Deterministic, quenched, and annealed parameter estimation for heterogeneous network models

At least two different approaches to define and solve statistical models for the analysis of economic systems exist: the typical, econometric one, interpreting the gravity model specification as the expected link weight of an arbitrary probability distribution, and the one rooted in statistical physics, constructing maximum-entropy distributions constrained to satisfy certain network properties. In a couple of recent companion papers, they have been successfully integrated within the framework induced by the constrained minimization of the Kullback-Leibler divergence: specifically, two broad classes of models have been devised, i.e., the integrated and conditional ones, defined by different, probabilistic rules to place links, load them with weights and turn them into proper, econometric prescriptions. Still, the recipes adopted by the two approaches to estimate the parameters entering into the definition of each model differ. In econometrics, a likelihood that decouples the binary and weighted parts of a model, treating a network as deterministic, is typically maximized; to restore its random character, two alternatives exist: either solving the likelihood maximization on each configuration of the ensemble and taking the average of the parameters afterwards or taking the average of the likelihood function and maximizing the latter one. The difference between these approaches lies in the order in which the operations of averaging and maximization are taken-a difference that is reminiscent of the quenched and annealed ways of averaging out the disorder in spin glasses. The results of the present contribution, devoted to comparing these recipes in the case of continuous, conditional network models, indicate that the annealed estimation recipe represents the best alternative to the deterministic one.