Limit of $\alpha n$-regular graphs

I have a quick question about a graph limit problem. If I have a sequence of dense $d_n$-regular graphs on $n$ vertices (assume that $\frac{d_n}{n} \rightarrow \alpha > 0$), what is the limiting graphon?

Of course, the term limiting graphon does not make sense unless I specify the mode of convergence. I would ideally want the convergence to be in the cut norm introduced in https://arxiv.org/pdf/math/0702004.pdf (page 14).

Any help will be appreciated!

• This depends a great deal on the structure of the graphs. Random regular graphs will converge to a different limit than random bipartite regular graphs, for example. – Ben Barber May 18 '18 at 10:28