Friendship Formulæ

One of the corollaries of an anomalous gateway between brain hemispheres is a persistent urge to apply logic to abstract concepts and abstraction to logical concepts. This theorem is a composition in the former: a rudimentary formula for resolving the strength of a friendship or relationship (used loosely).
 

F1: The Waterstone Identity:

Unravelling the symbology above, what the equation seeks to do is reconcile the unknown Ѵ (friendship bond strength) into its constituent parts (α, q, p). As the identity is purely hypothetical, the relative importance or weight of the three above named parts remains to be resolved, hence the use of the Beta 2 and Beta 3 coefficients (which are both necessarily positive).

Commencing with the two most clear factors governing friendship strength;  ‘q’ represents ‘Quality,’ and ‘p’ denotes ‘Proximity.’

Quality

It is generally accepted that the quality of interaction is imperative to the strength of a bond between two people. Picture an androgynous person named Alex, every week with whom you spend a sizeable 15 hours and converse over the phone for a further 5. Those 20 hours consist almost exclusively of going drinking/shopping and talking about sport/dieting. This is deemed ‘low quality’ interaction and, as such, the value for q would be closer to zero within its constrained bounds of zero and one. By contrast, were those 20 hours spent perhaps on exploring a new part of town, cultural pursuits and conversing on topics of life, relationships and the state of the world, a value closer to one would be appropriate and befitting of the obviously ‘higher quality’ interaction.

We see immediately that a higher quality of interaction begets a stronger friendship. Naturally, there is an association between the notions of quality and depth: thus low quality implies shallow depth and the outcome is a superficial friendship that is weak by virtue of it being built upon ramshackle foundations. By contrast, it is a rule of thumb that friendships founded upon profundity; explicitly because they have been built upon solid underlying foundations, are invariably stronger and more resilient to shocks.

Proximity

Proximity is the blanket term that extends over both the amount (time) and profile of interaction. The amount of time you spend interacting with someone and the means via which that interaction occurs is collected by the proximity component in the equation above.

This has been accomplished by first consigning subjective weights to the different modes of interaction – face to face (f), telephone (t) and online (o). Logically, face to face contact (f) is ascribed the highest weighting of 1.0 reflecting the precedence of spending time with someone in person over all other manners of interfacing. Following on, telephone conversation (t) is allocated a lower weight of 0.6 because it is less conducive to building a relational bond (because a phone conversation denies the interaction of body language, facial expression, physical proximity and shared experience which are largely exclusive to spending time together in person.) Lastly, continuous online communication such as Instant Messaging chat is assigned a dubious 0.3 weighting because, although it overcomes some conventional barriers such as shyness and insecurity; it detaches ‘realness’ further by removing the vocal expression through which emotion is conveyed (our voices oft betray how we’re feeling) and increasing the risk of insincerity (it’s easier to lie online).

Once we have estimated how many hours per month we spend interacting with a person face to face, on the phone and online, we multiply those amounts by their respective weights and then add the three products together – giving us proximity (p): the third component of our equation.

Alpha

Needless to say, evaluating a friendship’s strength on proximity and quality single-handedly is nonsensical as concepts of humanity are not suited to fixed factor models. Anything emotive is innately specious when placed in a milieu of logic. This is where Alpha, the equation’s ‘magical’ reconciling factor, comes into play.

Much like the error term in an ordinary regression, Alpha encapsulates all the other effects that lack primal significance  and evade numerical measurement but nevertheless, taken collectively, have a substantial impact on the strength of a friendship. For example, going through a life-changing experience with someone (such as being stranded in the jungle together for a month) will establish a bond of immense  strength, irrespective of the degrees of proximity and quality subsequent to the event. Further the regularity of contact and how long you’ve known the person: frequency* and duration* are two factors not overtly measured in the equation. The alpha term captures influences such as these.

*On a side note, frequency and duration were omitted from the formula because their impact is not uniform. Long friendships and those characterised by daily interaction aren’t necessarily stronger than the short and those of sporadic interaction. The relative importance of frequency and duration to bond strength fluctuates significantly enough from person to person to render the two factors subjective.

