Intro to MA 4027 Graph Theory By Ralucca

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

Overview • Motivation: network science (see network science academic certificate site: http: //faculty. nps. edu/rgera/Net. Sci/Certificate/ dist/index. html • Graph theory: – Origins • Complex networks (MA 4404) – Extending graph theory

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 3 measurements

Origins of graph theory Euler Graph Theory 4

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. 5

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?

A Modelling of Königsberg : Multigraph • A vertex : a region • An edge : a bridge between two regions e 1 X W Z Y W e 3 X e 6 e 2 e 4 Y e 5 e 7 Z • This is the first paper in graph theory. Graph Theory 7

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. 8

Analysis of Complex Networks

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 10

Example: The Internet 11

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 12

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 13

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. 14