Linked data is a W3C backed approach to sharing and publishing structured data on the web. Linked data typically uses web resolvable URIs to represent things in some knowledge domain. The URIs are persistent and durable and let us describe the entity with a structured data vocabulary or model such as schema.org. The linked data URIs can be connected to other entities within your own or someone else’s domain to indicate we are talking about the same, similar or related things. The structured data can be machine interpreted for use in applications.
Linked data does not necessarily need to be open data. While linked data has been embraced by the open data community and lends itself perfectly for publishing open data there are many good reasons for using it in the enterprise behind the public firewall. The structured data patterns are ideal for sharing data across business units and data silos internally in your organisation, and can be used as a core back-bone to your enterprise knowledge graph.
Of course you can also publish a subset of your enterprise linked data as an open data initiative if you think it will benefit the wider commuity.
While there are significant synergies and alignment between linked data and graph databases, you do not need a graph database to publish, consume or benefit from it. You can expose linked data from any data source via a service or API, as long as the data you expose uses URIs/IRIs to identify the entities being represented, and follows some simple linked data patterns. Similarly you can consume linked data and store it pretty much in any data store technology, extracting some value along the way.