Deterministic, quenched or annealed? Differences in the parameter estimation of heterogeneous network models

Analysing weighted networks requires modelling the binary and weighted properties simultaneously. We highlight three approaches for estimating the parameters responsible for them: econometric techniques treating topology as deterministic and statistical techniques either ensemble-averaging parameters or maximising an averaged likelihood over the topological randomness. In homogeneous models, equivalence holds; in heterogeneous network models, the local disorder breaks it, in a way reminiscent of the difference between `quenched' and `annealed' averages in the physics of disordered systems.