IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts....
Transcript of IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts....
![Page 1: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/1.jpg)
Introduction to Network TheoryLecture 1
Manuel Sebastian MarianiURPP Social Networks
Network Theory and Analytics | 18.09.18
![Page 2: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/2.jpg)
Outlook
L1: Introduction to Network Theory | 1. Outlook
![Page 3: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/3.jpg)
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
![Page 4: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/4.jpg)
Introductory example
L1: Introduction to Network Theory | 2. Introductory example
![Page 5: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/5.jpg)
The bridges of Königsberg 5
Is there a trail that transverses each bridge exactly once?XVIII Century
Euler, 1736: Geometry is unimportant, only degree ma ers.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
![Page 6: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/6.jpg)
The bridges of Königsberg 5
Is there a trail that transverses each bridge exactly once?XVIII Century
Euler, 1736: Geometry is unimportant, only degree ma ers.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
![Page 7: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/7.jpg)
The bridges of Königsberg 5
Is there a trail that transverses each bridge exactly once?XVIII Century
Euler, 1736: Geometry is unimportant, only degree ma ers.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
![Page 8: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/8.jpg)
The bridges of Königsberg 5
Is there a trail that transverses each bridge exactly once?XVIII Century
Euler, 1736: Geometry is unimportant, only degree ma ers.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
![Page 9: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/9.jpg)
The bridges of Kaliningrad 6
Nowadays it is possible to transverse exactly once each of theexisting bridges
XXI Century
■ 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
![Page 10: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/10.jpg)
The bridges of Kaliningrad 6
Nowadays it is possible to transverse exactly once each of theexisting bridges
XXI Century
■ 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
![Page 11: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/11.jpg)
Applications of Graph theory 7
■ Computer Science - graphs themselves are the objects ofinterest
■ 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
![Page 12: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/12.jpg)
Applications of Graph theory 7
■ Computer Science - graphs themselves are the objects ofinterest
■ 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
![Page 13: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/13.jpg)
Applications of Graph theory 7
■ Computer Science - graphs themselves are the objects ofinterest
■ 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
![Page 14: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/14.jpg)
Applications of Graph theory 7
■ Computer Science - graphs themselves are the objects ofinterest
■ 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
![Page 15: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/15.jpg)
Applications of Graph theory 7
■ Computer Science - graphs themselves are the objects ofinterest
■ 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
![Page 16: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/16.jpg)
Basic Concepts
L1: Introduction to Network Theory | 3. Basic Concepts
![Page 17: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/17.jpg)
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
![Page 18: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/18.jpg)
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
![Page 19: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/19.jpg)
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
![Page 20: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/20.jpg)
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
![Page 21: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/21.jpg)
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
![Page 22: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/22.jpg)
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 ofvertices
L1: Introduction to Network Theory | 3. Basic Concepts
![Page 23: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/23.jpg)
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 ofvertices
L1: Introduction to Network Theory | 3. Basic Concepts
![Page 24: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/24.jpg)
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 ofvertices
L1: Introduction to Network Theory | 3. Basic Concepts
![Page 25: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/25.jpg)
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 ofvertices
L1: Introduction to Network Theory | 3. Basic Concepts
![Page 26: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/26.jpg)
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 ofvertices
L1: Introduction to Network Theory | 3. Basic Concepts
![Page 27: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/27.jpg)
Graphs and networks 13
A graph is the mathematical object formally defined aboveGraph
A network is the representation of a real-world system. Nodesand links have a specific meaning within the context of the appli-cation. Also, they have a ributes
Network
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
![Page 28: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/28.jpg)
Graphs and networks 13
A graph is the mathematical object formally defined aboveGraph
A network is the representation of a real-world system. Nodesand links have a specific meaning within the context of the appli-cation. Also, they have a ributes
Network
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
![Page 29: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/29.jpg)
Graphs and networks 13
A graph is the mathematical object formally defined aboveGraph
A network is the representation of a real-world system. Nodesand links have a specific meaning within the context of the appli-cation. Also, they have a ributes
Network
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
![Page 30: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/30.jpg)
Simplest graphs 14
Trivial graph has only one vertex
Null graph has no edges
L1: Introduction to Network Theory | 3. Basic Concepts
![Page 31: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/31.jpg)
Path 15
Path is an alternating sequence of nodes and edges, beginning ata node and ending at a node. Paths do not visit any point morethan once
H - F - C - A - Dis a path
L1: Introduction to Network Theory | 3. Basic Concepts
![Page 32: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/32.jpg)
Walk 16
Walk allows nodes to be visited more than once. Path is a specialcase of walk
H - F - C - A - F - Dis a walk
L1: Introduction to Network Theory | 3. Basic Concepts
![Page 33: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/33.jpg)
Cycle 17
Cycle is a path that starts and ends in the same edge. Cycle is aspecial case of walk
H - F - C - A - D - G - His a cycle
L1: Introduction to Network Theory | 3. Basic Concepts
![Page 34: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/34.jpg)
Connectivity 18
■ A node is reachable from another node if there exists a path ofany length from one node to another.
