  Models of networks play a major role in explaining and reproducing
empirically observed patterns. Suitable models can be used to randomize an
observed network while preserving some of its features, or to generate
synthetic graphs whose properties may be tuned upon the characteristics of a
given population. In the present paper, we introduce the Fitness-Corrected
Block Model, an adjustable-density variation of the well-known Degree-Corrected
Block Model, and we show that the proposed construction yields a maximum
entropy model. When the network is sparse, we derive an analytical expression
for the degree distribution of the model that depends on just the constraints
and the chosen fitness-distribution. Our model is perfectly suited to define
maximum-entropy data-driven spatial social networks, where each block
identifies vertices having similar position (e.g., residence) and age, and
where the expected block-to-block adjacency matrix can be inferred from the
available data. In this case, the sparse-regime approximation coincides with a
phenomenological model where the probability of a link binding two individuals
is directly proportional to their sociability and to the typical cohesion of
their age-groups, whereas it decays as an inverse-power of their geographic
distance. We support our analytical findings through simulations of a stylized
urban area.
