so let's try to summarize some of the main results in themes from the course
so we began way back in Unit 1 by looking at fractals geometric fractals. Geometric
fractals are objects that are self-similar small parts of the object
or similar to the whole and that the self-similarity extends over many skills
as an example of a fractal is a fall.A late fall now where I am and so the
fall outside aren't looking so good but this one still has it shape if not
its color. so if you have a fall and you a break apart off you get a fall and
that shirt looks like it's self made up of little fall and so on and then the
counter example that we used was person you have a person and you brake a part
of the person off which you have doesn't look like a small little person it looks
like a creepy. OK so that's a quick reminder that. So anyway we then
said we can quantify described the self-similarity with a dimension d
defined in this way by looking at a magnification factors how much we have
to stretch out shapes and how many small copies there are in. The deal that we
need to make this equation true as a self-similarity dimension and we
calculated this for a number of geometric fractals like this triangle
and so on. so that's the idea of self-similarity is related to being
scale-free
objects and literacy it's scale-free so for example there is no typical size of
the bumps and a Koch curve there is no clue as to the scale. In contrast for
tomato or hand or pencil all of those have a typical size and that set the scale
another way to think about this is that in a fractal if you're shrunk you
wouldn't be able to tell because I know objects that satisfies scale you may be
in a tree that consistent branches that consists of branches that consists of
branches branches of
different sizes all the way up and down so there's no single branch that sets us
a scale.And we noted that real fractal physical objects like this are not
self-similar forever all the way down there is some lower cut-off scale.
And in unit 2 we set up how do we generate fractal and the answer is there are many different ways to do that. so
deterministic geometric iteration just repeating certain trent I G major
transportation transformation again and again and also we can do that with a
little bit of randomness in the rule or some irregularity or asymmetry we took a
little detour and talked about how fractal landscape can be generated and
then there's the chaos game which generated this for Penske triangle by
rolling dice and diffusion limited aggregation when a random walkers clump
together and form these
fractal into its shapes. The picture that emerged from this
exploration is that there are many simple ways to make fractals. Fractal
shapes may look at first to be quite complicated but they're actually rather
simple to build and moreover we can have both deterministic and random iterative
processes that make fractals. So processes have very different
character and can produce fractals. So I argued that in a sense we can think of
fractals as generic shapes in that they're easy and there are many
different simple ways to make them. so fractals a beautiful complicated but
maybe they shouldn't surprise us when we see them. In Unite 3 we look at the Box-Counting Dimensionq and
these are the main ideas here we looked at a number of boxes of a certain side
needed to cover an object and asked how does that change as the box size changes
and that leads to the idea of the box counting dimension and this is a tedious
but more concrete way of thinking about how you might define and calculate fractal
dimension. Was also in this unit that we saw the first instance of log-Log plots in
this particular case if applied the number of boxes versus the box size we
would see a straight line and that's something we saw after this again and
again through course we looked at lots and lots of Log-Log plot. has also
introduced or find a little more carefully this idea of scaling so that
if we have a relationship like this then and that's true for a wide range of
scales as he rest would be the box size but it could be something more general
then we say that as object or phenomena exhibits killing and the dimension as
exponent here telling us that there's something that's staying the same as the
scales changed that as we zoom in and out there some relationships and ratio
small to large that staying the same then in unit 4 we got a little bit more
abstract talking about physical fractal so far. For instance triangles and
Koch curve and so on but let's look at the mathematics a little bit more abstractly
so box counting led us to this equation and if this equation is true there is
self-similarity and we see a line on a log-log plot and then we reverse that
logic so if we're looking at some other phenomenon that's not necessarily a
geometric shape that we see linear behavior on a log-log plot that's a clue
an indication that there's self-similarity and so mathematically
the general formfor a power law is power law of x is A x power minus alpha
then we look at some properties of power law and some important properties are that
power law has long tails. They became much more slowly than for example an
exponential function. so here's a power-law and exponential and so the
exponential value exponential function here at 140 is essentially zero the
power law very small .0003 but not small. power law has long tail
and that means that large event extreme events are unlikely but not vanishingly
unlikely we will still see them as very different than exponential distribution
also power laws are scale free and one way to see that is to plot a power law on
different scales and we see that we have the same sort of shape this is another
way of saying that power laws don't continuous scale there's no clues as to
scale if I remove the labels from this access you wouldn't be able to figure
out what they were even if you knew what the power law was. additionally I argued that
power law is the only distribution that scale free so we see scale-free behavior we know there
has to be a power-law we see a power law we know there has to be some scale-free.
another interesting property of power-law is that for some values of the exponent
the average does not exist and for some other values of exponent the standard
deviation not exist and looked at a lot of plots like this I considered that St.
