After episode S6E5 of Game of Thrones I wondered how it was possible that the White Walker saw Bran?
In every other memory that Bran has tried to get the people's attention, they do not see him, so how is this possible?
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Spoiler: White Walkers (WW) are created by magic, their creators are the Children of the Forest. They are old Inhabitants of Westeros and they created the WW to protect themselves from mankind. Now, ages later, the WW and their leader Night's King, turned against their creators (a classic theme: creatures turn versus their creators). The WW though, as of same magic origin are able to see Bran as he's being in interaction with this magic: using Three Eyed Crows abilities, who's a descendant of the Children of the Forest.
Speculation, but the White Walkers were first created by being impaled while tied to a Weirwood Tree. Bran was Greenseering using a Weirwood Tree. As such, it may be that White Walkers are intrinsically linked to the magic of weirwoods and can feel the presence of their magic.
We can see that the Three-Eyed Raven/Crow is able to share visions with Bran and that they are able to see each other and interact within these visions. From this, we know that Greenseers can share visions.
From what we have seen I would say that the Night King is almost definitely a Greenseer. This would explain his ability to raise the dead as some form of Warging. I find it doubtful that the Children of The Forest would simply choose anyone to become a White Walker, it is possible that the victim being a Greenseer is a prerequisite for the spell to work. We know that Benjen had a Dragonglass dagger stabbed into his heart, just like the Night King but he is clearly not a White Walker. There may be many reasons for this but I would say that it is because Benjen is not a Greenseer.
So if we accept the fact that the Night King is a Greenseer then it is obvious that he could share a vision with Bran just like the Three Eyes Raven/Crow does.
This is mere speculation, but it may be that the Night King/White walker-you-mention has powers/abilities of a green seer. Like the Three-eyed-Raven who is also a grren-seer, has been able to see Bran in his green-sights and also touch him. It is possible that the Night King is someone like the Three-eyed-Raven or better.
I think it's clear that in this instance Bran did not travel into a past memory which he can not effect - he travelled to a present moment and as White Walkers were created by the Children of the Forest - perhaps their powers and Brans are all intertwined and the White Walkers can access that.