  Spectral clustering is one of the most popular clustering methods for finding
clusters in a graph, which has found many applications in data mining. However,
the input graph in those applications may have many missing edges due to error
in measurement, withholding for a privacy reason, or arbitrariness in data
conversion. To make reliable and efficient decisions based on spectral
clustering, we assess the stability of spectral clustering against edge
perturbations in the input graph using the notion of average sensitivity, which
is the expected size of the symmetric difference of the output clusters before
and after we randomly remove edges.
  We first prove that the average sensitivity of spectral clustering is
proportional to $\lambda_2/\lambda_3^2$, where $\lambda_i$ is the $i$-th
smallest eigenvalue of the (normalized) Laplacian. We also prove an analogous
bound for $k$-way spectral clustering, which partitions the graph into $k$
clusters. Then, we empirically confirm our theoretical bounds by conducting
experiments on synthetic and real networks. Our results suggest that spectral
clustering is stable against edge perturbations when there is a cluster
structure in the input graph.
