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I'm wondering (as a non-psychologist/cogsci programmer) what would be a good mathematical function to model the decay human relationships, or in other words, how people forget about people they don't interact anymore.

The background: I'm building an agent-based model where agents (representing people) interact, converging on a semi-stable network. If a link between two agents is not used, it should weaken, and eventually fade. Currently the growth and decay of a link between two agents is linear between 0..1 (more interactions -> stronger link, up to 1; less interactions -> link gradually weakens, evetually drops to 0). While my current decay function is linear, for the part that controls who interacts with who, I'm using an S-shaped function - there's a higher chance to interact with agents one knows, and who their "friends" know, but the sum of these values is then "bent" on an S-curve.

The concept of time in the model (counted in iterations of interactions) is relative, so it's not about minutes, days or years. I'm rather interested in a formal model that would capture the dynamics of forgetting people they don't see around anymore. I have a gut feeling it's not linear - one won't start to forget their mother, and the relationship won't weaken, if they don't see her for a little while. But if you just ask for directions on the street, chances are you'll forget the person rather quickly (unless he/she was hot or something; but the agents in my model don't have properties pertaining to such attributes).

A few alternatives could thus be: a sigmoid/S-curve (but the "steepness/curviness" would be a question still; green line below), logarithmic (also looks nice, but steepness of the curve would be a question; yellow), exponential (doesn't seem likely though; orange), perfect memory until arbitrary cutoff (probably not; red), compared to simply linear (blue). enter image description here

If an curve could be decided on for the decay, then I suppose I would also use it for link growth (e.g., on an S-curve a slow growth in the beginning up to a point (strangers) and towards higher values (friends past the get-to-know phase)). It might be useful to think of the lines on the plot as trajectories of a relationship, with the stength on the y-axis and relative time (either way) on x.

The question then: which one of these functions (and which parameters; or some other function) would be best to model the dynamics of how quickly people start to forget about/lose their connections with other people when they don't interact - or in other words, how a relationship weakens due to sparse contact? I assume there's literature on the subject, but I would have no idea where to start (googling generic keywords like relationships and decay function does not help), so a good answer could cite relevant sources or otherwise show the superiority of one function over the others.

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Here's a nice paper that explores the question in depth: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4626528/. The authors measure the emotional closeness as frequency of contact and number of different activities done together for a group of 25 people over the period of 18 months. As far as I can tell from the charts, the decay curve is closer to an exponent than to anything else for friends. However, for the kin the emotional closeness increases over time - though, again, in an exponential way.

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  • $\begingroup$ Link-only answers are frowned upon on this site. Would you mind elaborating on how the paper explores this question and how they came to the conclusion of exponential decay of social network links? $\endgroup$ – Seanny123 Jan 9 '17 at 1:02
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    $\begingroup$ @Seanny123 Thank you for the suggestion. I summarized the results of the paper. $\endgroup$ – DYZ Jan 9 '17 at 1:07
  • $\begingroup$ Not yet quite sure how to implement that, but it's definitely better than nothing; I had not come across this myself. $\endgroup$ – user3554004 Jan 12 '17 at 19:10
  • $\begingroup$ I would go for a piece-wise linear approximation of the exponential function: linear decay up to a certain threshold, followed by severing the tie. $\endgroup$ – DYZ Jan 12 '17 at 20:59

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