Hi I'm trying to understand how regimes and thresholds of Erdős–Rényi model are valid in symmetric stochastic block model. In Erdős–Rényi model $G(n,p)$ each edge is drawn independently with probability $p$ and in symmetric stochastic block model SSBM$(n,k,q_{in}, q_{out})$ we have $k$ clusters, the probability that there is an edge intra-cluster is $q_{in}$, between clusters is $q_{out}$, always independently. For $G(n,p)$ we know that $G(n,c \log(n)/n)$ is connected with high probability if and only if $c > 1$, $G(n, c/n)$ has a giant component (i.e., a component of size linear in n) if and only if $c > 1$.
In section $2.4$ http://www.princeton.edu/~eabbe/publications/abbe_FNT_final4_plain.pdf I've found:
For SSBM($n,k,q_{in}, q_{out}$), these results hold by essentially replacing c with the average degree, for example
For $a, b > 0$, SSBM($n, k, a \log n/n, b \log n/n$) is connected with high probability if and only if $\frac{a+(k−1)b}{k}$ > 1 (if a or b is equal to 0, the graph is of course not connected).
SSBM($n, k, a/n, b/n$) has a giant component (i.e., a component of size linear in n) if and only if d :=$\frac{a+(k−1)b}{k}$ > 1.
I think that considering the average degree in SSBM makes the graph more homogeneous, then similar to $G(n,p)$, so it works. Could you help me to make/find a more specific proof?