Graph mining - lesson 1 Introduction to graphs and networks - … · 2020. 5. 7. · A brief...
Transcript of Graph mining - lesson 1 Introduction to graphs and networks - … · 2020. 5. 7. · A brief...
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Graph mining - lesson 1Introduction to graphs and networks
Nathalie Vialaneix
[email protected]://www.nathalievialaneix.eu
M2 Statistics & EconometricsJanuary 8th, 2020
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A brief overview for this class...
Who am I? Statistician working in biostatistics at INRAE ToulouseMy research interests are: data mining, network inference andmining, machine learning
Purpose of this talk: presenting a few statistical tools for graphmining (graph structure, important vertices) and clustering
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Outline
A brief introduction to networks/graphs
Visualization
Global characteristics
Numerical characteristics calculation
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What is a network/graph?Mathematical object used to model relational data betweenentities.
A relation between two entities is modeled by an edge
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What is a network/graph?Mathematical object used to model relational data betweenentities.
The entities are called the nodes or the vertices
Arelation between two entities is modeled by an edge
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What is a network/graph?Mathematical object used to model relational data betweenentities.
A relation between two entities is modeled by an edge
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Where does graph theory come from?Seven Bridges of Königsberg: notable problem in mathematics.Königsberg set on both sides of the Pregel River and included twolarge islands.Question: Is there a walk through the city that crosses each bridgeonce and only once?
Image: Public Domain. CC BY-SA 3.0.
Leonhard Euler proved that the problem has no solution using amathematical proof which was the starting point of graph theory.
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Where does graph theory come from?Seven Bridges of Königsberg: notable problem in mathematics.Königsberg set on both sides of the Pregel River and included twolarge islands.Question: Is there a walk through the city that crosses each bridgeonce and only once?
Image: Public Domain. CC BY-SA 3.0.
Leonhard Euler proved that the problem has no solution using amathematical proof which was the starting point of graph theory.
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Examples of networks
Social networks:
Credits: Frauhoelle (CC BY-SA 2.0) and Caseorganic (CC BY-NC 2.0) on flickr
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Examples of networks
Blog co-citations and internet routes:
Credits: Porternovelli on flikr (CC BY-SA 2.0) and Matt Britt on wikimedia commons (CC BY 2.5)
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Examples of networks
Consumers/products graphs or co-purchase networks:
Credits: Loop© and http://www.annehelmond.nl
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Examples of networks(biological) Neural networks:
Credits: Soon-Beom et al. (CC BY-SA 3.0) and Andreashorn (CC BY-SA 4.0) on wikimedia commonsNathalie Vialaneix | Graph mining 6/30
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Examples of networksIngredient networks... and many others!
Credits: http://www.ladamic.com
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More complex relational models
Vertices...
can be labelled with a factor or a numeric variables or severalvariables (caracteristics attached to the entities in relation)
Edges...
I can be orientedI can be weightedI can be described by numerical attributes or factors
(caracteristics attached to the relation)
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More complex relational models
Vertices...
can be labelled with a factor or a numeric variables or severalvariables (caracteristics attached to the entities in relation)
Edges...
I can be orientedI can be weightedI can be described by numerical attributes or factors
(caracteristics attached to the relation)
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Many applications...
I Viral marketing: find a way to efficiently spread the informationabout a new product using social network informations
I Recommandation systems: recommand a product tosomeone based on his/her previous purchase andco-purchase information
I Biological network: acquire knowledge about biologicalnetworks (genes, metabolomic pathway...) in order tounderstand diseases associated with disfunctionning
I ...
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Standard issues associated with networksInference
Giving data, how to build a graph whose edges represent the directlinks between variables?Example: co-expression networks built from microarray data (vertices = genes;edges = significant “direct links” between expressions of two genes)
Graph mining (examples)
1. Network visualization: vertices are not a priori associated to agiven position. How to represent the network in a meaningfulway?
2. Network clustering: identify “communities” (groups of vertices that
are densely connected and share a few links with the other groups)
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Standard issues associated with networksInference
Giving data, how to build a graph whose edges represent the directlinks between variables?
Graph mining (examples)
1. Network visualization: vertices are not a priori associated to agiven position. How to represent the network in a meaningfulway?
Random positions or positions aiming at representingconnected vertices closer.
2. Network clustering: identify “communities” (groups of vertices that
are densely connected and share a few links with the other groups)
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Standard issues associated with networksInference
Giving data, how to build a graph whose edges represent the directlinks between variables?
Graph mining (examples)
1. Network visualization: vertices are not a priori associated to agiven position. How to represent the network in a meaningfulway?
2. Network clustering: identify “communities” (groups of vertices that
are densely connected and share a few links with the other groups)
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Notations for this class
Notations
In the following, a graph G = (V ,E,W) with:I V : set of vertices {x1, . . . , xn};I E: set of (undirected) edges. m = |E |;I W : weights on edges s.t. Wij ≥ 0, Wij = Wji and Wii = 0.
