  Mapper is an unsupervised machine learning algorithm generalising the notion
of clustering to obtain a geometric description of a dataset. The procedure
splits the data into possibly overlapping bins which are then clustered. The
output of the algorithm is a graph where nodes represent clusters and edges
represent the sharing of data points between two clusters. However, several
parameters must be selected before applying Mapper and the resulting graph may
vary dramatically with the choice of parameters.
  We define an intrinsic notion of Mapper instability that measures the
variability of the output as a function of the choice of parameters required to
construct a Mapper output. Our results and discussion are general and apply to
all Mapper-type algorithms. We derive theoretical results that provide
estimates for the instability and suggest practical ways to control it. We
provide also experiments to illustrate our results and in particular we
demonstrate that a reliable candidate Mapper output can be identified as a
local minimum of instability regarded as a function of Mapper input parameters.
