The SCM is a statistical ensemble of random graphs ${\displaystyle G}$  having ${\displaystyle n}$  vertices (${\displaystyle n=|V(G)|}$ ) labeled ${\displaystyle \{v_{j}\}_{j=1}^{n}=V(G)}$ , producing a probability distribution on ${\displaystyle {\mathcal {G}}_{n}}$  (the set of graphs of size ${\displaystyle n}$ ). Imposed on the ensemble are ${\displaystyle n}$  constraints, namely that the ensemble average of the degree ${\displaystyle k_{j}}$  of vertex ${\displaystyle v_{j}}$  is equal to a designated value ${\displaystyle {\widehat {k}}_{j}}$ , for all ${\displaystyle v_{j}\in V(G)}$ . The model is fully parameterized by its size ${\displaystyle n}$  and expected degree sequence ${\displaystyle \{{\widehat {k}}_{j}\}_{j=1}^{n}}$ . These constraints are both local (one constraint associated with each vertex) and soft (constraints on the ensemble average of certain observable quantities), and thus yields a canonical ensemble with an extensive number of constraints.^[2] The conditions ${\displaystyle \langle k_{j}\rangle ={\widehat {k}}_{j}}$  are imposed on the ensemble by the method of Lagrange multipliers (see Maximum-entropy random graph model).

The probability ${\displaystyle \mathbb {P} _{\text{SCM}}(G)}$  of the SCM producing a graph ${\displaystyle G}$  is determined by maximizing the Gibbs entropy ${\displaystyle S[G]}$  subject to constraints ${\displaystyle \langle k_{j}\rangle ={\widehat {k}}_{j},\ j=1,\ldots ,n}$  and normalization ${\displaystyle \sum _{G\in {\mathcal {G}}_{n}}\mathbb {P} _{\text{SCM}}(G)=1}$ . This amounts to optimizing the multi-constraint Lagrange function below:

${\displaystyle {\begin{aligned}&{\mathcal {L}}\left(\alpha ,\{\psi _{j}\}_{j=1}^{n}\right)\\[6pt]={}&-\sum _{G\in {\mathcal {G}}_{n}}\mathbb {P} _{\text{SCM}}(G)\log \mathbb {P} _{\text{SCM}}(G)+\alpha \left(1-\sum _{G\in {\mathcal {G}}_{n}}\mathbb {P} _{\text{SCM}}(G)\right)+\sum _{j=1}^{n}\psi _{j}\left({\widehat {k}}_{j}-\sum _{G\in {\mathcal {G}}_{n}}\mathbb {P} _{\text{SCM}}(G)k_{j}(G)\right),\end{aligned}}}$ 

where ${\displaystyle \alpha }$  and ${\displaystyle \{\psi _{j}\}_{j=1}^{n}}$  are the ${\displaystyle n+1}$  multipliers to be fixed by the ${\displaystyle n+1}$  constraints (normalization and the expected degree sequence). Setting to zero the derivative of the above with respect to ${\displaystyle \mathbb {P} _{\text{SCM}}(G)}$  for an arbitrary ${\displaystyle G\in {\mathcal {G}}_{n}}$  yields

${\displaystyle 0={\frac {\partial {\mathcal {L}}\left(\alpha ,\{\psi _{j}\}_{j=1}^{n}\right)}{\partial \mathbb {P} _{\text{SCM}}(G)}}=-\log \mathbb {P} _{\text{SCM}}(G)-1-\alpha -\sum _{j=1}^{n}\psi _{j}k_{j}(G)\ \Rightarrow \ \mathbb {P} _{\text{SCM}}(G)={\frac {1}{Z}}\exp \left[-\sum _{j=1}^{n}\psi _{j}k_{j}(G)\right],}$ 

the constant ${\displaystyle Z:=e^{\alpha +1}=\sum _{G\in {\mathcal {G}}_{n}}\exp \left[-\sum _{j=1}^{n}\psi _{j}k_{j}(G)\right]=\prod _{1\leq i<j\leq n}\left(1+e^{-(\psi _{i}+\psi _{j})}\right)}$ ^[3] being the partition function normalizing the distribution; the above exponential expression applies to all ${\displaystyle G\in {\mathcal {G}}_{n}}$ , and thus is the probability distribution. Hence we have an exponential family parameterized by ${\displaystyle \{\psi _{j}\}_{j=1}^{n}}$ , which are related to the expected degree sequence ${\displaystyle \{{\widehat {k}}_{j}\}_{j=1}^{n}}$  by the following equivalent expressions:

${\displaystyle \langle k_{q}\rangle =\sum _{G\in {\mathcal {G}}_{n}}k_{q}(G)\mathbb {P} _{\text{SCM}}(G)=-{\frac {\partial \log Z}{\partial \psi _{q}}}=\sum _{j\neq q}{\frac {1}{e^{\psi _{q}+\psi _{j}}+1}}={\widehat {k}}_{q},\ q=1,\ldots ,n.}$