The barista cheerfully remembers your regular order and has it started before you’ve said hello. Warmed inwardly, cup in hand, you step onto the sidewalk, and a passerby—the older woman who walks her sweet terrier every morning—greets you with a familiar smile. These are not friends, exactly, but people you are pleased to run into because they confer a sense of belonging. They are a part of your community, and you are a part of theirs. New York Times columnist Jane E. Brody refers to them as “consequential strangers” with whom we form convivial relationships through cordial gestures and brief, affable conversations. 

Repeated, casual encounters of these sorts etch a network of relations formed of “weak ties”—a phrase sociologist Mark Granovetter introduced in 1973 to co...