Thread by @DellAnnaLuca

@ghuubear @startuployalist For me & what I wanted to do (not structural engineering), I thought matrix calculus pointless - until I discovered that Google’s Pagerank is nothing else than the eigenvalue of the matrix of links representing the net.

Then it suddenly all made sense.

@ghuubear @startuployalist Short thread on what I wish my “matrix & vibrations course” teacher taught me:

1/ Matrixes also represent conductivity (rase of flow) between nodes in a distributed system

For example, how traffic moves from website to website (Pagerank is the eigenvalue of the matrix of links)

@ghuubear @startuployalist 2/ Matrixes also represent conductivity between nodes of an electrical grid, & are thus useful to determine how electricity flows.

Or they can represent traffic between crossroads in a city, with each node being a crossroad and each cell representing the conductivity of a street

@ghuubear @startuployalist 3/ For example, cell (5,3) might represent how broad is the street between crossroad E and crossroad C in a city.

(This example is further complicated by the fact that traffic speed is nonlinear respect to density of cars, but let's neglect this for a moment.)

@ghuubear @startuployalist 4/ Eigenvectors represent the flow through nodes at steady state.

In the example above, the eigenvector of the matrix representing the broadness of streets in a city tells us how many cars pass through each crossroad at steady state.

(I'm still neglecting nonlinearities)

@ghuubear @startuployalist 5/ In Pagerank, the matrix of outgoing links from pages of the net, the eigenvector represents the number of visitors on each page at steady state.

Why do we need Pagerank though, rather than using Google Analytics to know that information?

@ghuubear @startuployalist 6/ Well, Google Analytics is a set of observations, i.e. snapshots of flow in a timeframe. For a small website like mine (~10 visitors/day), variance is high. And in a tightly-connected system, variance → vibrations → waves → trouble measuring what happens when a node is added

@ghuubear @startuployalist 7/ Google Analytics estimates: what is happening now.
Pagerank (and matrixes, for the matter) estimates: how does the *steady state* of the system change when a new page / node is added, without needing any further measurement.

@ghuubear @startuployalist 8/ For reasons which are evident to those who are familiar with Taleb's writings, neglecting the dynamics is dangerous.

This is why matrixes are rarely to be used as *the* tool. However, they are *a* great tool to estimate future steady states of *linear* systems.

@ghuubear @startuployalist 9/ Some more real-world examples of what, under the hood, is basically an eigenvector:
- the shape of a guitar string vibrating
- the number of gossips each person in your circle is aware of at any given moment of time
- the daily amount of cash in each cash register of an island

@ghuubear @startuployalist (In tweet #1, “rate” and “eigenvector” instead of “rase” and “eigenvalue”, where is autocorrect when needed😊)

*eigenvector