If needed, attributes for the vertices will be denoted by fj(xi) (jthattribute for vertex i) and attributes for the edges (other than theweights) by gj(xi , xi′) (jth attribute for the edge (xi , xi′)).
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Notations for this class
Notations
In the following, a graph G = (V ,E,W) with:I V : set of vertices {x1, . . . , xn};I E: set of (undirected) edges. m = |E |;I W : weights on edges s.t. Wij ≥ 0, Wij = Wji and Wii = 0.
If needed, attributes for the vertices will be denoted by fj(xi) (jthattribute for vertex i) and attributes for the edges (other than theweights) by gj(xi , xi′) (jth attribute for the edge (xi , xi′)).
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Online graph datasets and ressources
I Mark Newman’s collection:http://www-personal.umich.edu/~mejn/netdata
I Stanford Large Network Dataset Collection (SNAP):http://snap.stanford.edu/data
I KONECT collection (Koblenz university):http://konect.uni-koblenz.de/networks
I Colorado Index of Complex Networks (ICON):https://icon.colorado.edu
Online course: http://barabasi.com/networksciencebook(Alberto Barabasi)
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Mining graphs/networks
I Visualizing and manipulating graphs in an interactive way:Gephi https://gephi.org, Tulip http://tulip.labri.fror Cytoscape http://cytoscape.org;
I Packages/librairies in data mining languages:I for Python: igraph, NetworkX and graph-toolI for R: igraph, statnet, bipartite and tnet. See also the CRAN
task view:https://cran.r-project.org/web/views/gR.html(graphical models)
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Special graphsFull graphs
A full graph with verticesV = {x1, . . . , xn} is thegraph with edge listE = {(xi , xj) : xi , xj ∈ V}
1
2
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Bipartite graphsA graph with verticesV = {x1, . . . , xn} partitionned intotwo groups {xi : f(xi) = 1} and{xi : f(xi) = −1} and such thatedges are a subset of {(xi , xj) :f(xi) = 1 and f(xj) = −1} (e.g.,purchase network)
−1
1
−1
1
−1
1
−1
1
−1
1
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Special graphsFull graphs
A full graph with verticesV = {x1, . . . , xn} is thegraph with edge listE = {(xi , xj) : xi , xj ∈ V}
1
2
34
5
6
Bipartite graphsA graph with verticesV = {x1, . . . , xn} partitionned intotwo groups {xi : f(xi) = 1} and{xi : f(xi) = −1} and such thatedges are a subset of {(xi , xj) :f(xi) = 1 and f(xj) = −1} (e.g.,purchase network)
−1
1
−1
1
−1
1
−1
1
−1
1
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Most standard ways to record a graph
I adjacency matrix: matrix W if the network is weighted or
Aij =
{1 if (xi , xj) ∈ E0 otherwise
if it is unweighted.
requires to store n2 values
I edge list: matrix B of dimension m × 2 (unweighted network)or m×3 (weighted network), Bk . = (xi , xj ,Wij) for a (xi , xj) ∈ E.requires to store 3m values
Since usually m � n2, the second solution is often prefered.
Other standard formats (readable by interactive software andallowing metadata) such as graphml (a graph version of XML)http://graphml.graphdrawing.org
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Most standard ways to record a graph
I adjacency matrix: matrix W if the network is weighted or
Aij =
{1 if (xi , xj) ∈ E0 otherwise
if it is unweighted.
requires to store n2 values
I edge list: matrix B of dimension m × 2 (unweighted network)or m×3 (weighted network), Bk . = (xi , xj ,Wij) for a (xi , xj) ∈ E.requires to store 3m values
Since usually m � n2, the second solution is often prefered.
Other standard formats (readable by interactive software andallowing metadata) such as graphml (a graph version of XML)http://graphml.graphdrawing.org
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Most standard ways to record a graph
I adjacency matrix: matrix W if the network is weighted or
Aij =
{1 if (xi , xj) ∈ E0 otherwise
if it is unweighted.
requires to store n2 values
I edge list: matrix B of dimension m × 2 (unweighted network)or m×3 (weighted network), Bk . = (xi , xj ,Wij) for a (xi , xj) ∈ E.requires to store 3m values
Since usually m � n2, the second solution is often prefered.
Other standard formats (readable by interactive software andallowing metadata) such as graphml (a graph version of XML)http://graphml.graphdrawing.org
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Most standard ways to record a graph
I adjacency matrix: matrix W if the network is weighted or
Aij =
{1 if (xi , xj) ∈ E0 otherwise
if it is unweighted.
requires to store n2 values
I edge list: matrix B of dimension m × 2 (unweighted network)or m×3 (weighted network), Bk . = (xi , xj ,Wij) for a (xi , xj) ∈ E.requires to store 3m values
Since usually m � n2, the second solution is often prefered.
Other standard formats (readable by interactive software andallowing metadata) such as graphml (a graph version of XML)http://graphml.graphdrawing.org
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Running example 1 “GOT”“Game of Thrones” coappearances network: weighted andundirected network with 107 vertices corresponding to uniquecharacters and 353 edges weighted by the number of times thetwo characters’ names appeared within 15 words of each other inthe Game of Thrones series by George R.R. Martin.
