Introduction to Network Theory Lecture 1 Manuel Sebastian Mariani URPP Social Networks Network Theory and Analytics | 18.09.18 Outlook L1: Introduction to Network Theory | 1. Outlook 1 Outlook 2 Introductory example 3 Basic Concepts 4 Representation 5 Network types 6 Simple network models 7 Exercise L1: Introduction to Network Theory | 1. Outlook Introductory example L1: Introduction to Network Theory | 2. Introductory example The bridges of Königsberg 5 XVIII Century Is there a trail that transverses each bridge exactly once? Euler, 1736: Geometry is unimportant, only degree maers. First paper in the history of graph theory. ■ Nodes: landmasses; edges: bridges ■ The number of bridges touching every landmass must be even ■ Only start and end nodes might have odd degrees ■ It is not possible to make such a trail L1: Introduction to Network Theory | 2. Introductory example The bridges of Königsberg 5 XVIII Century Is there a trail that transverses each bridge exactly once? Euler, 1736: Geometry is unimportant, only degree maers. First paper in the history of graph theory. ■ Nodes: landmasses; edges: bridges ■ The number of bridges touching every landmass must be even ■ Only start and end nodes might have odd degrees ■ It is not possible to make such a trail L1: Introduction to Network Theory | 2. Introductory example The bridges of Königsberg 5 XVIII Century Is there a trail that transverses each bridge exactly once? Euler, 1736: Geometry is unimportant, only degree maers. First paper in the history of graph theory. ■ Nodes: landmasses; edges: bridges ■ The number of bridges touching every landmass must be even ■ Only start and end nodes might have odd degrees ■ It is not possible to make such a trail L1: Introduction to Network Theory | 2. Introductory example The bridges of Königsberg 5 XVIII Century Is there a trail that transverses each bridge exactly once? Euler, 1736: Geometry is unimportant, only degree maers. First paper in the history of graph theory. ■ Nodes: landmasses; edges: bridges ■ The number of bridges touching every landmass must be even ■ Only start and end nodes might have odd degrees ■ It is not possible to make such a trail L1: Introduction to Network Theory | 2. Introductory example The bridges of Kaliningrad 6 XXI Century Nowadays it is possible to transverse exactly once each of the existing bridges ■ On the modern map of Kaliningrad: ■ Green bridges survived until today ■ Red bridges were destroyed in WWII ■ Blue bridges were built last Century L1: Introduction to Network Theory | 2. Introductory example The bridges of Kaliningrad 6 XXI Century Nowadays it is possible to transverse exactly once each of the existing bridges ■ On the modern map of Kaliningrad: ■ Green bridges survived until today ■ Red bridges were destroyed in WWII ■ Blue bridges were built last Century L1: Introduction to Network Theory | 2. Introductory example Applications of Graph theory 7 ■ Computer Science - graphs themselves are the objects of interest ■ Social Sciences - connections between people in society ■ Electrical Engineering - designing circuit connections ■ Epidemiology - contagion process in connected society ■ Chemistry - graphs represent molecular structure ■ … L1: Introduction to Network Theory | 2. Introductory example Applications of Graph theory 7 ■ Computer Science - graphs themselves are the objects of interest ■ Social Sciences - connections between people in society ■ Electrical Engineering - designing circuit connections ■ Epidemiology - contagion process in connected society ■ Chemistry - graphs represent molecular structure ■ … L1: Introduction to Network Theory | 2. Introductory example Applications of Graph theory 7 ■ Computer Science - graphs themselves are the objects of interest ■ Social Sciences - connections between people in society ■ Electrical Engineering - designing circuit connections ■ Epidemiology - contagion process in connected society ■ Chemistry - graphs represent molecular structure ■ … L1: Introduction to Network Theory | 2. Introductory example Applications of Graph theory 7 ■ Computer Science - graphs themselves are the objects of interest ■ Social Sciences - connections between people in society ■ Electrical Engineering - designing circuit connections ■ Epidemiology - contagion process in connected society ■ Chemistry - graphs represent molecular structure ■ … L1: Introduction to Network Theory | 2. Introductory example Applications of Graph theory 7 ■ Computer Science - graphs themselves are the objects of interest ■ Social Sciences - connections between people in society ■ Electrical Engineering - designing circuit connections ■ Epidemiology - contagion process in connected society ■ Chemistry - graphs represent molecular structure ■ … L1: Introduction to Network Theory | 