High　dimensional　embeddings　of　graph　data　into　hyperbolic　space　have　recently　been　shown　to　have　great　value　in　encoding　hierarchical　structures,　especially　in　the　area　of　natural　language　processing,　named　entity　recognition,　and　machine　generation　of　ontologies.　Given　the　striking　success　of　these　approaches,　we　extend　the　famous　hyperbolic　geometric　random　graph　models　of　Krioukov　et　al.　to　arbitrary　dimension,　providing　a　detailed　analysis　of　the　degree　distribution　behavior　of　the　model　in　an　expanded　portion　of　the　parameter　space,　considering　several　regimes　which　have　yet　to　be　considered.　Our　analysis　includes　a　study　of　the　asymptotic　correlations　of　degree　in　the　network,　revealing　a　non-trivial　dependence　on　the　dimension　and　power　law　exponent.　These　results　pave　the　way　to　using　hyperbolic　geometric　random　graph　models　in　high　dimensional　contexts,　which　may　provide　a　new　window　into　the　internal　states　of　network　nodes,　manifested　only　by　their　external　interconnectivity.