Therein we have an experimental identity:  bond strength is a function of proximity (interaction time and method), quality of said interaction, and an ‘x-factor’ which explains everything else.

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 F2: Utility (Extension)

 

When I was putting this concept together, the first iteration that occurred to me was utility (the satisfaction we get out of said friendship) because it should rationally correlates with strength.  But, through analysing data from my personal sample, it occurred that this inference wasn’t necessarily infallible. A coefficient exists which moderates this correlative association between the strength of a friendship and how gratifying it is – I will coin this coefficient as ‘recourse.’

‘Recourse’ is, quite simply, fundamental and we define it contextually here as how much the other party puts back in to a friendship/relationship (in terms of time,  energy, et cetera) relative to what we contribute.

On-balance or excess utility (U bar) can be modelled quite simply by using the recourse coefficient (r) to moderate the absolute value of bond strength (Ѵ).

The recourse coefficient, r, is constrained to plus/minus one, and the following methodology is used to estimate the r value:

r = +1   other party contributes and you don’t (0/100 balance)
r =   0   other party contributes equally (50/50 balance)
r =  -1   other party does not contribute and you do (100/0 balance)

Obviously, the absolute values ±1 are extreme situations of ‘unilateral friendship’ which are very seldom observed, safe for exceptional circumstances. More likely that there will be a small imbalance with r fluctuating between plus or minus 0.3. So a hypothetical friendship that was perfectly balanced or equitable would have r = 0, and therefore zero excess utility. Following this logic, where we contribute more than the other party and r is less than zero, we will have negative excess utility, and vice versa for positive r values.

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F3: Damage Extension

 

A powerful extension on the plain vanilla model is the alliterative dissolutive damage, which we will befittingly brand  ‘D².’ Essentially, dissolutive damage refers to the amount of psychological pain ensuing the breakdown of a friendship. It is the product of three factors multiplied across:

1. Proportional ‘Utility at Risk’ (UaR)

2. Emotional Sensitivity (es)

3. Attachment Propensity (ap)

The product of the latter two, ‘es’ and ‘ap’ is weighted by 4 so that it will show a result of 1 for a normal person, giving us a logical reference point (i.e. losing a friend worth 25% of their friendship utility pool would do 0.25 damage to a person with average/normal emotional sensitivity and attachment propensity.)

Utility at risk, given by the first term of the equation, is simply how much the friendship in question is worth relative to your entire pool of friendships. If the total utility of all your friendships was 200, and the dissolved friendship had a utility of 30, then UaR would be 15% or 0.15. Therefore the amount of damage increases the more the particular friend is ‘worth’ and/or the less total utility one possesses (which may stem from either or both of low utility friendships and a relatively low number of friendships.) It follows that a person with many acquaintances would not experience as much pain from losing an ‘average’ friend to a person who has fewer acquaintances.

Emotional sensitivity is a measure of impact exogeneous events have on the psychological state of an individual, bounded between zero and one in the equation. An emotional sensitivity near zero (stoicism) would suggest the dissolution of a friendship would have negligible impact on the person’s psychological state, yielding a low damage value. Taking the other extreme, someone who cries when Dumbledore dies in Harry Potter, we would bestow with an emotional sensitivity close to one’ and to whom a dissolution would do tremendous damage.

Finally, attachment propensity (which tends to covary with emotional sensitivity) concerns the individual’s susceptibility to becoming attached, or sentimentality if you will. Conceptually, this is self-explanatory: a ‘clingy’ person who attaches themselves like a barnacle on a hull has an attachement propensity close to one and is, by virtue of this strong dependency, liable to take significant damage from the dissolution of a friendship. Again, contrasting with a fiercely independent and nonchalant mindset which typifies low attachment propensity and results in a lessened level of damage.

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That concludes the analysis, though I will throw a contemplative spanner into the works. There is a hidden element that effects all three equations, but which hasn’t been specified on grounds of it being highly convoluted and epistemic. It relates specifically to certain generally accepted tenets: assumptions upon which both the analysis and friendship itself rest. I’d be much obliged if a correct guess were hazarded.

 P.X. Waterstone

El Amigo Invisible

~ by X on May 7, 2007.