Sparse sequence of random graphs

I have the following definition for sparse random graphs:

In the lecture it was said that actually this type of graphs have a lot of "hubs", i.e. a lot of vertices of high degree.

But this is confusing me. So could someone tell me what I am thinking wrong in the following, and tell me how this should be seen please?

By convergence (1.4.2) which is a everywhere convergence we deduce convergences in probability and in distribution. From convergence in distribution we get that the sequence $(P_k^n)_n$ of probability measures is tight(no escape to infinity), therefore there are no vertex with an infinite degree (since tightness of the sequence should guarantee that for each $n$ $P^n(\mathbb{N})=1$ and that $p(\mathbb{N})=1$).

So how does it comes that we have a lot of vertices with high degree?

thank you

It doesn't. If you consider a regular graph with degree $k$, this is a sparse degree sequence (your distribution is a Dirac), and there are no vertices of high degree by definition.