> So in theory once you hit a certain number of interactions, the people prone to snapping would have snapped.
That statement is presuming there's some fixed threshold of interactions or fixed set of people to interact with that most people achieve, before someone "snaps", at which point they are immediately incarcerated. I don't think there's anything resembling evidence, or even a theoretical model for criminal behaviour, suggesting that is the case --certainly not for crimes of passion, which are often not committed by people of all ages who hadn't previously murdered at any point in their lifetime.
> In theory, if murder is something that generally involves strong passions, wouldn't the closer relationships result in the violence when someone snaps from a betrayal or habitual disrespect?
Strong passions might erupt from interactions between a third person that one might have a mere acquaintance with, having an interaction with someone one had a closer relationship, no?
Let's assume that's true though. Let's say 80% of murders are crimes of passion, and 90% of crimes of passion are entirely derived from interactions within people one is so close to that regardless of population density you'd have more than enough interactions to hit some threshold. That still leaves 28% of murders that are far more likely to occur in more a densely populated environment... and if the rest of the murders are uniformly distributed, that would mean the divergences due to population density for those remaining murders would be far more significantly concentrated in denser populations.
Now, what the data actually shows is that there is one specific murder victim that is disproportionately likely to be murdered by someone they have a close relationship with: wives (sons are also disproportionately likely, but to a significantly lesser degree). In most cases the relationship between victim and offender is either unknown or a more casual "acquaintance": https://ucr.fbi.gov/crime-in-the-u.s/2011/crime-in-the-u.s.-...
> I think that's only true if we are weighting those interactions linearly and not including mitigating factors that could be more prevalent in one population or the other.
No. Over a decade long period, in a densely populated urban environment, if you look at the acquaintance-of-acquaintance or even acquaintance-of-friend populations, you could easily have a larger population than a small town or even a state. Even if you limit it to just "passionate", romantic relationships, the adage of, "I've dated eligible person in this town" is just something that is far more likely to occur in less densely populated areas.