The best open source alternative to neo4j is Wikibase, which is both free and Open Source. If that doesn't suit you, our users have ranked more than 10 alternatives to neo4j and 14 is open source so hopefully you can find a suitable replacement. Other interesting open source alternatives to neo4j are ArangoDB (Free, Open Source), GraphStack.io (Free, Open Source), RedisGraph (Free, Open Source) and Bitsy graph database (Free, Open Source).
Wikibase is the software that runs Wikidata, but is also usable for other projects beyond that. Wikibase allows the collaborative editing of structured data on top of MediaWiki. It is optimized for graph-like, schema-poor data, not for tabular and strongly pre-structured data.
RedisGraph is a property graph database module for Redis. It is the first graph database to use linear algebra on sparse adjacency matrices to implement graph operations, which lets it achieve high performance. It understands OpenCypher queries.