Approximate Fréchet Mean for Data Sets of Sparse Graphs
This work addresses the need for centrality measures in non-Euclidean metric spaces for graph data, which is incremental as it builds on existing Fréchet mean concepts with a specific pseudometric.
The authors tackled the problem of characterizing the central location of a set of graphs by proposing an approximate Fréchet mean algorithm, using a pseudometric based on the ℓ₂ norm between adjacency matrix eigenvalues to capture structural changes at multiple scales.
To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Fréchet mean. In this work, we equip a set of graph with the pseudometric defined by the $\ell_2$ norm between the eigenvalues of their respective adjacency matrix . Unlike the edit distance, this pseudometric reveals structural changes at multiple scales, and is well adapted to studying various statistical problems on sets of graphs. We describe an algorithm to compute an approximation to the Fréchet mean of a set of undirected unweighted graphs with a fixed size.