Graph databases in production: when does Neo4j actually beat SQL?

we have a highly social application, and queries like 'friends of friends' or finding common connections are incredibly slow with our current postgresql setup. recursive CTEs often take 10+ seconds for even moderately deep traversals. we've been hearing a lot about graph databases like neo4j promising much faster performance for these types of connected data problems. i'm trying to figure out the tipping point. when does the operational complexity of introducing a completely new database technology like neo4j, with its own query language and ecosystem, actually justify the performance benefits over trying to optimize sql-based approaches? has anyone truly seen neo4j significantly outperform sql for graph traversals in a way that warranted the architectural shift in production?