Word Embedding Analogies: Understanding King - Man + Woman = Queen

A surprising property of word vectors is that word analogies can often be solved with vector arithmetic. Most famously, \[\vec{king} - \vec{man} + \vec{woman} \approx \vec{queen}\] But why? Why do arithmetic operators apply to vectors generated by non-linear models such as word2vec? What conditions have to be satisfied by the training corpus for these analogies to hold in the vector space? There h