Reference: [Beveridge and Shan, 2016]
http://www.jstor.org/stable/10.4169Dataset available at: http://www.macalester.edu/~abeverid/data/stormofswords.csv(edgelist format)
Aemon
GrennSamwell
AerysJaime
Robert
Tyrion
Tywin
AlliserMance
Amory
Oberyn
AryaAnguy
BericBran
Brynden Cersei
Gendry
GregorJoffrey
Jon
Rickon
Roose
Sandor
ThorosBalon
Loras
BelwasBarristan
Illyrio
HodorJojen
LuwinMeera
Nan Theon
Brienne
BronnPodrickLothar
WalderCatelynEdmure
Hoster
Jeyne
LysaPetyr
Robb
Roslin
Sansa
Stannis
Elia
IlynMerynPycelle
Shae
Varys
Craster
Karl
DaarioDrogo
IrriDaenerys
AegonJorah
Kraznys
MissandeiRakharoRhaegar
Viserys
Worm
Davos
Cressen
Salladhor
Eddard
Eddison
Gilly
Qyburn
Renly
Tommen
Janos
Bowen
Kevan
Margaery
Myrcella
Dalla
Melisandre
Orell
QhorinRattleshirtStyr
ValYgritte
Jon Arryn
Lancel
OlennaMarillion
Robert ArrynEllaria
MaceRickard
Ramsay
Chataya
Shireen
Doran
Walton
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Running example 1 “GOT”“Game of Thrones” coappearances network: weighted andundirected network with 107 vertices corresponding to uniquecharacters and 353 edges weighted by the number of times thetwo characters’ names appeared within 15 words of each other inthe Game of Thrones series by George R.R. Martin.Reference: [Beveridge and Shan, 2016]
http://www.jstor.org/stable/10.4169
Dataset available at: http://www.macalester.edu/~abeverid/data/stormofswords.csv(edgelist format)
Aemon
GrennSamwell
AerysJaime
Robert
Tyrion
Tywin
AlliserMance
Amory
Oberyn
AryaAnguy
BericBran
Brynden Cersei
Gendry
GregorJoffrey
Jon
Rickon
Roose
Sandor
ThorosBalon
Loras
BelwasBarristan
Illyrio
HodorJojen
LuwinMeera
Nan Theon
Brienne
BronnPodrickLothar
WalderCatelynEdmure
Hoster
Jeyne
LysaPetyr
Robb
Roslin
Sansa
Stannis
Elia
IlynMerynPycelle
Shae
Varys
Craster
Karl
DaarioDrogo
IrriDaenerys
AegonJorah
Kraznys
MissandeiRakharoRhaegar
Viserys
Worm
Davos
Cressen
Salladhor
Eddard
Eddison
Gilly
Qyburn
Renly
Tommen
Janos
Bowen
Kevan
Margaery
Myrcella
Dalla
Melisandre
Orell
QhorinRattleshirtStyr
ValYgritte
Jon Arryn
Lancel
OlennaMarillion
Robert ArrynEllaria
MaceRickard
Ramsay
Chataya
Shireen
Doran
Walton
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Running example 1 “GOT”“Game of Thrones” coappearances network: weighted andundirected network with 107 vertices corresponding to uniquecharacters and 353 edges weighted by the number of times thetwo characters’ names appeared within 15 words of each other inthe Game of Thrones series by George R.R. Martin.Reference: [Beveridge and Shan, 2016]
http://www.jstor.org/stable/10.4169Dataset available at: http://www.macalester.edu/~abeverid/data/stormofswords.csv(edgelist format)
Aemon
GrennSamwell
AerysJaime
Robert
Tyrion
Tywin
AlliserMance
Amory
Oberyn
AryaAnguy
BericBran
Brynden Cersei
Gendry
GregorJoffrey
Jon
Rickon
Roose
Sandor
ThorosBalon
Loras
BelwasBarristan
Illyrio
HodorJojen
LuwinMeera
Nan Theon
Brienne
BronnPodrickLothar
WalderCatelynEdmure
Hoster
Jeyne
LysaPetyr
Robb
Roslin
Sansa
Stannis
Elia
IlynMerynPycelle
Shae
Varys
Craster
Karl
DaarioDrogo
IrriDaenerys
AegonJorah
Kraznys
MissandeiRakharoRhaegar
Viserys
Worm
Davos
Cressen
Salladhor
Eddard
Eddison
Gilly
Qyburn
Renly
Tommen
Janos
Bowen
Kevan
Margaery
Myrcella
Dalla
Melisandre
Orell
QhorinRattleshirtStyr
ValYgritte
Jon Arryn
Lancel
OlennaMarillion
Robert ArrynEllaria
MaceRickard
Ramsay
Chataya
Shireen
Doran
Walton
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Running example 1 “GOT”“Game of Thrones” coappearances network
Aemon
GrennSamwell
AerysJaime
Robert
Tyrion
Tywin
AlliserMance
Amory
Oberyn
AryaAnguy
BericBran
Brynden Cersei
Gendry
GregorJoffrey
Jon
Rickon
Roose
Sandor
ThorosBalon
Loras
BelwasBarristan
Illyrio
HodorJojen
LuwinMeera
Nan Theon
Brienne
BronnPodrickLothar
WalderCatelynEdmure
Hoster
Jeyne
LysaPetyr
Robb
Roslin
Sansa
Stannis
Elia
IlynMerynPycelle
Shae
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Jon Arryn
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Robert ArrynEllaria
MaceRickard
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Shireen
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Running example 2 “NVV”my facebook network (extracted from facebook in 2015) with 152vertices (my friends on facebook) and 551 edges (mutualfriendship between my friends)
Dataset available at: http://www.nathalievialaneix.eu/doc/txt/fbnet-el-2015.txt(edge list) and http://www.nathalievialaneix.eu/doc/txt/fbnet-name-2015.txt (metadata -initials- for the vertices)
