This study delves into the study of complex networks within a hyperbolic latent space model, presenting theoretical analysis of popular link prediction indices on hyperbolic random graphs. We investigate how different degrees of nodes influence link prediction heuristics. By modifying indices like the common neighbor and shortest path index, the study demonstrates theoretical and empirical improvements in both simulated and real-world networks. Additionally, we also explore embedding methods to recover hyperbolic geometry, introducing a modified hyperbolic ordinal embedding method.