The choice of free parameters in network models is subjective, since it depends on what topological properties are being monitored. However, we show that the Maximum Likelihood (ML) principle indicates a unique, statistically rigorous parameter choice, associated to a well defined topological feature. We then find that, if the ML condition is incompatible with the built-in parameter choice, network models turn out to be intrinsically ill-defined or biased. To overcome this problem, we construct a class of safely unbiased models. We also propose an extension of these results that leads to the fascinating possibility to extract, only from topological data, the `hidden variables' underlying network organization, making them `no more hidden'. We test our method on the World Trade Web data, where we recover the empirical Gross Domestic Product using only topological information.
Categories: cond-mat.dis-nn cond-mat.stat-mech physics.data-an physics.soc-ph