Mathy Friendships

On a scale of one to ten, how much do you like each of your friends? We don’t often have reason to carefully evaluate the strengths of our relationships. However, as a fan of quantification, I’ve created a visualization of the way my Friend_Feetfriendships change over time. It’s a joy when an acquaintance becomes a BFF. There’s also beauty in allowing a friendship to fade or blow up. The strengths of our relationships are not based fully on logic. Bonds are forged and broken through chemistry, physicality, proximity, and circumstance. By tracking the strengths of our relationships, we can gain insight into those enigmatic social forces. Also, it makes for some cool graphs.

How to quantify your friendships!

Over the 6 years I lived in Minneapolis for graduate school, I met a lot of people. Some of them became great friends! Some didn’t. I tracked the story of those friendships by rating the strength of each bond over time. To begin this project, I thought a lot about how to quantify a friendship. I didn’t want to rate each person on how much time we spent together, or how heartily we laughed. Those were too tied to circumstance. I didn’t want to rate each person on how much I LIKED them, or how much they seemed to like me, because that didn’t really address how close we were.

Friend Ships. Sketched by our guest Author!

Different types of Friend Ships. Sketched by our guest Author!

Instead, I use a scale of “bondedness,” a measure which (admittedly abstractly) describes the amount of closeness, comfort, trust, and love in a relationship. This scale goes from -1 to 10. 10 is the most bonded I can be with another person. We have each other’s backs no matter what, we can spend time together without discomfort or awkwardness, we can be open about our thoughts and feelings, and we are invested in each other’s well-being. The amount of bonding decreases as the value approaches 0. A 5 is someone who I can spend some enjoyable time with but I wouldn’t be too upset if they missed my birthday party. A 1 is someone I am very slightly bonded to; we occasionally chat but not about anything of interest. A 0 is someone with whom I feel no bond. Either we have met but not connected, or our relationship has dwindled to the point where there is no gravity between us. A -1 refers to bonds that are actively negative – someone who I get upset thinking about. I considered extending the scale down to -10 to capture the full range of bitterness, but this seemed unnecessary. I tend to remove myself from negative relationships, and they eventually become zeros on the bondedness scale. For ease of discussion, I’ll call the unit of bondedness a “bondie.”

By rating each of my Minneapolis-derived relationships four times a year, I was able to visualize the strength of those relationships over time. Those graphs are below, but before you look too closely, be warned: this data is fake. In examining my real friendship data and considering its practical meaning, I stumbled upon a paradox: we can estimate the strength of bonding between any two people simply by observing their interactions, but despite the obviousness of these bonds, it’s not socially acceptable to publicly rank the members of your social group (unless you’re a contestant on Survivor)! In fact, there are probably people out there who find this exercise of quantifying friendships to be horribly callous and uncouth. I disagree; this data is for science, not evil! To keep it that way, I’ve chosen to protect the identities of all involved and created a fake dataset that replicates the spirit and trends of the original.

This is the graph of friendship strength over time, where the x axis is true months.BondiesByTime

This is the same data, where the beginning of each relationship is aligned to zero on the x axis.


The bonds are born

In my dataset, I identified relationships borne of three different scenarios: necessity, choice, and romance. Relationships of necessity are with coworkers, classmates, and roommates – people I was forced to be in close proximity to. I’ll call these relationships “requisites.” Relationships of choice are with people who would be easy to avoid. We chose whether or not to see each other, and therefore growing these relationships typically took conscious effort. I’ll call these relationships “electives.” Romantic relationships had a romantic component that was either one-sided (eep) or mutual.

The requisites

First, let’s take a closer look at the requisites. We’ve all heard that shared experiences bond people together. What I observed is that proximity was polarizing!


Some very strong and long-lasting bonds formed in the requisite group, while an equal number eventually depreciated (or crashed and burned). Despite the fact that, on paper, I had a lot in common with the requisites, many of those relationships didn’t translate to long-term friendships after the period of forced proximity ended. This could reflect my personal preference for maintaining a relatively small group of very close friends. Further, most of the requisite relationships took place within larger social groups where complex interpersonal forces were at play. I suspect this particular bondedness pattern reflects the social and temporal pressures of graduate school and is not generalizable to new social situations.

