IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally deﬁned above Graph

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

















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


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


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■ …


















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)


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)


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)


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)


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


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


Graph 12






Graph 12






Graph 12






Graph 12






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)





Network






Network



Simplest graphs 14

Trivial graph has only one vertex

Null graph has no edges


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


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


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


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.


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)


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.


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 )


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!


Network types 24





3.2 Unweighted



Network types 24





3.2 Unweighted



Network types 24





3.2 Unweighted



Network types 24





3.2 Unweighted



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;




WWW, Firms



relations;




WWW, Firms



relations;




WWW, Firms



relations;




WWW, Firms



relations;


One-mode undirected unweighted 26


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


One-mode directed unweighted 28


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


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/


One-mode directed weighted 32



■ 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








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;




type;■ e.g.:







type;■ e.g.:







type;■ e.g.:







type;■ e.g.:





Bipartite networks: example 35

A supermarket chain wants to know which products arefrequently bought together.

They have the following data:


Bipartite network: Nodes 36


Bipartite network: Edges 37


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;


Bipartite networks: edge list 39

■ Blue nodes - receipts; Green nodes - products■ Edges exist only between nodes of different types.


One mode projection 40

Link all products that were bought together on the same receipt

Consider receipt F first




Now consider receipt G




Finally, consider receipt I



Resulting graph is unipartite, undirected, unweighted


Network of ingredients 44Network of ingredients that occur together more than by chance:

Teng, Lin, & Adamic (2011)


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












Simple network types 47

Fully connected network

■ All-to-all, well-mixedpopulation;

■ Amenable for analyticalcalculations;

■ In most situations: artificial;■ ki = N − 1■ Diameter: 1


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


Regular networks 49

One dimensional la ice

■ Traffic lanes;■ ki = 2κ

■ Diameter: ∝ N


Regular networks 50

Bi-dimensional la ice

■ Geographical data■ ki = 4κ

■ Diameter: ∝ N1/2


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;


References I 52

▶ Chin-Yuen Teng, Yu-Ru Lin, Lada A. Adamic, Reciperecommendation using ingredient networks, arXiv preprint:arXiv:1111.3919, 2012.


Manuel Sebastian Mariani

URPP Social Networks

B [email protected]

m h p://www.socialnetworks.uzh.ch


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.


http://snap.stanford.edu/data/index.html

igraph

networkx

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.


Manuel Sebastian Mariani

URPP Social Networks

B [email protected]

m h p://www.socialnetworks.uzh.ch

Documents

IntroductiontoNetworkTheory - UZH · L1: Introduction to Network Theory | 3. Basic Concepts. Graphsandnetworks 13 A graph is the mathematical object formally deﬁned above Graph