*The following notes are from "A First Course in Network Theory", … [Read More]
The following notes are from "A First Course in Network Theory", by Estrada and Knight".
Originated from lecture notes
Difficult to balance theory and application, catering to a highly diverse student cluster (in lectures_
Avoid heavy math bias (dammit)
No prerequisite graph theory (yay)
Assuming simple networks unless otherwise stated (bidirectional, unweighted, connecting unique adjacent nodes.)
Chapter 1: Introduction to Network Theory
1.1: Overview of Networks
A vital science, w a plethora of examples
Original mathematical contributions seemed frivolous in nature; and was for the pure and recreational mathematician
Why are networks so ubiquitous?
'being networked' is a fundamental characteristic of complex systems.
Aside from extreme scales (cosmological, quantum), all other forms are the study of complexity sciences
(aside: even quantum foam and the cosmos itself aren't imprevious to complexity science, more on that in a blog later)
Some disciplinary examples include
- Condensed Matter Physics
- Material Science
- Social sciences
**Interactions and relations are captured by 'edges' of a network (straight lines between nodes) **
Edges can represent...
Physical links (road, veins)
Physical interactions (actions between proteins, biological interactions)
'ethereal' connections (information transfer)
Geographic proximity (landscapes, cells & tissues)
Mass energy exchange (metabolic, reactions, trade)
Social (friendship, familial, business)
Conceptual (definitions, citations)
Functional (gene activation, brain functionality)
1.2 History of Graphs
Network theory began from graph theory, which started with Euler and the 'Koenigsberg bridge problem'.
Is it possible to travel through the city, and cross each bridge only once?
There were four areas of the city with bridges between them, so Euler abstracted the problem into areas , , , and , connected by edges representing bridges.
Example: Knight's Tour
Can a knight visit every square on a chessboard visiting each once and only once, returning to its original position?
This is an example of a circuit. Alexandre Vandermonde formalized the question by letting each square be given by , for , and gave the following definition:
A Knight's Tour is an ordered list in which if follows , then and , OR and .
A circuit visiting each 'vertex' (eg. square on chessboard) once and only once is a *Hamiltonian Circuit**.
The Schrodinger Equation
Consider a particle constrained on the x-axis, with mass , subjected to a force .
In Classical mechanics, we determined , the position. Then we find the velocity, and momentum . Finally, we determine the kinetic energy ... or any other dynamical quantity of interest.
From there we find using , for conservative systems.
Luckily, conservative systems are the only ones which occur at microscopic scales. We know that
which, along with initial conditions , we determine .
In Quantum Mechanics, the name of the game is different. We wish to find the particle's wave function, , which we do by solving the Schrodinger equation:
-more notes to be written soon.
- QM was not created by one individual, and retains 'scars' from its traumatic youth.
'I think I can safely say that nobody understands quantum mechanics.' -Feynman
One cannot understand what quantum mechanics means until one understands what it does. That's what this book is about.
May be 'forbiddingly mathematical'...but this is because it is useful, and not meant to be torture for undergraduate students.
But to become accustomed to the shovel, one must develop a few blisters. -David Griffith
- Do the fucking questions.
...yes Sensei Griffith.