Hello and welcome to Unit-6

This unit is a look at different mechanisms

that can generate power laws.

Before say little bit about this unit,

Let’s take a step back and

remind ourselves where we are on the course

So the first part of the course was about fractals

self-similar geometric objects

And in Unit-1 we talked about how to describe

those with the self-similarity dimension.

Unit-2 we looked at a number of different

mechanisms that can generate fractals.

And we saw that in a sense of

fractals are easy to make

There are number of different mechanisms,

deterministic and random that can produce fractals.

Unit-3 we dug deeper in the idea of dimension

and looked at the box counting dimension.

Unit-4 we started to shift from fractals

to power laws and scaling.

So the fractals have power laws

Power laws are fractals

In the sense that they are scale free

And so power law is a more general

not necessarily geometric manifestation

of self-similarity.

So Unit-4 we talked about that and some other

mathematical properties of power laws.

Unit-5, last unit was about in a sense

the statistics of power laws.

Given some data, how do you tell

if it’s a power law or not

How do you estimate the exponent alpha

And we’ve seen that it can be challenging

to both get a good estimate for alpha

and to tell if the data is really

well described by power law

or some other alternative that might be a better fit.

Steps bring us to this unit

Unit-6 where rather than sort of starting with data

and try doing backwards and do statistics,

we’re gonna consider a number of different

models and mechanisms that are capable

of generating power law or power law like behavior.

As we see throughout this unit also as many times

There are many different ways

of generating power laws.

There is a number of different mechanisms

for producing power laws.

Not like there is only one way to do it.

Just like there are lots of different ways

of making fractals.

So this is a broad topic

a lot to try to cover it in one unit.

So it will be a survey

I’ll present a few models described

in qualitative terms a few others

And hope to give you a sense that

there is some fun and interesting models

that produce power laws

and that there are a lot of different models

so that if you see a power law that doesn’t

immediately suggest one mechanism

there are multiple mechanisms

that could be responsible for it.

So we’ll get started by thinking

about some simple models

that are known as rich get richer models or

preferential attachment or cumulative advantage.

There are fun models with

some interesting mathematical properties.

And I think you’ll enjoy exploring them.