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

Recent years have witnessed the emergence of a new class of social networks, that require us to move beyond previously employed representations of complex graph structures. A notable example is that of the folksonomy, an online process where users collaboratively employ tags to resources to impart structure to an otherwise undifferentiated database. In a recent paper[1] we proposed a mathematical model that represents these structures as tripartite hypergraphs and defined basic topological quantities of interest. In this paper we extend our model by defining additional quantities such as edge distributions, vertex similarity and correlations as well as clustering. We then empirically measure these quantities on two real life folksonomies, the popular online photo sharing site Flickr and the bookmarking site CiteULike. We find that these systems share similar qualitative features with the majority of complex networks that have been previously studied. We propose that the quantities and methodology described here can be used as a standard tool in measuring the structure of tagged networks.

We analyze CiteULike, an online collaborative tagging system where users bookmark and annotate scientific papers. Such a system can be naturally represented as a tri-partite graph whose nodes represent papers, users and tags connected by individual tag assignments. The semantics of tags is studied here, in order to uncover the hidden relationships between tags. We find that the clustering coefficient can be used to analyze the semantical patterns among tags.