Introduction to Network TheoryLecture 1
Manuel Sebastian MarianiURPP 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
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
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
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
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
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
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
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
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
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
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
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
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 ofvertices
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 ofvertices
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 ofvertices
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 ofvertices
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 ofvertices
L1: Introduction to Network Theory | 3. Basic Concepts
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
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
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
Simplest graphs 14
Trivial graph has only one vertex
Null graph has no edges
L1: Introduction to Network Theory | 3. Basic Concepts
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
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
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
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
Representation
L1: Introduction to Network Theory | 4. Representation
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
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
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
Network types
L1: Introduction to Network Theory | 5. Network types
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
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
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
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
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
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
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
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
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
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
One-mode undirected unweighted 26
L1: Introduction to Network Theory | 5. Network types
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
One-mode directed unweighted 28
L1: Introduction to Network Theory | 5. Network types
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
One-mode undirected weighted 30
L1: Introduction to Network Theory | 5. Network types
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
One-mode directed weighted 32
L1: Introduction to Network Theory | 5. Network types
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
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
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
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
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
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
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
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
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
Bipartite network: Nodes 36
L1: Introduction to Network Theory | 5. Network types
Bipartite network: Edges 37
L1: Introduction to Network Theory | 5. Network types
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
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
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
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
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
One mode projection 43
Resulting graph is unipartite, undirected, unweighted
L1: Introduction to Network Theory | 5. Network types
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
Simple network models
L1: Introduction to Network Theory | 6. Simple network models
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
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
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
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
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
Regular networks 49
One dimensional la ice
■ Traffic lanes;■ ki = 2κ
■ Diameter: ∝ N
L1: Introduction to Network Theory | 6. Simple network models
Regular networks 50
Bi-dimensional la ice
■ Geographical data■ ki = 4κ
■ Diameter: ∝ N1/2
L1: Introduction to Network Theory | 6. Simple network models
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
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
Manuel Sebastian Mariani
URPP Social Networks
m h p://www.socialnetworks.uzh.ch
L1: Introduction to Network Theory | 6. Simple network models
Exercise
L1: Introduction to Network Theory | 7. Exercise
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
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
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