Geometries of mixed graphs in complex vector spaces. Hierarchies and clusters in complex networks

We introduce several geometric measures for mixed graphs represented by complex-valued Hermitian adjacency matrices.We define the communicability functions based on the exponential of the Hermitian adjacency matrix and define complex-valued position vectors. Then, we define a Euclidean distance as well as complex, and Euclidean angles between these positions vectors for mixed graphs. Further we introduce Kähler and Hermitian angles between different planes among position vectors and holomorphic and projection planes, respectively. We find several mathematical relations and inequalities between all these geometric parameters. To illustrate the usability of some of these indices in the study of real-world networks we study the Kähler angle for finding hierarchies and detecting hierarchical clusters of vertices in ecological food webs, networks of co-purchasing of political books, a neuronal network, an Internet trolls network, and a software collaboration graph. These applications give empirical evidence that the Kähler angle contains important information about the structure of mixed graphs which is relevant for real-world applications in the study of complex networks.