If you read through the cards on Rasputin and the Darkness it's implied that Rasputin detected the Darkness, ran simulations, realized it couldn't be defeated, turned off all the defenses and went into hibernation. Either to save itself or because it thought that was the only way anyone survived at all. It's also theorized that Rasputin fired on the Traveler to damage it enough that it couldn't run. Forcing it to stay and defend the Earth. This theory also says that it abandoned the Cabal, Fallen, Hive and Vex to the Darkness before it came to Earth and intended to do the same to the Humans.
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It only left the Fallen. The Vex were created by the Traveler.
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Also, the Hive are the creation of Oryx and the other 4 hive gods. Oryx and the other Hive gods were created by the darkness, therefore making the hive the embodiment of the darkness.