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Running example 2 “NVV”my facebook network (extracted from facebook in 2015) with 152vertices (my friends on facebook) and 551 edges (mutualfriendship between my friends)Dataset available at: http://www.nathalievialaneix.eu/doc/txt/fbnet-el-2015.txt(edge list) and http://www.nathalievialaneix.eu/doc/txt/fbnet-name-2015.txt (metadata -initials- for the vertices)
J.M
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Running example 2 “NVV”my facebook network (extracted from facebook in 2015)
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Running example 3 “FB”Amherst College https://www.amherst.edu facebook network :Snapshots of within-college social networks of the first 100colleges and universities admitted to thefacebook.com, inSeptember 2005. Vertices are annotated with metadata giving thetype of account (student, faculty, alumni, etc.), dorm, major,gender, and graduation year (2,235 vertices and 90,954 edges).
Reference: [Traud et al., 2012] http://arxiv.org/abs/1102.2166Dataset available at:https://escience.rpi.edu/data/DA/fb100 (Matlab© format;adjacency matrix + data frame with information on vertices: 7columns)
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Running example 3 “FB”Amherst College https://www.amherst.edu facebook network :Snapshots of within-college social networks of the first 100colleges and universities admitted to thefacebook.com, inSeptember 2005. Vertices are annotated with metadata giving thetype of account (student, faculty, alumni, etc.), dorm, major,gender, and graduation year (2,235 vertices and 90,954 edges).Reference: [Traud et al., 2012] http://arxiv.org/abs/1102.2166
Dataset available at:https://escience.rpi.edu/data/DA/fb100 (Matlab© format;adjacency matrix + data frame with information on vertices: 7columns)
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Running example 3 “FB”Amherst College https://www.amherst.edu facebook network :Snapshots of within-college social networks of the first 100colleges and universities admitted to thefacebook.com, inSeptember 2005. Vertices are annotated with metadata giving thetype of account (student, faculty, alumni, etc.), dorm, major,gender, and graduation year (2,235 vertices and 90,954 edges).Reference: [Traud et al., 2012] http://arxiv.org/abs/1102.2166Dataset available at:https://escience.rpi.edu/data/DA/fb100 (Matlab© format;adjacency matrix + data frame with information on vertices: 7columns)
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Running example 3 “FB”Amherst College https://www.amherst.edu facebook network
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Connected componentThe graph is said to be connected if any vertex can be reachedfrom any other vertex by a path along the edges.The connected components of a graph are all its connectedsubgraphs with maximum sizes.
Examples: GOT and FB are connected graphs. NVV is notconnected and contains 21 connected components, among whichthe largest has 122 vertices.
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Connected componentThe graph is said to be connected if any vertex can be reachedfrom any other vertex by a path along the edges.The connected components of a graph are all its connectedsubgraphs with maximum sizes.Examples: GOT and FB are connected graphs. NVV is notconnected and contains 21 connected components, among whichthe largest has 122 vertices.
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Outline
A brief introduction to networks/graphs
Visualization
Global characteristics
Numerical characteristics calculation
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Visualization tools help understand the graphmacro-structure
Purpose: How to display the vertices in a meaningful and aestheticway?
Standard approach: force directed placement algorithms (FDP)(e.g., [Fruchterman and Reingold, 1991])
I attractive forces: similar to springs along the edgesI repulsive forces: similar to electric forces between all pairs of
vertices
iterative algorithm until stabilization of the vertex positions.
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Visualization tools help understand the graphmacro-structure
Purpose: How to display the vertices in a meaningful and aestheticway?Standard approach: force directed placement algorithms (FDP)(e.g., [Fruchterman and Reingold, 1991])
I attractive forces: similar to springs along the edgesI repulsive forces: similar to electric forces between all pairs of
vertices
iterative algorithm until stabilization of the vertex positions.
