 Configuration model

For a prescribed degree sequence with even sum $M={\sum }_i{d}_i$, attach $d_i$ labelled half-edges to vertex $i$ and choose a uniform perfect matching of the $M$ half-edges. Each pair becomes an edge, giving a multigraph that may have self-loops or parallel edges. The number of pairings is $M!/[2^{M/2}(M/2)!]$. Every simple graph with that degree sequence has exactly ${\prod }_i{d}_i!$ pairing representations, so conditioning on simplicity produces the uniform simple-graph model.

The "Configuration model" can refer to different concepts depending on the context in which it's being used. Here are a few common interpretations of the term: 1. **Physics and Network Theory**: In statistical physics and complex networks, the configuration model is a way to generate random networks with a given degree distribution. The model allows for the construction of graphs that satisfy specific connectivity requirements, particularly focusing on how many edges (connections) each node (vertex) has.