I am not referring to the absence of universally accepted personality traits; what I mean is literal, that Haruhi and I are not like other organic lifeforms such as yourself.
To say that Yuki has trouble showing her emotions would be an understatement. She doesn't appear to be capable of communicating using facial expressions most of the time (barring subtle eyebrow movements), and most of her lines are devoid of emotional cues (eg. "No," to dissuade a classmate from covering a purchase, followed by "That," to indicate her selection). Even when talking about her feelings directly, she tends to underemphasize them (eg. claiming that she's "a little" interested in books despite practically being glued to them). The most obvious explanation for this is that she's an alien who just doesn't understand emotion... but that doesn't seem to be true at all.
A common mistake people make when looking at Yuki, or characters like her, or real life people like her, is to assume that since they don't show emotions, they don't have emotions. In Yuki's case, however, this is far from true. The most obvious counterpoint is, of course, her love of books. While her obsession with books may stem from her objective of collecting data, her preference for fiction indicates that this is likely not the only reason.
This comes to a head somewhat early on, when Haruhi orders Yuki and the protagonist, Kyon, to investigate the library for paranormal activity.
Yuki immediately hits the shelves, leaving Kyon to his own devices.
falling asleep verifying that there was, indeed, nothing strange about the library, he decides to leave... and quickly runs into a problem.
No matter what, he can't convince Yuki to put her books back. Resigned, he helps her obtain a library card so she can check the books out.
Though it seems like such a small thing, the fact that he helped her gain access to a wealth of books, to do what she loved most, caused this moment to become a treasured memory for her.
Even when she purged her own memories (more on this later), she couldn't bring herself to erase the moment Kyon helped her get a library card.
This, in itself, is quite interesting. Beyond highlighting Yuki's love for books, it shows that she can be a bit sentimental — rather emotional for an alien data organism.
If you're a particularly attentive reader, you might've noticed something odd about that last paragraph. Given that she's a highly competent individual, it's a bit strange that Kyon had to help her in the first place. At first blush, one might just chalk this up to sexism. However, an analysis of her speech patterns provides a different explanation. While some of her terseness seems to indicate a flat emotional affect, other examples (like saying "Book." in place of "Have you read the book I lent you?") seem rather bizarre in context. Ignoring her tone, it almost seems as if she's too nervous to speak, just barely managing to force the words out. This seems like a bit of a stretch, given her stoic demeanor and the comedic tone of the series, but it starts to make a lot more sense later on — again, more on this later. For the time being, the most direct evidence we have is this quote:
It is difficult to convey in words. Discrepancies may arise during the transmission of data. Regardless, listen.
I cannot count the number of times I've said exactly this (modulo phrasing, of course) when talking to someone. The fact that she even thought to say this betrays an atypical amount of self-conciousness (an emotion!). All of this evidence seems to indicate that Yuki suffers from some form of social anxiety. This, on top of her flat emotional affect, would make it very difficult to connect with her peers, even those in the SOS Brigade.
Obviously, this is a recipe for disaster. As it turns out, it would only take 595 years for things to go sour.
In the infamous Endless Eight arc, Haruhi tries her best to pack the last two weeks of summer vacation full of fun! Unfortunately, something's missing, but she doesn't quite know what... so she subconsciously uses her power to reset time and try again.
Fortunately, everyone's memories are reset (more or less) each time around, so for them, it just felt like two weeks had passed!
... except for one person. One person whose memory didn't quite work like a human's. One who has already shown that she can share memories forward and backward across time planes. One who would remember all 15,532 repetitions, all 595 years of floundering in this neverending endless summer vacation.
That person is, of course, none other than Yuki Nagato.
Of course, it's fine for her. She's a humanoid interface; she's practically an alien robot! She doesn't have feelings! She can just tough it out until Kyon finally remembers to do his homework. At one point, Kyon even asks if she's alright, and she says that she is!
But for once, her expression tells a very different story. It looks like her typical flat expression at first glance, but her eyelids are visibly drooping. Her eyebrows are furrowed, almost tilted in a way that indicates sadness. Her mouth's somewhere between a small smile and a bitter frown, to the point where it's hard to really pin down what it is. It's Yuki's most clear expression yet — and one of the worst she's had the misfortune of delivering. After 595 years of servicing Haruhi's summer dreams, she's done. She's tired. She's sick of Haruhi's shit.
And it only becomes clearer from there.
As long as I continue to exist, internal errors will continue to accumulate.
The Christmas season has finally arrived, and Haruhi's eager to get everyone into the spirit! She passes out party hats, tasks everyone with decorating the club room, and starts planning for a Christmas party. Everyone seems to be in high spirits. Even Kyon's grumpy exterior seems to be cracking!
Even Yuki's getting into the spirit.
Even Yuki put on a party hat.
It's a way for her to express her enthusiasm.
It's the only way for her to express her enthusiasm.
Her clubmates are noisly chatting away, smiling, laughing, having a good time.
She couldn't smile. It wasn't natural for her.
She couldn't laugh. She wouldn't know when to do so, and she'd probably sound stupid.
She couldn't chat. Even the thought of doing so was terrifying.
