  The $K$-means algorithm remains one of the most widely-used clustering
methods due to its simplicity and general utility. The performance of $K$-means
depends upon location of minima low in cost function, amongst a potentially
vast number of solutions. Here, we use the energy landscape approach to map the
change in $K$-means solution space as a result of increasing dataset outliers
and show that the cost function surface becomes more funnelled. Kinetic
analysis reveals that in all cases the overall funnel is composed of shallow
locally-funnelled regions, each of which are separated by areas that do not
support any clustering solutions. These shallow regions correspond to different
types of clustering solution and their increasing number with outliers leads to
longer pathways within the funnel and a reduced correlation between accuracy
and cost function. Finally, we propose that the rates obtained from kinetic
analysis provide a novel measure of clustering similarity that incorporates
information about the paths between them. This measure is robust to outliers
and we illustrate the application to datasets containing multiple outliers.
