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A Shared Nearest Neighbor (SNN) graph is a type of graph construction usingshared nearest neighbor information, which is a secondary similarity measurebased on the rankings induced by a primary $k$-nearest neighbor ($k$-NN)measure. SNN measures have been touted as being less prone to the curse ofdimensionality than conventional distance measures, and thus methods using SNNgraphs have been widely used in applications, particularly in clusteringhigh-dimensional data sets and in finding outliers in subspaces of highdimensional data. Despite this, the theoretical study of SNN graphs and graphLaplacians remains unexplored. In this pioneering work, we make the firstcontribution in this direction. We show that large scale asymptotics of an SNNgraph Laplacian reach a consistent continuum limit; this limit is the same asthat of a $k$-NN graph Laplacian. Moreover, we show that the pointwiseconvergence rate of the graph Laplacian is linear with respect to $(k/n)^{1/m}$with high probability.

Graph Laplacians on Shared Nearest Neighbor graphs and graph Laplacians on $k$-Nearest Neighbor graphs having the same limit