2. Introductory example Basic Concepts L1: Introduction to Network Theory | 3. Basic Concepts Nodes 9 ■ Set of nodes is called V ■ Fundamental units of which graphs are formed ■ Have many names: ■ Nodes ■ Vertices ■ Points ■ Actors ■ Represent objects ■ Individuals ■ Websites ■ Geographical Locations ■ Banks ■ ... ■ Are usually featureless (but not always) L1: Introduction to Network Theory | 3. Basic Concepts Nodes 9 ■ Set of nodes is called V ■ Fundamental units of which graphs are formed ■ Have many names: ■ Nodes ■ Vertices ■ Points ■ Actors ■ Represent objects ■ Individuals ■ Websites ■ Geographical Locations ■ Banks ■ ... ■ Are usually featureless (but not always) L1: Introduction to Network Theory | 3. Basic Concepts Edges 10 ■ Set of edges is called E ■ Second fundamental unit ■ Have many names: ■ Edges ■ Arcs ■ Lines ■ Ties ■ Represent connections between objects: ■ Friendship / follower / subscriber ■ Web-link ■ Geographical approachability ■ Loan ■ ... ■ Might have features (e.g. weight, see below) L1: Introduction to Network Theory | 3. Basic Concepts Edges 10 ■ Set of edges is called E ■ Second fundamental unit ■ Have many names: ■ Edges ■ Arcs ■ Lines ■ Ties ■ Represent connections between objects: ■ Friendship / follower / subscriber ■ Web-link ■ Geographical approachability ■ Loan ■ ... ■ Might have features (e.g. weight, see below) L1: Introduction to Network Theory | 3. Basic Concepts Graph 11 ■ Graph is an ordered pair G = (V , E) ■ In networks, network size; In graph theory, order of the graph: |V | ■ In graph theory, size of the graph: |E L1: Introduction to Network Theory | 3. Basic Concepts Graph 12 ■ Graph is an ordered pair G = (V , E) ■ E consists of 2-element subsets of V ■ Vertices belonging to an edge are called ends or end vertices of the edge ■ Vertices connected by an edge are called neighbouring or adjacent. ■ Some vertices may not belong to any edge, but all edges belong to a pair of vertices L1: Introduction to Network Theory | 3. Basic Concepts Graph 12 ■ Graph is an ordered pair G = (V , E) ■ E consists of 2-element subsets of V ■ Vertices belonging to an edge are called ends or end vertices of the edge ■ Vertices connected by an edge are called neighbouring or adjacent. ■ Some vertices may not belong to any edge, but all edges belong to a pair of vertices L1: Introduction to Network Theory | 3. Basic Concepts Graph 12 ■ Graph is an ordered pair G = (V , E) ■ E consists of 2-element subsets of V ■ Vertices belonging to an edge are called ends or end vertices of the edge ■ Vertices connected by an edge are called neighbouring or adjacent. ■ Some vertices may not belong to any edge, but all edges belong to a pair of vertices L1: Introduction to Network Theory | 3. Basic Concepts Graph 12 ■ Graph is an ordered pair G = (V , E) ■ E consists of 2-element subsets of V ■ Vertices belonging to an edge are called ends or end vertices of the edge ■ Vertices connected by an edge are called neighbouring or adjacent. ■ Some vertices may not belong to any edge, but all edges belong to a pair of vertices L1: Introduction to Network Theory | 3. Basic Concepts Graph 12 ■ Graph is an ordered pair G = (V , E) ■ E consists of 2-element subsets of V ■ Vertices belonging to an edge are called ends or end vertices of the edge ■ Vertices connected by an edge are called neighbouring or adjacent. ■ Some vertices may not belong to any edge, but all edges belong to a pair of vertices L1: Introduction to Network Theory | 3. Basic Concepts Graphs and networks 13 Graph A graph is the mathematical object formally defined above Network A network is the representation of a real-world system. Nodes and links have a specific meaning within the context of the appli- cation. Also, they have aributes Graph theory versus network theory ■ Different research questions ■ Graph techniques can be used to analyse networks ■ All networks are graphs (but the opposite is not true) L1: Introduction to Network Theory | 3. Basic Concepts Graphs and networks 13 Graph A graph is the mathematical object formally defined above Network A network is the representation of a real-world system. Nodes and links have a specific meaning within the context of the appli- cation. Also, they have aributes Graph theory versus network theory ■ Different research questions ■ Graph techniques can be used to analyse networks ■ All networks are graphs (but the opposite is not true) L1: Introduction to Network Theory | 3. Basic Concepts Graphs and networks 13 Graph A graph is the mathematical object formally defined above Network A network is the representation of a real-world system. Nodes and links have a specific meaning within the context of the appli- cation. Also, they have aributes Graph theory versus network theory ■ Different research questions