Akatsuki's various relationships in the Log Horizon series.
Akatsuki and Shiroe were acquainted with each other before the Catastrophe, as they had worked together on quests before. She understood his leadership and strategic skills as well as his experience and time in the game, leading her to seek him out for a potion that could change her appearance (since her avatar was a tall man and it was difficult for her to navigate that body). Since then, she has been his "ninja", devoted to helping her "lord" Shiroe in whatever way she can.
However, she soon begins seeing him in a romantic light and loves him for his intelligent personality, kindness, acceptance of her as a person (whereas most people saw her as a little child), maturity, and leadership. It is not fully elaborated on how Shiroe feels for Akatsuki and whether he sees her as a romantic interest (he is oblivious to any romantic interest that some of the female characters feel for him), but it is evident he sees her as someone important to him and someone he can rely on and wants to be with.
Although Akatsuki does not like Naotsugu's perverted side (which usually consist of crude comments about women's panties and about men being open or closed perverts etc.), she gets along with him very quickly. The two work well as a team - in the battle against the Genius at Shibuya, they combine their skills into a single attack - and both deeply respect Shiroe's leadership.
It should be noted that in the anime, Akatsuki kicks him far more than she does in the light novel. Although she is alluded to have kicked him quite a bit for his perversion, she is only shown doing it once. A majority of her kicks in the first season of the anime serve to censor Naotsugu's lines rather than simply cutting them out. By the second season, it serves largely as a gag.
Unlike with Naotsugu, Akatsuki respects and looks up to Nyanta. She seems him as a worthy ally with terrific skills. She also deeply appreciates his cooking and takes his advice to heart.
Akatsuki initially got along well with Minori up until they realized that they were both rivals for Shiroe's affection. Afterwards, Akatsuki easily became jealous of Minori, but berated herself for having those thoughts and reveals that deep down, she respects Minori and doesn't see Minori as a bad person. They were willing to work together to bake a cake for Shiroe (although Nyanta's Fortune Recipes and the anime give different reasons why); in the NFR manga, the two of them celebrate when Shiroe says the cake tasted good.
Tetora can easily see that she and Minori both have feelings for Shiroe, and oftentimes tries to stir up trouble for her own amusement; for example, she suggests that he was in a relationship with Kanami (which he wasn't), and appears inside of his clothes at one point, rousing the anger of the two girls. However, other than that, Akatsuki seems to get along with her.