The electives

The electives looked a lot different. In this group, less than 20% of relationships ever hit bottom – the rest were pretty strong and continued growing over time.


After 36 months, the average requisite bond was 4.3 and the average elective bond was 6.8! That said, data selection skewed these results since I only included people who I bonded with to some degree. People in the requisite group whose bondedness rose and then precipitously fell might never have been included in the elective data set since, without an external force, we may have never bonded at all.

Bonding speed

I was interested in seeing if the speed of bond formation differed between the requisite and elective groups. However, each group’s average rate of bond growth during the first year wasn’t very informative (an average of ~0.47 bondies/month for both groups combined, if you were curious), since the distributions for both groups were so different and the sample size is so low. What WAS interesting was comparing those slopes to the bondedness rating after 3 years.


There seemed to be a positive relationship between the ultimate strength of the bond and the speed with which that bond develops. Hey, that makes sense!

The romances!

The third type of relationship in my dataset was of a more… volatile nature. My romance-tinged relationships were characterized by tumultuous bondedness ratings.


More often than not, these relationships ended up at -1 bondies for some amount of time. I’ll note that these bondedness ratings are from MY perspective alone. I would be both interested and horrified to see the bondedness ratings from the opposite perspective! To what extent would they match up, and what would the mismatches reveal about our relationship’s twists and turns?

Fading away

During the time I lived in Minneapolis, many of my friends moved away to pursue exciting and prestigious opportunities elsewhere. In the absence of regular contact, these bonds sadly tended to fade.


It’s been my experience that relationships that start out with high bondedness will never reach zero, no matter how long we’ve been apart, and that they can easily bounce back up! But still, visualizing the wane of the bondies is upsetting (I’d like to think my friendships stay strong in perpetuity, not matter how much circumstance may try to tear us apart). In my dataset, separation caused bondedness to fall by 0.12 bondies per month.

There was a trend that relationships with higher bondedness prior to separation faded more slowly after the move, which makes sense. The people I am more bonded to are the people who I will probably continue to interact with on a regular basis.


What did I learn?

These data tell a story about what it was like for me to move to a new city where I knew no one, go back to school, and ultimately build a community. In retrospect, I highly value the pressure cooker of graduate school because interacting with a lot of really smart and interesting people allowed me to easily separate those people who were excellent friend matches for me (the lines rising to the top of the requisite graph) frFriend_Toesom those who, for whatever reason, just weren’t. Also, I learned that people who I bond with quickly tend to be long-lasting buddies. This data COULD be used for forecasting, allowing me to wisely invest my social energy in relationships with maximum “payoff.” For example, I could set a bonding speed cutoff, where if I haven’t reached, say, 3 bondies in 6 months (a slope of 0.5 bondies/month), then I should stop putting effort into the new relationship because most likely we’ll never reach the level of great friends. Don’t get me wrong: the idea of rejecting a relationship just because an equation recommended it makes me cringe! However, I think we all make similar calculations intrinsically as we navigate our complex social world.

It was particularly interesting to quantify the fade of bonding that occurs when people move away. I’m struck by the fact that I lose 0.12 bondies per month after the separation. That sounds pretty slow, but it adds up: after 2 years, the friendship has lost 3 bondies! This data has galvanized me to put more effort into my long-distance relationships to head off that drift.

Getting past the squeamishness I felt rating my friendships was hard. It’s fascinating that the social equations we constantly compute in our own brains must stay hidden there for fear of offending others. If you can put this squeamishness aside, should you track your own friendships? Sure, you might learn something about yourself! Everyone has different goals for their relationships, different categories of friends, different numbers of friends, different variables to consider, and different friendship-initiating factors in their lives. It’s very likely your graphs would look nothing like mine. And that’s what makes people, and data, beautiful!


Author’s bio: Tess Kornfield has a PhD in neuroscience and is currently a postdoc at UC Berkeley studying Alzheimer’s disease, neurodegeneration, and aging using cool imaging techniques like PET and MRI. She loves her friends and she loves quantifying things. How much? About a 10/10 and 9/10, respectively. Find her on Twitter @scyspy .


About Samantha from SocialMath

Applied Mathematician and writer of
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