My function is to observe only.
That's the refrain that carried her through 595 years of unending torment. 595 years of living Hell. 595 years of inaction.
She took up a snowflake decoration.
Snow. Heavily implied to be the true origin of her name.
A beautiful thing. A tiny thing. An exciting thing.
A transient thing. A short-lived thing.
Something that could melt in an instant.
Its beauty lasted for a single moment before fading into a puddle of frigid water, never to be seen again.
It won't last for long.
You should enjoy it while you can.
Faced with the threat of pain,
faced with the threat of torment,
faced with the threat of her emotions being locked away,
faced with the threat of being perceived as an emotionless robot for all eternity,
faced with the threat of melting,
she stole Haruhi's power
and rebuilt the world...
[runs up to Kyon and gingerly grabs his sleeve, stopping him from leaving]
Yuki Nagato is a character from The Disappearance of Haruhi Suzumiya. On the surface, she seems like an unassuming, shy bookworm, but in reality, she's a gentle, caring bookworm... when she opens up to you, that is.
She's only human, after all.
After altering the world to remove all traces of supernatural elements, Yuki finished the job by wiping her own memories, letting her pain, her torment, her torture at the hands of the SOS Brigade all fade into gentle oblivion. As far as she was concerned, she was just an ordinary human, complete with the ability to express herself!
Of course, just in case she'd royally screwed up, she'd created a failsafe, with one critical component worth mentioning: Kyon was the only one who kept his memories. This went about as well as you'd expect, and before long, Kyon found himself scrambling to the Literary Club room for some sort of mental relief, finding it entirely unoccupied save for a much more expressive Yuki. After a rather strange false start prompted by Kyon's panic, the two start to talk. Two things quickly become apparent:
Both of these, especially the latter, combined with the fact that she doesn't seem to have made any friends before December (when Kyon showed up), indicate that the new Yuki suffers from social anxiety. As a matter of fact, I don't have to speculate about this — she straight up confirms it herself when recounting how Kyon helped her get a library card:
There were lots of people that day, and the librarians looked like they were really busy. I didn't want to bother them. I-I have a hard time in situations where I have to talk to people and... ask for something. So I just stood there, feeling sorry for myself, and then someone came up to me.
But really, does this matter for the greater question of Yuki's character? After all, at the end of the day, Kyon uses Yuki's failsafe to restore the world, then goes back and forth in time to stop her from resetting it in the first place. The new Yuki is no more, the original is here to stay. Besides, Yuki might've just given herself a stutter for the sheer cuteness of it or something. So what if the new Yuki was socially anxious? So what if the new Yuki expressed emotion?
However, recall that even the original Yuki had a hard time talking to people — neither Yuki was able to obtain a library card on their own. It's entirely possible that both ran into the same issue — social anxiety — but only the new Yuki was capable of explaining it. In addition, consider my earlier point about the original Yuki's strangely clipped speech, how sometimes, it almost felt like she was trying to force something, anything to come out. Just looking at the series, this seems pretty farfetched, but when you start to consider the new Yuki's behaviour, everything starts falling into place. The original's blunt, terse speech is the new Yuki's speech, passed through a sieve that removes almost all emotional expression, all filler words, all obvious signs of imperfection, resulting in constructions like "Can I have that one?" → "Um.... c-c-can... th-that o..." → "That." to ask a vendor for a mask. While different on the surface level, deep down, both Yuki seem to suffer from the same issues with verbal communication.
I am responsible for everything that happened.
The original Yuki's most expressive moment is almost definitely at the ending of The Disappearance of Haruhi Suzumiya (I think this scene was elsewhere in the light novels?). After Kyon awakens in the hospital, apparently having fallen down the stairs, he decides to hang out on the rooftop. A few minutes later, Yuki approaches, immediately admitting responisiblity for everything she put Kyon through. Though her tone is as flat as ever, and her diction makes her seem like the objective alien robot she was designed to be, there's no denying that there's a twinge of guilt in her words. Upon hearing Kyon apologize for the pain she's been through, the highlights in her eyes shift, and she lowers her head, apparently disgraced.
And then, he asks...
'Yuki'... it means 'snow', doesn't it?
She lifts her head once more, only to find snow falling gently onto the rooftop. She becomes enraptured by it, staring up at the snow in awe. Though her face appears blank, there's a subtle hint of wonderment in her expression as the snow lightly covers her, Kyon, the roof, and the city around her.
There's no longer any room for doubt: though she doesn't show it, Yuki is full of emotion — sadness, pain, discomfort, a desire to change, nervousness, self-consciousness, regret, guilt, wonder, hope, and even happiness. The very act of her altering the world to be able to express herself was itself an expression of her emotions. Otherwise, she would've had no reason to go through with it.
This is part of the beauty of Yuki. Though she's an alien, in many ways, she's strangely human — not just on the inside, but on the outside as well. None of her emotional expression issues or anxiety issues are exclusive to aliens, nor are they typical of humanoid interfaces. Though they initially seem to be a convenient way to inform the audience that she is, indeed, inhuman, they end up serving to make her one of the most human characters in the Haruhi Suzumiya series.