Facebook, which operates the largest online social network, reported Wednesday that its profits doubled in the third quarter, to $621 million, after excluding expenses related to stock options. The company’s revenue rose 60 percent, to $2.02 billion, compared with last year’s third quarter. Most of that, about $1.8 billion, came from advertising. Perhaps most important for Facebook’s future as an advertising-driven company in a world of iPads and Android smartphones, the company noted that mobile ads accounted for 49 percent of its advertising revenue, up from 41 percent in the second quarter. Facebook said prices for mobile ads remained high, and users were clicking on them in their news feeds more frequently.