*The following notes are from "A First Course in Network Theory", by Estrada and Knight".*

#Preface

- 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

- Chemistry

- Condensed Matter Physics

- Material Science

- Engineering

- Biology

- Psychology

- Economics

- 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**.

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