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Visualization tools help understand the graphmacro-structure
Purpose: How to display the vertices in a meaningful and aestheticway?Standard approach: force directed placement algorithms (FDP)(e.g., [Fruchterman and Reingold, 1991])
I attractive forces: similar to springs along the edges
I repulsive forces: similar to electric forces between all pairs ofvertices
iterative algorithm until stabilization of the vertex positions.
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Visualization tools help understand the graphmacro-structure
Purpose: How to display the vertices in a meaningful and aestheticway?Standard approach: force directed placement algorithms (FDP)(e.g., [Fruchterman and Reingold, 1991])
I attractive forces: similar to springs along the edgesI repulsive forces: similar to electric forces between all pairs of
vertices
iterative algorithm until stabilization of the vertex positions.
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Visualization tools help understand the graphmacro-structure
Purpose: How to display the vertices in a meaningful and aestheticway?Standard approach: force directed placement algorithms (FDP)(e.g., [Fruchterman and Reingold, 1991])
I attractive forces: similar to springs along the edgesI repulsive forces: similar to electric forces between all pairs of
vertices
iterative algorithm until stabilization of the vertex positions.
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Visualization softwareI package igraph1 [Csardi and Nepusz, 2006] (static
representation with useful tools for graph mining)
I free software Gephi2 (interactive software, supportszooming and panning)
1http://igraph.sourceforge.net/
2http://gephi.org
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Visualization softwareI package igraph1 [Csardi and Nepusz, 2006] (static
representation with useful tools for graph mining)
I free software Gephi2 (interactive software, supportszooming and panning)
1http://igraph.sourceforge.net/
2http://gephi.org
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Outline
A brief introduction to networks/graphs
Visualization
Global characteristics
Numerical characteristics calculation
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Density / TransitivityDensity: Number of edges divided by the number of pairs ofvertices. Is the network densely connected?
Transitivity (sometimes called clustering coefficient): Number oftriangles divided by the number of triplets connected by at leasttwo edges. What is the probability that two people with a commonfriend are also friends?
Examples
Example 1: GOTI density ' 6.2%, transitivity ' 32.9%.
Example 2: NVVI density ' 4.8%, transitivity ' 56.2%;I LCC: density ' 7.2%, transitivity ' 56%.
Example 3: FBI density ' 3.6%, transitivity ' 23.3%.
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Density / TransitivityDensity: Number of edges divided by the number of pairs ofvertices. Is the network densely connected?
Examples
Example 1: GOTI 107 vertices, 352 edges⇒ density = 352
107×106/2 ' 6.2%.
Example 2: NVVI 152 vertices, 551 edges⇒ density ' 4.8%;I largest connected component: 122 vertices, 535 edges⇒
density ' 7.2%.
Example 3: FBI 2235 vertices, 9.0954 × 104 edges⇒ density ' 3.6%.
Transitivity (sometimes called clustering coefficient): Number oftriangles divided by the number of triplets connected by at leasttwo edges. What is the probability that two people with a commonfriend are also friends?
Examples
Example 1: GOTI density ' 6.2%, transitivity ' 32.9%.
Example 2: NVVI density ' 4.8%, transitivity ' 56.2%;I LCC: density ' 7.2%, transitivity ' 56%.
Example 3: FBI density ' 3.6%, transitivity ' 23.3%.
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Density / TransitivityDensity: Number of edges divided by the number of pairs ofvertices. Is the network densely connected?Transitivity (sometimes called clustering coefficient): Number oftriangles divided by the number of triplets connected by at leasttwo edges. What is the probability that two people with a commonfriend are also friends?
Density is equal to 44×3/2 = 2/3 ; Transitivity is equal to 1/3.
Examples
Example 1: GOTI density ' 6.2%, transitivity ' 32.9%.
Example 2: NVVI density ' 4.8%, transitivity ' 56.2%;I LCC: density ' 7.2%, transitivity ' 56%.
Example 3: FBI density ' 3.6%, transitivity ' 23.3%.
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Density / TransitivityDensity: Number of edges divided by the number of pairs ofvertices. Is the network densely connected?Transitivity (sometimes called clustering coefficient): Number oftriangles divided by the number of triplets connected by at leasttwo edges. What is the probability that two people with a commonfriend are also friends?
Examples
Example 1: GOTI density ' 6.2%, transitivity ' 32.9%.
Example 2: NVVI density ' 4.8%, transitivity ' 56.2%;I LCC: density ' 7.2%, transitivity ' 56%.
Example 3: FBI density ' 3.6%, transitivity ' 23.3%.
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Density / TransitivityDensity: Number of edges divided by the number of pairs ofvertices. Is the network densely connected?Transitivity (sometimes called clustering coefficient): Number oftriangles divided by the number of triplets connected by at leasttwo edges. What is the probability that two people with a commonfriend are also friends?
Examples
Example 1: GOTI density ' 6.2%, transitivity ' 32.9%.
Example 2: NVVI density ' 4.8%, transitivity ' 56.2%;I LCC: density ' 7.2%, transitivity ' 56%.
Example 3: FBI density ' 3.6%, transitivity ' 23.3%.
