Graph databases, specifically Neo4j, are designed to manage complex interdependencies in IT network management by storing interconnected data that is not purely hierarchic. Unlike relational databases, graph databases make it easier to understand network data and can capture the relationships between devices more effectively. This allows for better diagnosis of failures and improved scalability, especially when dealing with large numbers of machines. The developer's experience with Neo4j has been positive, citing its excellent community support and ease of use, which led them to leverage Neo4j in developing their own network management solution.