  Graph learning is often a necessary step in processing or representing
structured data, when the underlying graph is not given explicitly. Graph
learning is generally performed centrally with a full knowledge of the graph
signals, namely the data that lives on the graph nodes. However, there are
settings where data cannot be collected easily or only with a non-negligible
communication cost. In such cases, distributed processing appears as a natural
solution, where the data stays mostly local and all processing is performed
among neighbours nodes on the communication graph. We propose here a novel
distributed graph learning algorithm, which permits to infer a graph from
signal observations on the nodes under the assumption that the data is smooth
on the target graph. We solve a distributed optimization problem with local
projection constraints to infer a valid graph while limiting the communication
costs. Our results show that the distributed approach has a lower communication
cost than a centralised algorithm without compromising the accuracy in the
inferred graph. It also scales better in communication costs with the increase
of the network size, especially for sparse networks.
