How to store datasets of lexical connections? I'm investigating the possibility of storing the semantic-lexical connections (such as the relationships to the other words such as phrases and other dependencies, its strength, part of speech, language, etc.) in order to provide analysis of the input text.

I assume this has been already done. If so, to avoid reinventing the wheel, is there any efficient method to store and manage such data in some common format which has been already researched and tested?
I would think you could use a graph database, perhaps Neo4J or Titan or something of that nature.  Or, if you want a simple file format, you could use one of the many formats that exist for representing graphs.  You can find a list and overview of some of them here.

Another option would be to store them in RDF using a triplestore like Jena.