Eigenspectral Analysis of Hermitian Adjacency Matrices for the Analysis of Group Substructures

ABSTRACT

In this paper we propose the use of the eigensystem of complex adjacency matrices to analyze the structure of asymmetric directed weighted communication.

The use of complex Hermitian adjacency matrices allows to store more data relevant to asymmetric communication, and extends the interpretation of the resulting eigensystem beyond the principal eigenpair. This is based on the fact, that the adjacency matrix is transformed into a linear self-adjoint operator in Hilbert space.

Subgroups of members, or nodes of a communication network can be characterised by the eigensubspaces of the complex Hermitian adjacency matrix. Their relative ‘traffic-level’ is represented by the eigenvalue of the subspace, and their members are represented by the eigenvector components. Since eigenvectors belonging to distinct eigenvalues are orthogonal the subgroups can be viewed as independent with respect to the communication behavior of the relevant members of each subgroup.

As an example for this kind of analysis the EIES data set is used. The substructures and communication patterns within this data set are described.

Keywords:

• Hermitian matrix
• eigensystem
• perturbation
• social network analysis
• rank prestige index