  Finding the mean of sampled data is a fundamental task in machine learning
and statistics. However, in cases where the data samples are graph objects,
defining a mean is an inherently difficult task. We propose a novel framework
for defining a graph mean via embeddings in the space of smooth graph signal
distributions, where graph similarity can be measured using the Wasserstein
metric. By finding a mean in this embedding space, we can recover a mean graph
that preserves structural information. We establish the existence and
uniqueness of the novel graph mean, and provide an iterative algorithm for
computing it. To highlight the potential of our framework as a valuable tool
for practical applications in machine learning, it is evaluated on various
tasks, including k-means clustering of structured aligned graphs,
classification of functional brain networks, and semi-supervised node
classification in multi-layer graphs. Our experimental results demonstrate that
our approach achieves consistent performance, outperforms existing baseline
approaches, and improves the performance of state-of-the-art methods.
