  In the study of economic networks, econometric approaches interpret the
traditional Gravity Model specification as the expected link weight coming from
a probability distribution whose functional form can be chosen arbitrarily,
while statistical-physics approaches construct maximum-entropy distributions of
weighted graphs, constrained to satisfy a given set of measurable network
properties. In a recent, companion paper, we integrated the two approaches and
applied them to the World Trade Web, i.e. the network of international trade
among world countries. While the companion paper dealt only with
discrete-valued link weights, the present paper extends the theoretical
framework to continuous-valued link weights. In particular, we construct two
broad classes of maximum-entropy models, namely the integrated and the
conditional ones, defined by different criteria to derive and combine the
probabilistic rules for placing links and loading them with weights. In the
integrated models, both rules follow from a single, constrained optimization of
the continuous Kullback-Leibler divergence; in the conditional models, the two
rules are disentangled and the functional form of the weight distribution
follows from a conditional, optimization procedure. After deriving the general
functional form of the two classes, we turn each of them into a proper family
of econometric models via a suitable identification of the econometric function
relating the corresponding, expected link weights to macroeconomic factors.
After testing the two classes of models on World Trade Web data, we discuss
their strengths and weaknesses.
