  In network data analysis, it is becoming common to work with a collection of
graphs that exhibit \emph{heterogeneity}. For example, neuroimaging data from
patient cohorts are increasingly available. A critical analytical task is to
identify communities, and graph Laplacian-based methods are routinely used.
However, these methods are currently limited to a single network and do not
provide measures of uncertainty on the community assignment. In this work, we
propose a probabilistic network model called the ``Spiked Laplacian Graph''
that considers each network as an invertible transform of the Laplacian, with
its eigenvalues modeled by a modified spiked structure. This effectively
reduces the number of parameters in the eigenvectors, and their sign patterns
allow efficient estimation of the community structure. Further, the posterior
distribution of the eigenvectors provides uncertainty quantification for the
community estimates. Subsequently, we introduce a Bayesian non-parametric
approach to address the issue of heterogeneity in a collection of graphs.
Theoretical results are established on the posterior consistency of the
procedure and provide insights on the trade-off between model resolution and
accuracy. We illustrate the performance of the methodology on synthetic data
sets, as well as a neuroscience study related to brain activity in working
memory.
  Keywords: Hierarchical Community Detection, Isoperimetric Constant,
Mixed-Effect Eigendecomposition, Normalized Graph Cut, Stiefel Manifold
