Expected $k$-shell distribution of a random graph Given a random graph $G$ from the Erdős–Rényi model $\mathcal{G}(n,m)$ is there a closed form solution for the distribution of the $k$-shell sizes (i.e. the expected number of nodes in each shell)? Alternatively, is there a closed form bound for the degeneracy of a such random graph? I have looked around for relevant results, but all I keep finding are exponential random graph models that can replicate random graphs with a given $k$-shell distirbution.
 A: "Sudden Emergence of a Giant $k$-core in a Random Graph" by Pittel, Spencer, and Wormald, answers all your questions (at least for constant $k$).
The exact results are Theorems 1-3 in the paper above, so let me summarize. For each $k \ge 3$, the $k$-core appears in $G(n,m)$ when $m \sim c_k n/2$, where $c_k = k + \sqrt{k \log k} + O(\log k)$. At this point, it already contains a linear number of vertices. For $c > c_k$, the size of the $k$-core of $G(n,cn/2)$ is asymptotic to $n f_k(c)$, where $f_k$ is a known function of $c$.
The actual values of $c_k, f_k(c)$, and other parameters are hard to describe without just quoting the paper. They are similar in spirit to the fixed-point type expressions for the size of a giant component in $G(n,m)$, but are defined in terms of tail probabilities of a $\text{Poisson}(\lambda)$ distribution.
For $k=2$, a giant $2$-core appears with the giant component, but a $2$-core already exists before that: it is just the set of all vertices that are part of some cycle. The paper "Counting connected graphs inside-out" by Pittel and Wormald shows that in $G(n,cn/2)$ the $2$-core is a $(1-t)$-fraction of the vertices in the giant component, where $t$ is the unique root of $t e^{-t} = ce^{-c}$ in $(0,1)$. There are only $O(\log n)$ vertices in the $2$-core that are not part of the giant component.
