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This paper presents a generalization of the random dot product model fornetworks whose edge weights are drawn from a parametrized probabilitydistribution. We focus on the case of integer weight edges and show that manypreviously studied models can be recovered as special cases of thisgeneralization. Our model also determines a dimension--reducing embeddingprocess that gives geometric interpretations of community structure andcentrality. The dimension of the embedding has consequences for the derivedcommunity structure and we exhibit a stress function for determiningappropriate dimensions. We use this approach to analyze a coauthorship networkand voting data from the U.S. Senate.

A Random Dot Product Model for Weighted Networks