  Population mobility can be studied readily and cheaply using cellphone data,
since people's mobility can be approximately mapped into tower-mobile
registries. We model people moving in a grid-like city, where edges of the grid
are weighted and paths are chosen according to overall weights between origin
and destination. Cellphone users leave sparse signals in random nodes of the
grid as they move by, mimicking the type of data collected from the
tower-cellphone interactions. From this noisy data we seek to build a model of
the city, {\it i.e.} to predict probabilities of paths from origin to
destination. We focus on the simplest case where users move along shortest
paths (no loops, no going backwards). In this simplified setting, we are able
to infer the underlying weights of the edges (akin to road transitability) with
an inverse statistical mechanic model.