■ A graph is connected if there exists a path of any lengthbetween any pair of nodes.
■ A connected component is a subgraph, in which all nodes arereachable from every other.
L1: Introduction to Network Theory | 3. Basic Concepts
![Page 35: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/35.jpg)
Representation
L1: Introduction to Network Theory | 4. Representation
![Page 36: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/36.jpg)
Adjacency matrix 20
A = {aij}Ni ,j=1 =
{1 if there is an edge from i to j ,0 otherwise
(1)
L1: Introduction to Network Theory | 4. Representation
![Page 37: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/37.jpg)
Edgelist 21
Note that this edgelist must said to be undirected, otherwise it isnot full, and more edges must be added to the list, from target tosources.
L1: Introduction to Network Theory | 4. Representation
![Page 38: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/38.jpg)
Adjacency matrix vs. Edge list 22
Adjacency matrix Edge listMemory O(N2) O(E )Lookup specific edge Fast, O(1) SlowIterate over all edges Slow, O(N2) FastFind neighbours of a node Time O(N) Time O(E )Be er for Dense graphs Sparse graphsAdding new vertices Hard EasyAdding new edges O(1) O(1) or O(E )
L1: Introduction to Network Theory | 4. Representation
![Page 39: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/39.jpg)
Network types
L1: Introduction to Network Theory | 5. Network types
![Page 40: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/40.jpg)
Network types 24
1. By mode of nodes:1.1 One mode
1.2 Two nodes2. By direction of edges:
2.1 Directed2.2 Undirected
3. By weights of edges:3.1 Weighted
3.2 Unweighted
Any combination is possible!
L1: Introduction to Network Theory | 5. Network types
![Page 41: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/41.jpg)
Network types 24
1. By mode of nodes:1.1 One mode
1.2 Two nodes2. By direction of edges:
2.1 Directed2.2 Undirected
3. By weights of edges:3.1 Weighted
3.2 Unweighted
Any combination is possible!
L1: Introduction to Network Theory | 5. Network types
![Page 42: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/42.jpg)
Network types 24
1. By mode of nodes:1.1 One mode
1.2 Two nodes2. By direction of edges:
2.1 Directed2.2 Undirected
3. By weights of edges:3.1 Weighted
3.2 Unweighted
Any combination is possible!
L1: Introduction to Network Theory | 5. Network types
![Page 43: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/43.jpg)
Network types 24
1. By mode of nodes:1.1 One mode
1.2 Two nodes2. By direction of edges:
2.1 Directed2.2 Undirected
3. By weights of edges:3.1 Weighted
3.2 Unweighted
Any combination is possible!
L1: Introduction to Network Theory | 5. Network types
![Page 44: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/44.jpg)
Network types 24
1. By mode of nodes:1.1 One mode
1.2 Two nodes2. By direction of edges:
2.1 Directed2.2 Undirected
3. By weights of edges:3.1 Weighted
3.2 Unweighted
Any combination is possible!
