  Although theoretical properties such as expressive power and over-smoothing
of graph neural networks (GNN) have been extensively studied recently, its
convergence property is a relatively new direction. In this paper, we
investigate the convergence of one powerful GNN, Invariant Graph Network (IGN)
over graphs sampled from graphons.
  We first prove the stability of linear layers for general $k$-IGN (of order
$k$) based on a novel interpretation of linear equivariant layers. Building
upon this result, we prove the convergence of $k$-IGN under the model of
\citet{ruiz2020graphon}, where we access the edge weight but the convergence
error is measured for graphon inputs.
  Under the more natural (and more challenging) setting of
\citet{keriven2020convergence} where one can only access 0-1 adjacency matrix
sampled according to edge probability, we first show a negative result that the
convergence of any IGN is not possible. We then obtain the convergence of a
subset of IGNs, denoted as IGN-small, after the edge probability estimation. We
show that IGN-small still contains function class rich enough that can
approximate spectral GNNs arbitrarily well. Lastly, we perform experiments on
various graphon models to verify our statements.
