  The exponential family of random graphs represents an important and
challenging class of network models. Despite their flexibility, conventionally
used exponential random graphs have one shortcoming. They cannot directly model
weighted networks as the underlying probability space consists of simple graphs
only. Since many substantively important networks are weighted, this limitation
is especially problematic. We extend the existing exponential framework by
proposing a generic common distribution for the edge weights and rigorously
analyze the associated phase transitions and critical phenomena. We then apply
these general results to get concrete answers in exponential random graph
models where the edge weights are uniformly distributed.
