On Joint Diagonalisation for Dynamic Network Analysis

TL;DR

This paper introduces the application of joint diagonalisation to analyze dynamic contact networks by identifying multiple modes in their evolution, enabling the construction of time-specific average graphs.

Contribution

It is the first to apply joint diagonalisation to network analysis, revealing multi-modal behaviors and temporal structures in contact networks.

Findings

Multi-modal distribution of deviations indicates multiple underlying modes.

JD can decompose network behavior into time-specific static graphs.

Application to real-world contact networks demonstrates practical utility.

Abstract

Joint diagonalisation (JD) is a technique used to estimate an average eigenspace of a set of matrices. Whilst it has been used successfully in many areas to track the evolution of systems via their eigenvectors; its application in network analysis is novel. The key focus in this paper is the use of JD on matrices of spanning trees of a network. This is especially useful in the case of real-world contact networks in which a single underlying static graph does not exist. The average eigenspace may be used to construct a graph which represents the `average spanning tree' of the network or a representation of the most common propagation paths. We then examine the distribution of deviations from the average and find that this distribution in real-world contact networks is multi-modal; thus indicating several \emph{modes} in the underlying network. These modes are identified and are found to correspond to particular times. Thus JD may be used to decompose the behaviour, in time, of contact networks and produce average static graphs for each time. This may be viewed as a mixture between a dynamic and static graph approach to contact network analysis.