Petersburg coin tossing game as an example. so here's an example with alpha
2.5 there is an average approaches a limit but the standard deviation does
not it spikes upwards and then tells down and continue to do that no matter
how many data points you have in your dataset and how to how many experiments
you do
then unit 5 we looked at empirical power laws and this was a little bit more
technical and involve some statistics in brief the main points were at estimating
that exponent for power laws is tricky. you have a dataset and you think it's
described by a power law. estimating reliably the exponent is a difficult statistic
task and in particular using a least-squares fit on a log-log plot
which is sort of the logical thing to do
turns out to not be reliable way so instead of, alternative is to use what's
known as a maximum likelihood estimator and we talked about the formula for that
and a little bit about where that formula comes from. in addition to
knowing how to get the best exponent we also want to know how good if it is the
best fit power law. once we have the best fit does it describe a lot of the data or
just a little bit and so we talk some of this was a little bit more technical
about how to estimate p value for model as shift by bootstrapping and also power
law behavior often has a lower cut-off not a straight line for all values and
so there's a principle way not arbitrary way to estimate that lower cut
off by trying different values and choosing one that leads to the best fit
additionally it's important to compare alternative models so maybe your data is
pretty well fit by power law but it might be better to fit by something else. maybe a law
of normal stretch or exponential or something. And so the way to do that is
to find optimal parameters for the alternatives and estimate their p values
and probably better still to calculate a likelihood ratio so the details for all
of this are in this important paper by Clauset Shalizi and Newman where this
is discussed much more much more fully
but then that led us to ask some questions right so doesn't always matter
if it's a power-law well of course it depends on the particular scientific
questions you're interested in in some cases it might be important mainly just
to establish that your data has a long tail that it decays slower than
exponential and it's worth noting that people talk also that heavy tailed and
fat tailed distributions in addition to long tailed distributions so maybe the
particular application you're interested in its not that important to say is it a
power law or not the main issue is does it have a long tail. So again it's
important to think about the questions are trying to answer as you're doing
your data analysis. So it's one thing to say yeah my data seems to be
reasonably well fit by a power law just like you might say oh there some sort
of linear trend in a dataset but it's very different to say my data is a power
law or my data is linear not just has a linear trend and so in these settings
you might want to say well is there any theory under well understood theory
based on first principles that were predictive power law behavior. So in general
establishing that dataset is a power long as opposed to merely as a
reasonably well fit by a power law can be a difficult task
ok. then unit 6 we said well how would one generate power law. what are some
processes that generate power law distribution this was similar to Unit 2 when
looked at what are some proceduces that generated fractals and the
take-home message is the same there are many different ways of generating power
laws. so we talked about rich get richer models preferential attachment certainly
isn't combining exponential distributions multiplicative processes
that have a lower threshold and then a couple optimization scheme
a network say designed to optimize certain qualities would would lead
naturally to something with a power distribution something that was scale
free. So the main point is there are many different ways of generating
power law. So what does this all mean what's the upshot well I think power
law are unarguably interesting. they have long tail behavior that somewhat
unusual scale-free so they're definitely noteworthy but they're not that unusual
they're not that outside of the realm of the possible because they're many
different ways of making them so in particular if we know that something as
power law distributed that doesn't alone just merely knowing that it's a
power law doesn't imply any particular mechanism and in particular their very
different mechanism that can lead give rise to power laws so random process
these and optimality considerations both give rise to power law. in addition
thinking back to the previous unit many empirical power laws may actually people
have said oh this is a power law based on their data may actually be lognormal
or something else because many researchers having pared their power law
claims two alternatives
So in unit 7 and 8 any we looked at some particular applications of ideas of scaling
In first in unit 7 we look in metabolic scaling and the starting point here was the
empirical relationship that the metabolic rate scales with a
three-quarter power law and this is Kleiber law and this is a puzzle
because if we were to assume that metabolism is determined by service area
features need to dissipate heat then we would expect an exponent of 2/3
It is 2/3 and not 3/4 So in late 1990s West Brown and Enquist
that comes up with a mechanism that explains why we would see 3/4
only actually observed and not 2/3to third which is what one would predict from
surface area and the main crux of the argument is that metabolism is
determined by optimal fractal ie cell similar vascular networks. and their
theory leads to this prediction which is very well burnout experimentally and
interestingly they also has another predictions that biological rates heart
rate respiration rate with scale of third quarter
biological times like like times might scale as a him to the plus one quarter
these are also pretty well burnout experimentally and then lastly we looked
at urban scaling and said that many properties of city is to scale more or
less and here are just two examples road length and GDP and if we look at a bunch
of properties of cities we see this sort of curious clustering that some city
Properties LLC preliminarily often with an expert at around 1.15 somewhere
linear as maybe not surprising and then some are sub linear often with an
exponent clusters around 0.85 to be infrastructure sorts of things in these
tend to be socio-economic outputs of some sort
so we talked a little bit about possible explanations for this and I think there
is the beginnings of a compelling theory but maybe isn't quite there yet this is
very recent work just a couple of years old
additionally there's a lot of variation in the data so these scaling behaviors
that are senior certainly not exact laws and any sense but there's definitely a
trend and there's definitely variation around the trend both of those are
interesting things to study and to think about but it's curious to me and I think
the other is that there are these fairly robust scaling relationships across
cities on different continents and scaling relationships
hold over time as well if you look at one city or set of cities decade by
decade say so this is an ongoing an active area of research and I think it's
particularly exciting place to watch her to watch for developments in complex
systems over the next couple years
alright so that's what we covered and let me just finally try to highlight
some things and things that I kept coming back to my mind as I was thinking
about the ideas of course right. so first saying something is fractal or for
that matter a power-law scaling is not either-or category I would say so
objects are more or less fractal like
described by our laws to a lesser or greater extent as opposed to an
either-or yes it is a power law and you know it is
statistical estimation for power law is tricky and I think many of the claims of
power law is strong claims saying it is a power law are not merely that it's like
they very well may run a little bit misleading. there are many many ways to
generate fractal and power law and so what this means is that if you knew that
something is a power law that's important and interesting but that in
and of itself does not necessarily reveal itself
why? because very different models very different processes can produce the
same outcome
random process a deterministic process something involving optimality
merely knowing something is a power law does not I would say that I've reveal
essence. So another way to think about this is observing that something
as radical a power law is usually not the end of research but it's actually
the beginning often a very fruitful line of research says hey there's a somewhat
unusual pattern here that exit that's just that there's some commonality
across a lot of scales now let's try to learn more about that process and see
what we can figure out.