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Density / TransitivityDensity: Number of edges divided by the number of pairs ofvertices. Is the network densely connected?Transitivity (sometimes called clustering coefficient): Number oftriangles divided by the number of triplets connected by at leasttwo edges. What is the probability that two people with a commonfriend are also friends?
Examples
Example 1: GOTI density ' 6.2%, transitivity ' 32.9%.
Example 2: NVVI density ' 4.8%, transitivity ' 56.2%;I LCC: density ' 7.2%, transitivity ' 56%.
Example 3: FBI density ' 3.6%, transitivity ' 23.3%.
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Density / TransitivityDensity: Number of edges divided by the number of pairs ofvertices. Is the network densely connected?Transitivity (sometimes called clustering coefficient): Number oftriangles divided by the number of triplets connected by at leasttwo edges. What is the probability that two people with a commonfriend are also friends?
Examples
Example 1: GOTI density ' 6.2%, transitivity ' 32.9%.
Example 2: NVVI density ' 4.8%, transitivity ' 56.2%;I LCC: density ' 7.2%, transitivity ' 56%.
Example 3: FBI density ' 3.6%, transitivity ' 23.3%.
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Diameter and 6 degrees of separationDiameter (of a connected graph): length of the longest shortestpath between two vertices in the graph.
0e+00
5e+05
1e+06
2 4 6
shortest path length
coun
t
shortest path length distribution − FB
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Diameter and 6 degrees of separationDiameter (of a connected graph): length of the longest shortestpath between two vertices in the graph.
The diameter of this graph is 2.
0e+00
5e+05
1e+06
2 4 6
shortest path length
coun
t
shortest path length distribution − FB
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Diameter and 6 degrees of separationDiameter (of a connected graph): length of the longest shortestpath between two vertices in the graph.
The diameter of this graph is 2.
0e+00
5e+05
1e+06
2 4 6
shortest path length
coun
t
shortest path length distribution − FB
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Diameter and 6 degrees of separationDiameter (of a connected graph): length of the longest shortestpath between two vertices in the graph.
The diameter of this graph is 2.
0e+00
5e+05
1e+06
2 4 6
shortest path length
coun
t
shortest path length distribution − FB
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Diameter and 6 degrees of separationDiameter (of a connected graph): length of the longest shortestpath between two vertices in the graph.
The diameter of this graph is 2.
0e+00
5e+05
1e+06
2 4 6
shortest path length
coun
t
shortest path length distribution − FB
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Diameter and 6 degrees of separationDiameter (of a connected graph): length of the longest shortestpath between two vertices in the graph.
The diameter of this graph is 2.
0e+00
5e+05
1e+06
2 4 6
shortest path length
coun
t
shortest path length distribution − FB
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Diameter and 6 degrees of separationDiameter (of a connected graph): length of the longest shortestpath between two vertices in the graph.
The diameter of this graph is 2.
0e+00
5e+05
1e+06
2 4 6
shortest path length
coun
t
shortest path length distribution − FB
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Diameter and 6 degrees of separationDiameter (of a connected graph): length of the longest shortestpath between two vertices in the graph.
The diameter of this graph is 2.
0e+00
5e+05
1e+06
2 4 6
shortest path length
coun
t
shortest path length distribution − FB
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Diameter and 6 degrees of separationDiameter (of a connected graph): length of the longest shortestpath between two vertices in the graph.
6 degrees of separation
From a volume of novels by Frigyes Karinthy: all living things andeverything else in the world is six or fewer steps away from eachother (i.e., a chain of “a friend of a friend” statements can be madeto connect any two people in a maximum of six steps).Hypothesis tested by Milgram with a letter chain (1967): ' 2 − 10intermediates to reach a target person from any starting people(known as the “small world experiment”).
0e+00
5e+05
1e+06
2 4 6
shortest path length
coun
t
shortest path length distribution − FB
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Diameter and 6 degrees of separationDiameter (of a connected graph): length of the longest shortestpath between two vertices in the graph.
Examples of diameter
Example 1: GOT diameter = 6 (unweighted) and 85 (weighted)Example 2: NVV diameter in LCC: 18Example 3: FB diameter = 7
0e+00
5e+05
1e+06
2 4 6
shortest path length
coun
t
shortest path length distribution − FB
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Diameter and 6 degrees of separationDiameter (of a connected graph): length of the longest shortestpath between two vertices in the graph.
0
500
1000
1500
2000
2 4 6
shortest path length
coun
t
shortest path length distribution − GOT (unweighted)
0e+00
5e+05
1e+06
2 4 6
shortest path length
coun
t
shortest path length distribution − FB
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Diameter and 6 degrees of separationDiameter (of a connected graph): length of the longest shortestpath between two vertices in the graph.
0
100
200
300
0 20 40 60 80
shortest path length
coun
t
shortest path length distribution − GOT (weighted)
0e+00
5e+05
1e+06
2 4 6
shortest path length
coun
t
shortest path length distribution − FB
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Diameter and 6 degrees of separationDiameter (of a connected graph): length of the longest shortestpath between two vertices in the graph.
