1. Intro to MA 4027 Graph Theory
By: Ralucca Gera, NPS
Excellence Through Knowledge

2. Overview
Motivation: network science (see network science academic certificate site:
Graph theory:
Origins
Complex networks (MA 4404)
Extending graph theory

3. Why Network Science?
Newest science (20 years old or so) and a very active field, relevant to the type and amount of data available nowadays
Applicable to the study of the structural evolution of large networks
It studies networks holistically
Modeling phenomena around us using networks can be done in multiple ways and at different levels/depths
Can be used both for passive and active measurements

4. Origins of graph theory Euler

5. The Origin of Graph Theory
The Seven Bridges of Königsberg (the problem that is at the origin of graph theory) was posed by Leonhard Euler in 1735 (also prefigured the idea of topology) The citizens of Königsberg supposedly walked about on Sundays trying to find a route that crosses each bridge Königsberg of exactly once, and return to the starting point.

6. Königsberg Bridges (now Kaliningrad, Russia)
Is it possible to find
a route that:
Starts and finishes at the same place?
Crosses each bridge exactly once?

7. A Modelling of Königsberg : Multigraph
A vertex : a region
An edge : a bridge between two regions
This is the first paper in graph theory. e1 e2 e3 e4 e6 e5 e7 Z Y X W X Y Z W
Graph Theory

8. Goals for the Graph Theory course
Course Goal:
To learn to model, analyze (with proofs) and interpret the data using graphs.
Course Objectives:
The understanding of fundamental definitions and properties of graphs.
The ability to read and write rigorous mathematical proofs involving graphs.
Recognition of the numerous applications of graph theory.

9. Analysis of Complex Networks

10. Transitioning to complex networks
The need for adapting graph theoretical concepts the development of new tools to compare complex networks as they model the world around
as the networks have shifted from simple and small to complex and extremely large (data explosion),
as the modeling transitioned from static graphs to dynamic graphs (like geometry to calculus),
as objects to be studied were of one type, and now there is a variety of data types

11. Example: The Internet

12. Types of questions to answer
Example: The Internet
What: measure Internet-scale Topology
Why: network security, infrastructure protection,
How: using measures on graphs to understand network evolution
Why is it hard to measure it?
requires multiple days for even a partial map
incomplete
dynamic

13. From Simple to Complex Networks
Simple graphs (the ones we have seen so far):
have a small number of vertices,
which interact according to well understood laws
usually static in time (at least on small time intervals).
Complex networks (no established definition):
very large and contain mixt type of data
evolve (In 1990 the WWW had only one page. Now it has a few billion pages)
generally display organization with no apparent external organizing principle being applied, and no internal control

14. Goals for Complex Networks
Goals of studying complex networks
to extract emergent properties
to understand the function of such complex systems
to be able to predict changes in the network
to control how the network evolves
To understand a complex system we need to understand the network that models it.