  Graphons have traditionally served as limit objects for dense graph
sequences, with the cut distance serving as the metric for convergence.
However, sparse graph sequences converge to the trivial graphon under the
conventional definition of cut distance, which make this framework inadequate
for many practical applications. In this paper, we utilize the concepts of
generalized graphons and stretched cut distance to describe the convergence of
sparse graph sequences. Specifically, we consider a random graph process
generated from a generalized graphon. This random graph process converges to
the generalized graphon in stretched cut distance. We use this random graph
process to model the growing sparse graph, and prove the convergence of the
adjacency matrices' eigenvalues. We supplement our findings with experimental
validation. Our results indicate the possibility of transfer learning between
sparse graphs.
