*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**.