  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.
