Tracy-Widom Distribution & Mn Diversity

I can’t remember where I was first introduced to the article discussing the Tracy-Widom Distribution; though the analogy was prescient.

At what population does a set of islands become over-populated?

At what point does this set become under-populated?

What is necessary to retain equilibrium?

Applying such a set of statistical tools to the topic of narrative systems and message proliferation, a similar set of questions can be asked:

At what point does a diversity become self-replenishing?

At what point is diversity  unsustainable?

What is the equilibrium necessary to retain control?

Background

In practical terms, Tracy–Widom is the crossover function between the two phases of weakly versus strongly coupled components in a system.

In a mathematical sense, we’re dealing with matrices, and are beyond the capacity of common parlance. I don’t understand it all, but I’m learning, and will share as I better comprehend. Moving on,

Metropolitan and rural areas are scrambling to make more space for “diversity” at all levels of business and governance.

This is not without tension, and varying levels of magnitude, as some areas struggle more than others.

Nonetheless, what we are looking to take into account are the following:

Individual Actors (IA) each possessing will to survive and persist, differentiated by nondescript categories (NDC).

Each IA belongs to a group of IAs who share an NDC.

We are asking, at what point does population share & growth allow a minority population to persist and grow?

The Models

Using hypothetical populations NDC1 & NDC2, and a very preliminary model.

Population = NDC1X + NDC2Y

In this equation, were it run every year, estimates that NDC1 grows by X% each year, while NDC2 grows by Y% each year.

if X were greater than Y,  what would it take for NDC2’s growth rate to catch up to NDC1’s growth rate?

Beyond Basic

In our original model P = NX + NY, we use the variable X and Y as cover for a vast array of behavioral activity.

What are the engines powering population growth? What are metrics indicative of those powers?

Home-ownership, employment, education, nutritional access and so much more may contribute to these growth rates. These are data points we can use.

Now we need to make composite models that take these variables into account – so that we can estimate a rolling growth rate.

Comparative Modeling

Once we have created an initial composite model, hypothetically something similar to:

P = NDC1(AX+BY+CZ) + NDC2(AY + BX + CZ)

Hopefully now we have a formula more responsive to, and indicative of, changes in communities for each year that it is applied. And by applying this model to multiple years we can gain further insight when graphing .

Furthermore, we can make adjustments to the model, to isolate and compare factors. For example what are the differences between the output of the above equation and the following:

P = NDC1(AX+ BX+ BY+CZ) + NDC2(AY + .4BX + BY + CZ)

How might such an adjustment affect short and long-term growth in the population?

By digging deeper into the metrics, hopefully we can better comprehend what we need to do in our own communities to create a healthy andsustainable diversity of conversation.

The Goal

The goal is to answer the following questions:

What metrics may indicate growth rates within and between communities? 

What is the magnitude of effect for any of the metrics found?

Can we create a meaningful model to reflect historical growth rates of diverse populations?

This conversation has not yet taken into account Bayesian methodologies; which are subject to further model adjustments. And requires a more comprehensive understanding of the Tracy-Widom distribution and supporting math.

Future posts will further explore these conversations.