# Modeling Relationships

1. Jul 15, 2009

### John Creighto

According to the following link a relationship lasts on average 3-5 years.

If we take this as fact I'm curious what kind of mathematical model might be well, suited to describe this. The simplest model would be of the form:

Let
C - be the people in a relationship
S - be the people not i a relationship
H - be the probability of hooking up per time
D - be the probability of being dumped per time

Then one may assume a simple mathematical model:

$$\dot{C}=-D C + H S$$
$$\dot{S}=D C - H S$$

This model would of course have many problems. There is one known parameter (The average relationship length. There are two unkonwn parameters to fit D, H and the two initial states. However, if we know the average number of people in a relationship over several years. Say people at ages, 21, 22, 23, 24 Then we can can estimate the paramters D, H for this four year period of time.

The question now remains is how to model the age dependency on the average relationship length. On idea would be to perhaps partition relationships into several groups, say: short, medium and long term and find the average length of time a relationship lasts in each of those groups. Longer term relationships would have a lower probability of forming but last longer so this could explain why relationships last longer as people get older.

Another suggestion would be that perhaps there is a dependency between the number of relationships and the length of relationships. For instance, the more relationships people have the more they may be willing to settle down (note some may argue the opposite is true). My final suggestion would be some kind of demographic mix. That is if two people who are both looking for long term relationships hook up the relationship is likely to last longer then if two people who aren't looking for long term relationships hook up. The final suggestion would be some kind of metric based on computability.

2. Jul 15, 2009

### John Creighto

Availability

One suggestion that would give an age dependency on relationship length is availability. That is people are more likely to beak up if they have more options. Similarly people are less likely to hook up if they have less options.

Let:

$$D$$ - be the probability of breaking up if there were an equal amount of singles and couples
$$1-\alpha _{CD}$$ be the probability of breaking up per time if you thought you couldn't find anyone else.
$$1+\alpha _{SD}$$ be the probability of breaking up per time if everyone is available.

$$H$$ - be the probability of hooking up per time if there were an equal amount of singles and couples
$$1-\alpha _{SH}$$ be the probability of hooking up per time if everyone was available.
$$1+\alpha _{SD}$$ be the probability of hooking up per time with someone completly random if you thought no one else was available.

$$\theta_D= 1- \alpha _{CD} {C \over (S+C)|S-C|} + \alpha _{SD} {S \over (S+C)|S-C|} \right)$$
$$\theta_H= 1 + \alpha _{SH} {S \over (S+C)|S-C|} - \alpha _{CH} {C \over (S+C)|S-C|} \right)$$

$$\dot{C}=-D \Theta_D C + H \theta_H S$$

$$\dot{S}=D \Theta_D C - H \theta_H S$$

Last edited: Jul 15, 2009