Thursday, November 4, 2010

The Mathematics of Social Life

We can think of a person's personality as a vector in Fn. The dimensions of F represent the dimensions of the goals and interests that one might entertain—what one values, broadly speaking. Some people are more interested in pursuing money than others; some people have more an inclination towards spirituality than others; some people give more importance to their family than others. Thus, we can say that money, spirituality, and family represent some of the nearly infinite dimensions of this space (but we think of n as finite for simplicity). The zero vector would signify balance within all dimensions.

Like R, F is continuous, infinite, and contains postive and negative values; just as value traits come in shades of grey, have no known limits, and always work in pairs of opposites. Although personality doesn't just include values, they are its most important component. Values are what drive action and shape our life's trajectory, which is what we're interested in modeling.

These personality vectors represent a static snapshot of a person's personality. However, people's goals and interests are, of course, constantly changing: at any given moment, a person has a certain place she's at in life, and a certain place she'd like to go. This defines what we'll call a change vector in Fn. The vector points towards the new personality vector one wants to have, and the magnitude represents the rate at which the change is happening. Continuing the previous example, we can say that a person with a penitant heart who wants to focus more attention on his family rather than on money would have a change vector that points 1) less on the money dimension 2) more on the spirtuality dimension and 3) more on the family dimension.

Over the course of our lives, the personality vector traces a trajectory through Fn that describes how our goals and interests have changed. These trajectories are useful in describing, characterizing, and categorizing people's lives. When we say, for example, that someone has meandered through life, we may mean that the distance traveled along the trajectory is much larger than the distance traveled in Fn.

The change vectors are first derivatives of our life's trajectory. At each moment in time, the vectors are tangent to the trajectory, and mark where we're immediately heading. Because our personalities have inertia, our life's trajectory follows the vector unless the vector changes: until we receive some new information, or have some new experience, our thought pattern will not change.

A society is a well-defined set of people. Societies can be both big and small, ranging from small towns, to nations, to the entire world. Societies can also be defined in terms other than geography, such as national origin, occupation, or marital status.

Societies do not have a uniform distribution of personalities. Instead, they tend to cluster in certain regions of the space (this is where stereotypes come from). We can describe the distribution of personalities of a society with a density map on Fn.

Not all people within a society are equally important to us; we tend to think that some have more of a claim on our consideration than others. A social network is thus defined as a graph of all social claims on oneself that one considers meaningful. The graph is organized as a series of concentric rings, with us at the center and each progressively farther rings representing more distant/less valued relationships.

Hypothesis on Nature vs. Nurture
When we are born, we have an initial personality vector in Fn. This is the nature component, since it limits the possibility of our life's trajectory to within the neighborhood of that point. However, once that point is established, we can move in any direction. This is the nurture component.

An analogy is a cow tied by a rope to a tree: the cow can only move as far as the rope will let her, but within the radius of the rope she can move in any direction.

It's hard to say whether babies are born with change vectors as well, i.e. whether they have a conception of what they want to do with life. But even if they did, it would be almost inseparable from the immediate influence that the family exerts on the newborn.

Interaction is comprised of two components: liking and influence. They are separate matters, since we are often influenced by people that we do not necessarily like.

We often gauge how much we like something or someone by how much it aligns with our goals and interests. That is, liking is a measure of the similarity between two vectors, i.e. the angle between them. An angle of 0 measures complete liking and an angle of pi measures complete dislike. If a vector is perpendicular to another, then it is neither heading in a similar direction on any dimension, nor is it heading away; perpendicular vectors thus signal indifference.

Note that we are talking about personalities, not genes. People with very similar personalities may look completely different, talk different, and manifest their goals and interests in different ways—but their goals and interests are similar.

Liking, like influence, happens not just through face-to-face encounters with other people, but also by learning about the work they produce. Our work crystallizes our personality in time, making it possible for distant, unknown people to like and be influenced by us, even without directly meeting. Reading the works of an author reveals the idea vector of the author at that time; learning about the social works of a great leader is a way of meeting their vector.

