  The extraction of a clear and simple footprint of the structure of large,
weighted and directed networks is a general problem that has many applications.
An important example is given by origin-destination matrices which contain the
complete information on commuting flows, but are difficult to analyze and
compare. We propose here a versatile method which extracts a coarse-grained
signature of mobility networks, under the form of a $2\times 2$ matrix that
separates the flows into four categories. We apply this method to
origin-destination matrices extracted from mobile phone data recorded in
thirty-one Spanish cities. We show that these cities essentially differ by
their proportion of two types of flows: integrated (between residential and
employment hotspots) and random flows, whose importance increases with city
size. Finally the method allows to determine categories of networks, and in the
mobility case to classify cities according to their commuting structure.
