How would you design a LinkedIn-like system that instantly shows whether a profile is a first-, second-, or third-level connection for a user at very large scale?

"A recruiter reached out to me early in the year, gave me good prep material, and Google skipped the usual screen based on my resume, which already felt different. The loop itself was split into two remote rounds and then three in-person rounds, so it was a real five-interview final rather than the usual single onsite block I expected. What stood out most was that they were not only testing EM skills. They were looking for googliness, broader general competencies, and what I kept thinking of as cognitive skill, especially in how deep they pushed on system design. I felt both system design interviewers wanted me to go past the usual black-box architecture answers and explain how I would actually implement the core logic. I ended up failing the must-pass coding round, which was a very unusual code review and debugging exercise, and the recruiter told me I would need to wait 12 months before interviewing again."