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.