Influence is a question of what influences the change vectors. Since influence can occur to varying degrees, we can think of it as lying on a spectrum. One one extreme, we have complete influence, where both parties fully and equally influence each other. In complete influence, two people leave the interaction with the same new "goal" personality vector, one that is the sum of their previous individual personality vectors: components of the vector moving in opposite directions would cancel out, and components moving in the same direction would build on each other. However, because personality change is continuous, and occurs over time, the change in personality vector is not instantaneous. Instead, after the interaction, both parties leave with change vectors that now point towards their new goal personality vector. Over time, if these change vectors are unaltered, then their personalities will become the same.

On the other end of the spectrum, we have a complete lack of influence, where the interaction does not affect the change vectors at all. Most interaction is either of this latter type (since we can't afford to constantly reevaluate ourselves every time we interact), or somewhere in between.

Because individuals have free will, personality change is a stochastic event, and hard to pin down. However, we do know that a person is more likely to be influenced one direction or another based on the distribution of her social network within Fn.

We can simplify the process of influence by thinking of it as a random event where a person chooses to be influenced from someone in her social network. That is, every t periods a person decides to be influenced. She samples someone at random from her social network and is influenced, let's say completely influenced. Because her social network is unevenly distributed, she is more likely to sample certain people over others, causing her to be more influenced in a certain direction. The family, as a child's first social network, has the first say in influencing the child's change vector. However, as the child begins her process of socialization within larger society, the familial claims weaken and other people gain stronger influence.

Social Life
When we find ourselves in good company, it is because the personalities of the people involved complement each other. Similarly, in tight knit circles of friends, the personality vectors are linearly dependent on each other.

Social cliques are subspaces of Fn. These cliques are, by definition, closed and bounded, and represent clusters of a socially defined personality type.

Large social gatherings, such as parties, are also home to the formation of subspaces. Like-minded people tend to gravitate towards each other and form social circles, and at times these circles don't even mix. A good party, though, is one where the maximal subspace of the party includes every member, meaning that everyone has interest in each other.

We can think of all the people in the world as also representing a bounded region in Fn. The edges of the space represent the boundaries of the present human experience. As people have interacted more and more throughout history, they have provided more and more opportunities for ideas to change, pushing personality vectors in new directions and expanding the human experience.

The region of high density of a density map on all of humanity is the zeitgeist of the era. It represents the certain general paradigm of thinking, acting, and being for people at that time.


  1. Overall, this seems like a fairly robust model, and it might even work well in producing quantitative social analysis. I do have one issue, specifically with the way you represented the social network. It seems to me that the value of the relationships here is equivalent to the amount of influence that person has. It would then make sense to establish a set of standard differential equations with variable coefficients relating any two people, then have a separate vector space for these coefficient vectors and a set of vectors for each person corresponding to the rest of the population. This set then represents the entirety of the subjects social life, and complete knowledge of it would determine the change vector fully (incomplete knowledge would allow for approximation). If desired, this formulation would still allow for extraction of what you called the social network from this space, by simply using some metric to determine influence quantitatively from the coefficient vectors and then, if you want objective networks, rather than separate ones for each individual, setting up an equivalence relation.

  2. Um, balance in Fn would not be a zero vector. It would be a 45 degree angle from the x-axis in n=2 for example.

    This is an interesting metaphor, but I'm curious about what you would discover using this as a model that you don't already know. E.g. you know that social cliques are groups of people with similar interests. You know they tend to exclude people with different interests. You know that some people in society are more important to you than others.

    The real test of a model's value is what it tells you that you didn't already know.

  3. @LDerek: I'm talking about balance within a dimension. I imagine that personality traits have both positive and negative aspects (recklessness and timidity could be the positive and negative extremes of a scale called confidence), so zero on that dimension would mean "balancing" between those two ends. If one has this kind of balance on all dimensions, then the personality vector is a zero vector.

    Your point that this model does not have much explanatory power is valid. My intention was just to develop a humorous thought experiment. At the same time, though, after writing this post I feel I think about these issues more clearly/precisely.

    Thank you for your comment.