In this artwork, how is the Influence discussed?
The Triumph of Bacchus received a number of rather grand and elaborate idealized treatments in Renaissance art, of which Titian's Bacchus and Ariadne, then in the Spanish royal collection, was an imaginative variant. Usually Bacchus was processing in a chariot drawn by leopards, with a retinue of satyrs and revellers, including his guardian Silenus.  The use of the title for Velázquez's painting is almost ironic given the very different treatment here. 
One inspiration for Velázquez is Caravaggio's treatments of religious subjects combining central figures in traditional iconographical robes with subsidiary figures in contemporary dress, and Ribera's naturalistic portraits of figures from antiquity, sometimes depicted as beggars.  Entertainments hosted by Bacchus appear as an occasional subject in art from the Renaissance onwards, as one type of the wider subject of the Feast of the gods in art: around 1550 Taddeo Zuccari painted a large feast at the Wedding of Bacchus and Ariadne in fresco in the Villa Giulia, Rome, Some paintings show Bacchus with revellers in contemporary modern dress, as in the Frangipane illustrated.Mark Wallinger argued that The Triumph of Bacchus prefigured Las Meninas and stated, "Velázquez presents us with a complexity of focal points. [...] The look [the two liggers on the left of Bacchus] direct at the viewer slices clean through 350 years in the most disconcerting way. [...] However one might describe them, we are made complicit in the meaning of the work."
Silenus satyr Mark Wallinger Ribera's Spanish Taddeo Zuccari beggar left Titian Villa Giulia Las Meninas Spanish royal collection Bacchus Bacchus and Ariadne