L1: Introduction to Network Theory | 5. Network types
![Page 45: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/45.jpg)
Unipartite networks 25
Unipartite networks (one mode)■ All nodes are of the same nature;■ E.g.: Social networks, Internet,
WWW, Firms
Unipartite graph can also be:■ Undirected unweighted: aij ∈ {0, 1}, A is symmetric -
Simplification;■ Directed unweighted: aij ∈ {0, 1}, A is asymmetric - Followers;■ Undirected weighted: aij ∈ R, A is symmetric - Contact;■ Directed weighted: aij ∈ R, A is asymmetric - Economic
relations;
L1: Introduction to Network Theory | 5. Network types
![Page 46: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/46.jpg)
Unipartite networks 25
Unipartite networks (one mode)■ All nodes are of the same nature;■ E.g.: Social networks, Internet,
WWW, Firms
Unipartite graph can also be:■ Undirected unweighted: aij ∈ {0, 1}, A is symmetric -
Simplification;■ Directed unweighted: aij ∈ {0, 1}, A is asymmetric - Followers;■ Undirected weighted: aij ∈ R, A is symmetric - Contact;■ Directed weighted: aij ∈ R, A is asymmetric - Economic
relations;
L1: Introduction to Network Theory | 5. Network types
![Page 47: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/47.jpg)
Unipartite networks 25
Unipartite networks (one mode)■ All nodes are of the same nature;■ E.g.: Social networks, Internet,
WWW, Firms
Unipartite graph can also be:■ Undirected unweighted: aij ∈ {0, 1}, A is symmetric -
Simplification;■ Directed unweighted: aij ∈ {0, 1}, A is asymmetric - Followers;■ Undirected weighted: aij ∈ R, A is symmetric - Contact;■ Directed weighted: aij ∈ R, A is asymmetric - Economic
relations;
L1: Introduction to Network Theory | 5. Network types
![Page 48: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/48.jpg)
Unipartite networks 25
Unipartite networks (one mode)■ All nodes are of the same nature;■ E.g.: Social networks, Internet,
WWW, Firms
Unipartite graph can also be:■ Undirected unweighted: aij ∈ {0, 1}, A is symmetric -
Simplification;■ Directed unweighted: aij ∈ {0, 1}, A is asymmetric - Followers;■ Undirected weighted: aij ∈ R, A is symmetric - Contact;■ Directed weighted: aij ∈ R, A is asymmetric - Economic
relations;
L1: Introduction to Network Theory | 5. Network types
![Page 49: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/49.jpg)
Unipartite networks 25
Unipartite networks (one mode)■ All nodes are of the same nature;■ E.g.: Social networks, Internet,
WWW, Firms
Unipartite graph can also be:■ Undirected unweighted: aij ∈ {0, 1}, A is symmetric -
Simplification;■ Directed unweighted: aij ∈ {0, 1}, A is asymmetric - Followers;■ Undirected weighted: aij ∈ R, A is symmetric - Contact;■ Directed weighted: aij ∈ R, A is asymmetric - Economic
relations;
L1: Introduction to Network Theory | 5. Network types
![Page 50: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/50.jpg)
One-mode undirected unweighted 26
L1: Introduction to Network Theory | 5. Network types
![Page 51: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/51.jpg)
One-mode undirected unweighted 27■ All connections are mutual and of the same strength■ Adjacency matrix: symmetric, ∀i , j : aij ∈ {0, 1}■ e.g.: Friendship network of Facebook users
L1: Introduction to Network Theory | 5. Network types
![Page 52: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/52.jpg)
One-mode directed unweighted 28
L1: Introduction to Network Theory | 5. Network types
![Page 53: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/53.jpg)
One-mode directed unweighted 29■ Connections are not mutual, but of the same strength■ Adjacency matrix: non-symmetric, ∀i , j : aij ∈ {0, 1}■ e.g.: Follower network of Twi er users
h p://sites.davidson.eduL1: Introduction to Network Theory | 5. Network types
![Page 54: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/54.jpg)
One-mode undirected weighted 30
L1: Introduction to Network Theory | 5. Network types
![Page 55: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/55.jpg)
One-mode undirected weighted 31
■ All connections are mutual, but of different strength■ Adjacency matrix: symmetric, ∀i , j : aij ∈ R■ e.g.: Cooperation network between individuals in ICIC
(1919-1927)
h p://www.martingrandjean.ch/intellectual-cooperation-multi-level-network-analysis/
L1: Introduction to Network Theory | 5. Network types
![Page 56: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/56.jpg)
One-mode directed weighted 32
L1: Introduction to Network Theory | 5. Network types
![Page 57: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/57.jpg)
One-mode directed weighted 33
■ Connections are not mutual and of different strength■ Adjacency matrix: non-symmetric, ∀i , j : aij ∈ R■ e.g.: Affinity network of EU countries at Eurovision 2009-2012
L1: Introduction to Network Theory | 5. Network types
![Page 58: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/58.jpg)
One-mode directed weighted 33
■ Connections are not mutual and of different strength■ Adjacency matrix: non-symmetric, ∀i , j : aij ∈ R■ e.g.: Affinity network of EU countries at Eurovision 2009-2012
L1: Introduction to Network Theory | 5. Network types
![Page 59: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/59.jpg)
One-mode directed weighted 33
■ Connections are not mutual and of different strength■ Adjacency matrix: non-symmetric, ∀i , j : aij ∈ R■ e.g.: Affinity network of EU countries at Eurovision 2009-2012
