Soft configuration model
In applied mathematics, the soft configuration model is a random graph model subject to the principle of maximum entropy under constraints on the expectation of the degree sequence of sampled graphs. Whereas the configuration model uniformly samples random graphs of a specific degree sequence, the SCM only retains the specified degree sequence on average over all network realizations; in this sense the SCM has very relaxed constraints relative to those of the CM. The SCM for graphs of size has a nonzero probability of sampling any graph of size, whereas the CM is restricted to only graphs having precisely the prescribed connectivity structure.
Model formulation
The SCM is a statistical ensemble of random graphs having vertices labeled, producing a probability distribution on . Imposed on the ensemble are constraints, namely that the ensemble average of the degree of vertex is equal to a designated value, for all. The model is fully parameterized by its size and expected degree sequence. These constraints are both local and soft, and thus yields a canonical ensemble with an extensive properties|extensive] number of constraints. The conditions are imposed on the ensemble by the method of Lagrange multipliers.Derivation of the probability distribution
The probability of the SCM producing a graph is determined by maximizing the Gibbs entropy subject to constraints and normalization. This amounts to optimizing the multi-constraint Lagrange function below:where and are the multipliers to be fixed by the constraints. Setting to zero the derivative of the above with respect to for an arbitrary yields
the constant being the partition function normalizing the distribution; the above exponential expression applies to all, and thus is the probability distribution. Hence we have an exponential family parameterized by, which are related to the expected degree sequence by the following equivalent expressions: