  The theory of graphons is an important tool in understanding properties of
large networks. We investigate a power-law random graph model and cast it in
the graphon framework. The distinctively different structures of the limit
graph are explored in detail in the sub-critical and super-critical regimes. In
the sub-critical regime, the graph is empty with high probability, and in the
rare event that it is non-empty, it consists of a single edge. Contrarily, in
the super-critical regime, a non-trivial random graph exists in the limit, and
it serves as an uncovered boundary case between different types of graph
convergence.
