Somewhat more benignly, some college students use fake names in an effort to protect their Facebook content from the eyes of future employers.

Mr. Sullivan declined to say what portion of the company’s now one billion plus users were fake. The company quantified the problem last June, in responding to an inquiry by the Securities and Exchange Commission. At that time, the company said that of its 855 million active users, 8.7 percent, or 83 million, were duplicates, false or “undesirable,” for instance, because they spread spam.

Mr. Sullivan said that since August, the company had put in place a new automated system to purge fake “likes.” The company said it has 150 to 300 staff members to weed out fraud.

Flags are raised if a user sends out hundreds of friend requests at a time, Mr. Sullivan explained, or likes hundreds of pages simultaneously, or most obvious of all, posts a link to a site that is known to contain a virus. Those suspected of being fakes are warned. Depending on what they do on the site, accounts can be suspended.

In October, Facebook announced new partnerships with antivirus companies. Facebook users can now download free or paid antivirus coverage to guard against malware.

“It’s something we have been pretty effective at all along,” Mr. Sullivan said.

Facebook’s new aggressiveness toward fake “likes” became noticeable in September, when brand pages started seeing their fan numbers dip noticeably. An average brand page, Facebook said at the time, would lose less than 1 percent of its fans.

But the thriving market for fakery makes it hard to keep up with the problem. Gaston Memorial, for instance, first detected a fake page in its name in August; three days later, it vanished. The fake page popped up again on Oct. 4, and this time filled up quickly with the loud denunciations of the Obama administration. Dallas P. Wilborn, the hospital’s public relations manager, said her office tried to leave a voice-mail message for Facebook but was disconnected; an e-mail response from the social network ruled that the fake page did not violate its terms of service. The hospital submitted more evidence, saying that the impostor was using its company logo.