0
250
500
750
1000
1250
0 5 10 15
shortest path length
coun
t
shortest path length distribution − NVV
0e+00
5e+05
1e+06
2 4 6
shortest path length
coun
t
shortest path length distribution − FB
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Diameter and 6 degrees of separationDiameter (of a connected graph): length of the longest shortestpath between two vertices in the graph.
0e+00
5e+05
1e+06
2 4 6
shortest path length
coun
t
shortest path length distribution − FB
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girth, cohesion
I girth: number of vertices in the shortest circle (equal to 3 iftransitivity is not equal to 0)
I cohesion (of a connected graph): minimum number of verticesto remove in order to disconnect the graph
Examples
Example 1: GOT girth = 3 and cohesion = 1Example 2: NVV girth = 3 and cohesion = 1Example 3: FB girth = 3 and cohesion = 1
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girth, cohesion
I girth: number of vertices in the shortest circle (equal to 3 iftransitivity is not equal to 0)
I cohesion (of a connected graph): minimum number of verticesto remove in order to disconnect the graph
Examples
Example 1: GOT girth = 3 and cohesion = 1Example 2: NVV girth = 3 and cohesion = 1Example 3: FB girth = 3 and cohesion = 1
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Outline
A brief introduction to networks/graphs
Visualization
Global characteristics
Numerical characteristics calculation
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Extracting important vertices: hubsvertex degree: number of edges adjacent to the vertex∣∣∣{xj : (xi , xj) ∈ E, j , i}
∣∣∣. The weighted version∑
j,i Wij is called thestrength.Vertices with a high degree are called hubs: measure of the vertexpopularity.
degrees
JaimeTyrion
Tywin
Jon
RobbSansa
(unweighted)
degreesV222 456
V1423 467V1700 467
(degree larger than 400)
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Extracting important vertices: hubsvertex degree: number of edges adjacent to the vertex∣∣∣{xj : (xi , xj) ∈ E, j , i}
∣∣∣. The weighted version∑
j,i Wij is called thestrength.Vertices with a high degree are called hubs: measure of the vertexpopularity.
degreesJaime 24Tyrion 36Tywin 22
Jon 26Robb 25
Sansa 26
(degree larger than 20)
degreesV222 456
V1423 467V1700 467
(degree larger than 400)
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Extracting important vertices: hubsvertex degree: number of edges adjacent to the vertex∣∣∣{xj : (xi , xj) ∈ E, j , i}
∣∣∣. The weighted version∑
j,i Wij is called thestrength.Vertices with a high degree are called hubs: measure of the vertexpopularity.
strength
Tyrion
Jon
(weighted)
degreesV222 456
V1423 467V1700 467
(degree larger than 400)
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Extracting important vertices: hubsvertex degree: number of edges adjacent to the vertex∣∣∣{xj : (xi , xj) ∈ E, j , i}
∣∣∣. The weighted version∑
j,i Wij is called thestrength.Vertices with a high degree are called hubs: measure of the vertexpopularity.
S.L
M.P
V.G
N.E
Hubs (degree larger than 25) are two students who have been keptback one year at school.
degreesV222 456
V1423 467V1700 467
(degree larger than 400)
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Extracting important vertices: hubsvertex degree: number of edges adjacent to the vertex∣∣∣{xj : (xi , xj) ∈ E, j , i}
∣∣∣. The weighted version∑
j,i Wij is called thestrength.Vertices with a high degree are called hubs: measure of the vertexpopularity.
degreesS.L 31
M.P 29V.G 27N.E 27
(degree larger than 25)
degreesV222 456
V1423 467V1700 467
(degree larger than 400)
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Extracting important vertices: hubsvertex degree: number of edges adjacent to the vertex∣∣∣{xj : (xi , xj) ∈ E, j , i}
∣∣∣. The weighted version∑
j,i Wij is called thestrength.Vertices with a high degree are called hubs: measure of the vertexpopularity.
V222V1423
V1700
degreesV222 456
V1423 467V1700 467
(degree larger than 400)
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Extracting important vertices: hubs
vertex degree: number of edges adjacent to the vertex∣∣∣{xj : (xi , xj) ∈ E, j , i}∣∣∣. The weighted version
∑j,i Wij is called the
strength.Vertices with a high degree are called hubs: measure of the vertexpopularity.
degreesV222 456
V1423 467V1700 467
(degree larger than 400)
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Degree distribution
In real graphs (WWW, social networks...), the degree distribution isoften found to fit a power law: P(degree = k) ∼ k−γ for a γ > 0.
0.000
0.025
0.050
0.075
0.100
0.125
0 10 20 30
degrees
dens
ity
degree distribution − GOT (unweighted)
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Degree distribution
In real graphs (WWW, social networks...), the degree distribution isoften found to fit a power law: P(degree = k) ∼ k−γ for a γ > 0.
