Though he plays a fairly sizable role at the beginning of the novel, Paul Larkin disappears around halfway through. In a way, his absence from the action does more to characterize him than his presence.
First off, Paul is a rather ineffectual businessman who took over Hank's ore mines after the Equalization of Opportunity Bill passed. So Paul was never really a big presence in the Washington Looter Club. He's similar to Philip Rearden in that he wasn't really involved in politics. Both sort of hung on the fringes of things and were used by other looters because of their ties to Hank Rearden.
This lack of a presence in Washington is likely the reason for Paul's absence later on in the novel. He got squeezed out of the increasingly cutthroat Washington circles. Paul seemed like a nice enough guy, but he was very weak:
"I wish we didn't have to hurt anybody."
"That is an anti-social attitude," drawled Taggart. . . .
"Good," said Taggart.
"I can't be expected to buck the trend of the whole world, can I?" Larkin seemed to plead, but the plea was not addressed to anyone. (188.8.131.52-63)
Paul's wishy-washy behavior, and the fact that he let Washington dictate his business for him, eventually helped run Rearden ore and the entire Great Lakes shipping industry into the ground. Paul demonstrates the frighteningly vast consequences that can result from a weak-minded person content to follow.