  We introduce a metric space of clusterings, where clusterings are described
by a binary vector indexed by the vertex-pairs. We extend this geometry to a
hypersphere and prove that maximizing modularity is equivalent to minimizing
the angular distance to some modularity vector over the set of clustering
vectors. In that sense, modularity-based community detection methods can be
seen as a subclass of a more general class of projection methods, which we
define as the community detection methods that adhere to the following two-step
procedure: first, mapping the network to a point on the hypersphere; second,
projecting this point to the set of clustering vectors. We show that this class
of projection methods contains many interesting community detection methods.
Many of these new methods cannot be described in terms of null models and
resolution parameters, as is customary for modularity-based methods. We provide
a new characterization of such methods in terms of meridians and latitudes of
the hypersphere. In addition, by relating the modularity resolution parameter
to the latitude of the corresponding modularity vector, we obtain a new
interpretation of the resolution limit that modularity maximization is known to
suffer from.
