$E$ with both extremities in $V'$ is called the subgraph of $G$ induced by $V'$.

**Random graph in the microcanonical ensemble**: If $V=\{1,\cdots,N\}$, there are ${{ N(N-1)/2 \choose L }}$ graphs with vertex set $V$ and $L$ edges (making a total of $2^{N(N-1)/2}$ graphs with vertex set $V$). Saying that all ${{ N(N-1)/2 \choose L }}$ are equiprobable turns the set of graphs on $N$ vertices and $L$ edges into a probability space whose elements we call *random graphs in the microcanonical ensemble*. This is the ensemble we use below for numerical simulations.

**Random graph in the canonical ensemble**: Given a number $p \in
[0,1]$, we introduce $\frac{N(N-1)}{2}$ independent random variables $\varepsilon_{i,j}$, $1\leq i < j \leq N$, each taking value $1$ with probability $p$ and $0$ with probability $1-p$. Saying that $\{i,j\}$ is an edge of $G$ if and only if $\varepsilon_{i,j}=1$ turns the set of all $2^{N(N-1)/2}$ graphs with vertex set $V$ into a probability space whose elements we call *random graphs in the canonical ensemble*. This is the ensemble we use below for analytical computations.

**Connectivity, $\alpha$**: In the sequel, we are interested in the large $N$ limit with a finite limit $\alpha$ for $\frac{2L}{N}$ (microcanonical ensemble) 