induces a hyperspherical geometric embedding of the signed network, and derive communicability-based metrics that satisfy the axioms of a distance even in the presence of negative edges. We then apply these metrics to solve several problems in the data analysis of signed graphs within a unified framework. These include the partitioning of signed graphs, dimensionality reduction, finding hierarchies of alliances in signed networks, and quantifying the degree of polarization between the existing factions in social systems represented by these types of graphs.

翻译：符号图是一种新兴的数据表示方法，适用于存在对抗性交互的多种场景，包括生物、生态和社会系统中的数据。本文提出了符号图的可沟通性概念，并深入探讨了其数学性质。我们证明了可沟通性能够诱导符号网络形成超球面几何嵌入，并推导出基于可沟通性的度量方法，这些度量即使在存在负边的情况下也满足距离公理。随后，我们在统一框架中应用这些度量来解决符号图数据分析中的若干问题，包括符号图的分割、降维、发现符号网络中的联盟层级结构，以及量化此类图所表征的社会系统中现有派系之间的极化程度。