  One of the most fundamental concepts in statistics is the concept of sample
mean. Properties of the sample mean that are well-defined in Euclidean spaces
become unwieldy or even unclear in graph spaces. Open problems related to the
sample mean of graphs include: non-existence, non-uniqueness, statistical
inconsistency, lack of convergence results of mean algorithms, non-existence of
midpoints, and disparity to midpoints. We present conditions to resolve all six
problems and propose a Majorize-Minimize-Mean (MMM) Algorithm. Experiments on
graph datasets representing images and molecules show that the MMM-Algorithm
best approximates a sample mean of graphs compared to six other mean
algorithms.
