  In the last few years we have witnessed the emergence, primarily in on-line
communities, of new types of social networks that require for their
representation more complex graph structures than have been employed in the
past. One example is the folksonomy, a tripartite structure of users,
resources, and tags -- labels collaboratively applied by the users to the
resources in order to impart meaningful structure on an otherwise
undifferentiated database. Here we propose a mathematical model of such
tripartite structures which represents them as random hypergraphs. We show that
it is possible to calculate many properties of this model exactly in the limit
of large network size and we compare the results against observations of a real
folksonomy, that of the on-line photography web site Flickr. We show that in
some cases the model matches the properties of the observed network well, while
in others there are significant differences, which we find to be attributable
to the practice of multiple tagging, i.e., the application by a single user of
many tags to one resource, or one tag to many resources.