L1: Introduction to Network Theory | 5. Network types
![Page 60: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/60.jpg)
Bipartite networks 34
Bipartite networks (two modes)■ Nodes are of two well-differentiated nature■ Node of one type can only be connected to a node of another
type;■ e.g.:
■ Recommender systems (product/user)■ Goods (buyer/product; buyer/seller; manufacturer/contractor)
Bipartite graph can also be:■ Unweighted: aij ∈ {0, 1}, A is
rectangular■ Weighted: aij ∈ R, A is rectangular;
L1: Introduction to Network Theory | 5. Network types
![Page 61: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/61.jpg)
Bipartite networks 34
Bipartite networks (two modes)■ Nodes are of two well-differentiated nature■ Node of one type can only be connected to a node of another
type;■ e.g.:
■ Recommender systems (product/user)■ Goods (buyer/product; buyer/seller; manufacturer/contractor)
Bipartite graph can also be:■ Unweighted: aij ∈ {0, 1}, A is
rectangular■ Weighted: aij ∈ R, A is rectangular;
L1: Introduction to Network Theory | 5. Network types
![Page 62: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/62.jpg)
Bipartite networks 34
Bipartite networks (two modes)■ Nodes are of two well-differentiated nature■ Node of one type can only be connected to a node of another
type;■ e.g.:
■ Recommender systems (product/user)■ Goods (buyer/product; buyer/seller; manufacturer/contractor)
Bipartite graph can also be:■ Unweighted: aij ∈ {0, 1}, A is
rectangular■ Weighted: aij ∈ R, A is rectangular;
L1: Introduction to Network Theory | 5. Network types
![Page 63: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/63.jpg)
Bipartite networks 34
Bipartite networks (two modes)■ Nodes are of two well-differentiated nature■ Node of one type can only be connected to a node of another
type;■ e.g.:
■ Recommender systems (product/user)■ Goods (buyer/product; buyer/seller; manufacturer/contractor)
Bipartite graph can also be:■ Unweighted: aij ∈ {0, 1}, A is
rectangular■ Weighted: aij ∈ R, A is rectangular;
L1: Introduction to Network Theory | 5. Network types
![Page 64: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/64.jpg)
Bipartite networks 34
Bipartite networks (two modes)■ Nodes are of two well-differentiated nature■ Node of one type can only be connected to a node of another
type;■ e.g.:
■ Recommender systems (product/user)■ Goods (buyer/product; buyer/seller; manufacturer/contractor)
Bipartite graph can also be:■ Unweighted: aij ∈ {0, 1}, A is
rectangular■ Weighted: aij ∈ R, A is rectangular;
L1: Introduction to Network Theory | 5. Network types
![Page 65: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/65.jpg)
Bipartite networks: example 35
A supermarket chain wants to know which products arefrequently bought together.
They have the following data:
L1: Introduction to Network Theory | 5. Network types
![Page 66: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/66.jpg)
Bipartite network: Nodes 36
L1: Introduction to Network Theory | 5. Network types
![Page 67: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/67.jpg)
Bipartite network: Edges 37
L1: Introduction to Network Theory | 5. Network types
![Page 68: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/68.jpg)
Bipartite networks: adjacency matrix 38■ Blue nodes - reciepts; Green nodes - products■ Edges exist only between nodes of different types.■ Adjacency matrix for bipartite networks: block-matrix;
L1: Introduction to Network Theory | 5. Network types
![Page 69: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/69.jpg)
Bipartite networks: edge list 39
■ Blue nodes - receipts; Green nodes - products■ Edges exist only between nodes of different types.
L1: Introduction to Network Theory | 5. Network types
![Page 70: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/70.jpg)
One mode projection 40
Link all products that were bought together on the same receipt
Consider receipt F first
L1: Introduction to Network Theory | 5. Network types
![Page 71: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/71.jpg)
One mode projection 41
Link all products that were bought together on the same receipt
Now consider receipt G
L1: Introduction to Network Theory | 5. Network types
![Page 72: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/72.jpg)
One mode projection 42
Link all products that were bought together on the same receipt
Finally, consider receipt I
L1: Introduction to Network Theory | 5. Network types
![Page 73: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/73.jpg)
One mode projection 43
Resulting graph is unipartite, undirected, unweighted
L1: Introduction to Network Theory | 5. Network types
![Page 74: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/74.jpg)
Network of ingredients 44Network of ingredients that occur together more than by chance:
Teng, Lin, & Adamic (2011)
L1: Introduction to Network Theory | 5. Network types
![Page 75: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/75.jpg)
Simple network models
L1: Introduction to Network Theory | 6. Simple network models
![Page 76: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/76.jpg)
What are network models? 46
■ A model is an abstract, idealised description of reality that stillcaptures a specific trait
■ Network models are constructed to represent complexsystems: social, physical, information, etc.