0.0000
0.0025
0.0050
0.0075
0.0100
0 200 400
degrees
dens
ity
degree distribution − GOT (weighted)
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Degree distribution
In real graphs (WWW, social networks...), the degree distribution isoften found to fit a power law: P(degree = k) ∼ k−γ for a γ > 0.
0.00
0.02
0.04
0.06
0.08
0 10 20 30
degrees
dens
ity
degree distribution − NVV
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Degree distribution
In real graphs (WWW, social networks...), the degree distribution isoften found to fit a power law: P(degree = k) ∼ k−γ for a γ > 0.
0.000
0.002
0.004
0.006
0 100 200 300 400
degrees
dens
ity
degree distribution − FB
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Degree distribution
In real graphs (WWW, social networks...), the degree distribution isoften found to fit a power law: P(degree = k) ∼ k−γ for a γ > 0.
+
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++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
++++++++++
+−7
−6
−5
−4
0 2 4 6
log(k)
log(
P(d
egre
e =
k))
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Extracting important vertices: betweennessvertex betweenness: number of shortest paths between all pairs ofvertices that pass through the vertex. Betweenness is a centralitymeasure indicating which vertices are the most important toconnect the network.
Robert
Tyrion
betweenness degreeV177 30797 253V222 30649 456V340 49127 299
V1173 30778 313V1423 60272 467V1700 37868 467
(betweenness larger than 30,000; hubs had a degree larger than400)
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Extracting important vertices: betweennessvertex betweenness: number of shortest paths between all pairs ofvertices that pass through the vertex. Betweenness is a centralitymeasure indicating which vertices are the most important toconnect the network.
betweenness degreeRobert 1166 18Tyrion 1164 36
(betweenness larger than 1000; hubs had a degree larger than 20)
betweenness degreeV177 30797 253V222 30649 456V340 49127 299
V1173 30778 313V1423 60272 467V1700 37868 467
(betweenness larger than 30,000; hubs had a degree larger than400)
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Extracting important vertices: betweennessvertex betweenness: number of shortest paths between all pairs ofvertices that pass through the vertex. Betweenness is a centralitymeasure indicating which vertices are the most important toconnect the network.
B.M
L.F
Vertices with largest betweenness (larger than 3000) are publicfigures.
betweenness degreeV177 30797 253V222 30649 456V340 49127 299
V1173 30778 313V1423 60272 467V1700 37868 467
(betweenness larger than 30,000; hubs had a degree larger than400)
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Extracting important vertices: betweennessvertex betweenness: number of shortest paths between all pairs ofvertices that pass through the vertex. Betweenness is a centralitymeasure indicating which vertices are the most important toconnect the network.
betweennessB.M 3439L.F 3146
betweenness degreeV177 30797 253V222 30649 456V340 49127 299
V1173 30778 313V1423 60272 467V1700 37868 467
(betweenness larger than 30,000; hubs had a degree larger than400)
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Extracting important vertices: betweennessvertex betweenness: number of shortest paths between all pairs ofvertices that pass through the vertex. Betweenness is a centralitymeasure indicating which vertices are the most important toconnect the network.
V177
V222V340V1173V1423
V1700
betweenness degreeV177 30797 253V222 30649 456V340 49127 299
V1173 30778 313V1423 60272 467V1700 37868 467
(betweenness larger than 30,000; hubs had a degree larger than400)
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Extracting important vertices: betweennessvertex betweenness: number of shortest paths between all pairs ofvertices that pass through the vertex. Betweenness is a centralitymeasure indicating which vertices are the most important toconnect the network.
betweenness degreeV177 30797 253V222 30649 456V340 49127 299
V1173 30778 313V1423 60272 467V1700 37868 467
(betweenness larger than 30,000; hubs had a degree larger than400)
Nathalie Vialaneix | Graph mining 29/30
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Other centrality measure: eccentricity and closenessvertex eccentricity: shortest path length from the farthest othervertex in the graph (the smallest eccentricity is the radius)vertex closeness: inverse of the average length of the shortestpaths from this vertex to all the other vertices in the graph:
1∑j,i spl(i,j)
eccentricity closeness
betweenness
Radius
Example 1: GOT radius = 3Example 2: NVV radius in LCC: 9Example 3: FB radius = 4
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Other centrality measure: eccentricity and closeness
vertex eccentricity: shortest path length from the farthest othervertex in the graph (the smallest eccentricity is the radius)vertex closeness: inverse of the average length of the shortestpaths from this vertex to all the other vertices in the graph:
1∑j,i spl(i,j)
Radius
Example 1: GOT radius = 3Example 2: NVV radius in LCC: 9Example 3: FB radius = 4
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Beveridge, A. and Shan, J. (2016).Network of thrones.Math Horizons, 23(4):18–22.
Csardi, G. and Nepusz, T. (2006).The igraph software package for complex network research.InterJournal, Complex Systems.
Fruchterman, T. and Reingold, B. (1991).Graph drawing by force-directed placement.Software, Practice and Experience, 21:1129–1164.
Traud, A., Mucha, P., and Porter, M. (2012).Social structure of facebook networks.Physica A, 391(16):4165–4180.
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