Section 7 - Tipping Point

Tipping Points

Tipping points are different from Exponential Growth

Percolation Models

Does water percolate through soil
P - Probability of percolating
If P<59.2 then no percolation

Forest Fires

Percolating Banks
- Models failing banks - Generally you add more details for the banks, like assets, liabilities, etc.

Information Percolating
- If I pass a piece of information will it pass through to everyone

Works well for networks

Can work for technology

Contagion Models

SIS - Susceptible - Infected - Susceptible
Diffusion Model - uses a Transmission Rate - tau

what if 2 people meet
W = number of people  with Disease
(N-W) = number of people w/out Disease

tau(W/N)((N-W)/N) = probability of two people meeting and it spreading

Add the contact rate: c

multiple above by N*C

When W is small you get
When W = N/2

This is not Tipping, this is natural diffusion

SIS Model

Now once they get over the disease, they can get infected again

Just add a new factor to the end of the equation

     For SIS, we have added the -aW component which are cured people who can get infected again.
which can be simplified to:

     so for small Wt, then if ct>a, then disease moves on.  If ct<a the disease will not spread.

now call R(0) = ct/a,  if >1, then it spreads, if <1, then not  (R(0) is the basic reproduction number)

Classifying Tipping Points

Dynamical System

  In this graphi, there are 2 stable points, but the one labeled U is unstable and any small movement will push it one way or the other.

 

He will call this a "DIRECT TIP" - Small action or event has a large effect on the long run.

 

Contextual Tipping Points
- Where the environment changes forcing to a new class



Between/Within Class
Changing between classes

Measuring Tips

Utilize change in the likelihood of outcomes

Diversity Index - Social Science
-  Amount of Information
is based upon how different the outcomes are between the classes.

Diversity = Summation(1/Summation(P^2))


Entropy - from physics
-  Number of types
Entropy = - summation(P*log(2) (P))