■ In this course, we focus on network models of complexsocio-economic systems
L1: Introduction to Network Theory | 6. Simple network models
![Page 77: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/77.jpg)
What are network models? 46
■ A model is an abstract, idealised description of reality that stillcaptures a specific trait
■ Network models are constructed to represent complexsystems: social, physical, information, etc.
■ In this course, we focus on network models of complexsocio-economic systems
L1: Introduction to Network Theory | 6. Simple network models
![Page 78: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/78.jpg)
What are network models? 46
■ A model is an abstract, idealised description of reality that stillcaptures a specific trait
■ Network models are constructed to represent complexsystems: social, physical, information, etc.
■ In this course, we focus on network models of complexsocio-economic systems
L1: Introduction to Network Theory | 6. Simple network models
![Page 79: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/79.jpg)
Simple network types 47
Fully connected network
■ All-to-all, well-mixedpopulation;
■ Amenable for analyticalcalculations;
■ In most situations: artificial;■ ki = N − 1■ Diameter: 1
L1: Introduction to Network Theory | 6. Simple network models
![Page 80: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/80.jpg)
Simple network types 48
Star network
■ Extremely centralised;■ Can represent topology of
computer network(client-server)
■ k0 = N − 1, ki = 1∀i > 0■ Diameter: 2
L1: Introduction to Network Theory | 6. Simple network models
![Page 81: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/81.jpg)
Regular networks 49
One dimensional la ice
■ Traffic lanes;■ ki = 2κ
■ Diameter: ∝ N
L1: Introduction to Network Theory | 6. Simple network models
![Page 82: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/82.jpg)
Regular networks 50
Bi-dimensional la ice
■ Geographical data■ ki = 4κ
■ Diameter: ∝ N1/2
L1: Introduction to Network Theory | 6. Simple network models
![Page 83: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/83.jpg)
Why these models are important? 51
■ These models represent some real-world structures (computernetworks, geographical data, traffic lanes);
■ Can be used for analysis and modelling of the networks■ Estimation of: connectivity, average (or maximum) load on lanes
or server, etc.■ Can be used for prediction of future behavior;
L1: Introduction to Network Theory | 6. Simple network models
![Page 84: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/84.jpg)
References I 52
▶ Chin-Yuen Teng, Yu-Ru Lin, Lada A. Adamic, Reciperecommendation using ingredient networks, arXiv preprint:arXiv:1111.3919, 2012.
L1: Introduction to Network Theory | 6. Simple network models
![Page 85: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/85.jpg)
Manuel Sebastian Mariani
URPP Social Networks
m h p://www.socialnetworks.uzh.ch
L1: Introduction to Network Theory | 6. Simple network models
![Page 86: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/86.jpg)
Exercise
L1: Introduction to Network Theory | 7. Exercise
![Page 87: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/87.jpg)
Degree distribution 55
■ Download one unipartite unweighted network fromhttp://snap.stanford.edu/data/index.html, ideally composed of∼ 1000 to 10, 000 nodes.
■ Describe the meaning of the nodes and the edges.■ Analyze the network with a network-analysis package, using
your favorite programming language.■ Recommended: igraph, networkx.
L1: Introduction to Network Theory | 7. Exercise
![Page 88: IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally defined above Graph](https://reader033.fdocuments.net/reader033/viewer/2022042918/5f5c97130d6fc602c976879f/html5/thumbnails/88.jpg)
Degree distribution 56
■ Plot the selected network’s degree distribution P(k). Is itbe er to plot it on a linear scale, or on a log-log scale? Discuss.
■ Compare with the expectation for a random graph:
PER(k) = N pk (1 − p)N−k−1.
(Find the normalization factor N .)■ Are the observed and expected distribution similar? Discuss
the meaning of the result.
L1: Introduction to Network Theory | 7. Exercise