Modularity with a more accurate baseline model

We derive an expression for the exact probability Pr[i∼j] of a link between a node i with degree d_{i} and a node j with degree d_{j} in a graph belonging to the class of Erdős-Rényi G(N,L) random graphs with N nodes and L links. The probability Pr[i∼j] is commonly approximated as d_{i}d_{j}/2L and appears in the formula of Newman's modularity, which plays a crucial rule in community detection in networks. We show that, when applied to graphs not belonging to the class of Erdős-Rényi random graphs, our formula for Pr[i∼j] is considerably more accurate than d_{i}d_{j}/2L and leads to the detection of different clusters or partitions than the original modularity formula.