He recalled the many hours he spent in engrossing conversation with his friend Eudora Welty, who was known not only for her Pulitzer Prize-winning fiction but also for her capacity for friendship. Together they edited “The Norton Book of Friendship,” an anthology of works on the topic. “The notion of doing nothing but spending time in each other’s company has, in a way, become a lost art,” replaced by volleys of texts and tweets, Mr. Sharp said. “People are so eager to maximize efficiency of relationships that they have lost touch with what it is to be a friend.”

By his definition, friends are people you take the time to understand and allow to understand you.

Because time is limited, so, too, is the number of friends you can have, according to the work of the British evolutionary psychologist Robin I.M. Dunbar. He describes layers of friendship, where the topmost layer consists of only one or two people, say a spouse and best friend with whom you are most intimate and interact daily. The next layer can accommodate at most four people for whom you have great affinity, affection and concern and who require weekly attention to maintain. Out from there, the tiers contain more casual friends with whom you invest less time and tend to have a less profound and more tenuous connection. Without consistent contact, they easily fall into the realm of acquaintance. You may be friendly with them but they aren’t friends.

“There is a limited amount of time and emotional capital we can distribute, so we only have five slots for the most intense type of relationship,” Mr. Dunbar said. “People may say they have more than five but you can be pretty sure they are not high-quality friendships.”

Such boasting implies they have soul mates to spare in a culture where we are taught that leaning on someone is a sign of weakness and power is not letting others affect you. But friendship requires the vulnerability of caring as well as revealing things about yourself that don’t match the polished image in your Facebook profile or Instagram feed, said Mr. Nehamas at Princeton. Trusting that your bond will continue, and might even be strengthened, despite your shortcomings and inevitable misfortunes, he said, is a risk many aren’t willing to take.

According to medical experts, playing it safe by engaging in shallow, unfulfilling or nonreciprocal relationships has physical repercussions. Not only do the resulting feelings of loneliness and isolation increase the risk of death as much as smoking, alcoholism and obesity; you may also lose tone, or function, in the so-called smart vagus nerve, which brain researchers think allows us to be in intimate, supportive and reciprocal relationships in the first place.

“It’s huge to have good vagal tone, because it modulates our instinctive fight, flight or freeze response,” said Amy Banks, a psychiatrist at the Wellesley Centers for Women who specializes in the growing field of interpersonal neurobiology and is the author of “Wired to Connect: The Surprising Link Between Brain Science and Strong, Healthy Relationships.”