The study analyzed friendship ties among 84 subjects (ages 23 to 38) in a business management class by asking them to rank one another on a five-point continuum of closeness from “I don’t know this person” to “One of my best friends.” The feelings were mutual 53 percent of the time while the expectation of reciprocity was pegged at 94 percent. This is consistent with data from several other friendship studies conducted over the past decade, encompassing more than 92,000 subjects, in which the reciprocity rates ranged from 34 percent to 53 percent.

Mr. Pentland said it could be that “the possibility of nonreciprocal friendship challenges one’s self-image.” But the problem may have more to do with confusion over what friendship is. Ask people to define friendship — even researchers like Mr. Pentland who study it — and you’ll get an uncomfortable silence followed by “er” or “um.”

“Friendship is difficult to describe,” said Alexander Nehamas, a professor of philosophy at Princeton, who in his latest book, “On Friendship,” spends almost 300 pages trying to do just that. “It’s easier to say what friendship is not and, foremost, it is not instrumental.”