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Social Network Analysisfor Computer Scientists
Frank Takes
LIACS, Leiden University
https://liacs.leidenuniv.nl/~takesfw/SNACS
Lecture 2 — Advanced network concepts and centrality
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 1 / 74
Recap
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 2 / 74
Networks
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 3 / 74
Notation
Concept Symbol
Network (graph) G = (V ,E )
Nodes (objects/vertices/actors/entities) V
Links (relationships/edges/ties/connections) E
Directed — E ⊆ V × VUndirected
Number of nodes — |V | n
Number of edges — |E | m
Degree of node u deg(u)
Distance from node u to v d(u, v)
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 4 / 74
Real-world networks
1 Sparse networks density
2 Fat-tailed power-law degree distribution degree
3 Giant component components
4 Low pairwise node-to-node distances distance
5 Many triangles clustering coefficient
Many examples: communication networks, citation networks,collaboration networks (Erdos, Kevin Bacon), protein interactionnetworks, information networks (Wikipedia), webgraphs, financialnetworks (Bitcoin) . . .
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 5 / 74
Advanced concepts
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 6 / 74
Advanced concepts
Assortativity
Reciprocity
Power law exponent
Planar graphs
Complete graphs
Subgraphs
Trees
Spanning trees
Diameter
Bridges
Graph traversal
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 7 / 74
Assortativity
Assortativity: extent to which “similar” nodes attract each otherValue close to -1 if dissimilar nodes more often attract each otherValue close to 1 if similar nodes more often attract each other
Degree assortativity: nodes with a similar degree connect morefrequently
Attribute assortativity: nodes with similar attributes attract eachother
Influence on connectivity of network, spreading of information, etc.
Social networks: homophily
Complex networks: mixing patterns
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 8 / 74
Assortativity
Assortativity: extent to which “similar” nodes attract each otherValue close to -1 if dissimilar nodes more often attract each otherValue close to 1 if similar nodes more often attract each other
Degree assortativity: nodes with a similar degree connect morefrequently
Attribute assortativity: nodes with similar attributes attract eachother
Influence on connectivity of network, spreading of information, etc.
Social networks: homophily
Complex networks: mixing patterns
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 8 / 74
Degree assortativity
Figure: Degree assortativity (left) and degree disassortativity (right)
Image: Estrada et al., Clumpiness mixing in complex networks, J. Stat. Mech. Theor. Exp. P03008 (2008).
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 9 / 74
Reciprocity
Reciprocity: measure of the likelihood of nodes in a directed networkto be mutually linked
Let m<−> be the number of links in the directed network for whichthere also exists a symmetric counterpart:
m<−> = |{(u, v) ∈ E : (v , u) ∈ E}|
Reciprocity r is then the fraction of links that is symmetric:
r =m<−>m
Measures the extent to which relationships are mutual
Useful to compare between networks
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 10 / 74
Power law degree distribution
Source: http://konect.cc/networks/citeseer/
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 11 / 74
Power law exponent in undirected networks
The probabibility pk of a node having degree k depends on the powerlaw exponent γ:
pk ∼ k−γ
This means thatlog pk ∼ −γ log k
And as such, the straight line in log-log scale plots is observed.
In real-world networks, γ has a value of around 2 to 3
Useful to compare between similar networks
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 12 / 74
Power law exponent in directed networks
Source: A. Barabasi, Network Science, 2016.
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 13 / 74
Planar graphs
Planar graphs can be visualized such that no two edges cross eachother
Image: Zafarani et al., Social Media Mining, 2014.
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 14 / 74
Complete graphs
In complete graphs, all pairs of nodes are connected
The number of edges m is equal to 12 · n · (n − 1)
Figure: Complete graphs of size 1, 2, 3 and 4
Image: Zafarani et al., Social Media Mining, 2014.
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 15 / 74
Trees
A tree is a graph without cycles
A set of disconnected trees is called a forest
A tree with n nodes has m = n − 1 edges
Image: Zafarani et al., Social Media Mining, 2014.
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 16 / 74
Trees
Image: M. Lima, Book of trees: Visualizing branches of knowledge, 2014.
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 17 / 74
Subgraphs
Given a graph G = (V ,E )
Subgraph G ′ = (V ′,E ′) with V ′ ⊆ V and E ′ ⊆ (E ∩ (V ′ × V ′))(subset of the nodes and edges of the original network, commonlyused when defining communities or clusters)
Subgraph G ′ = (V ,E ′) with E ′ ⊆ E(only edges are left out, commonly used when modelling networkevolution)
Special subgraphs: spanning trees
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 18 / 74
Spanning trees
A spanning tree is a tree and subgraph of a graph that covers allnodes of the graph
In weighted graphs, a minimal spanning tree is one of minimal edgeweight
Image: Zafarani et al., Social Media Mining, 2014.
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 19 / 74
Diameter
Distance d(u, v) = length of shortest path from u to v
Diameter D(G ) = maxu,v∈V d(u, v) = maximal distance
Eccentricity e(u) = maxv∈V d(u, v) = length of longest shortest pathfrom u
Diameter D(G ) = maxu∈V e(u) = maximal eccentricity
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 20 / 74
Diameter
Distance d(u, v) = length of shortest path from u to v
Diameter D(G ) = maxu,v∈V d(u, v) = maximal distance
Eccentricity e(u) = maxv∈V d(u, v) = length of longest shortest pathfrom u
Diameter D(G ) = maxu∈V e(u) = maximal eccentricity
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 20 / 74
Bridges
Bridge: an edge whose removal will result in an increase in thenumber of connected components
Also called cut edges, with applications in community detection
Image: Zafarani et al., Social Media Mining, 2014.
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 21 / 74
Graph traversal
Given a network, how can we explore it?
Specifically: exploration starting from a particular source
Node adjacency: two nodes are adjacent if there is an edgeconnecting them
Neighborhood: set of nodes adjacent to a node v ∈ V :
N(v) = {w ∈ V : (v ,w) ∈ E}
Techniques to iteratively explore neighborhoods: DFS and BFS
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 22 / 74
Graph traversal: DFS
Depth First Search (DFS)
Image: Zafarani et al., Social Media Mining, 2014.
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 23 / 74
Graph traversal: BFS
Source: A. Barabasi, Network Science, 2016.
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 24 / 74
Graph traversal: BFS
Breadth First Search (BFS)
Graph traversal in level-order
Image: Zafarani et al., Social Media Mining, 2014.
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 25 / 74
Graph traversal: BFS
Breadth First Search (BFS)
From source node, create a rooted spanning tree of the graph
Graph traversal in level-order
Often implemented using a queue
BFS considers traversing each of the m edges once, so O(m)
Important for computing various centrality measures
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 26 / 74
Centrality
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 27 / 74
Centrality
Given a social network, which person is most important?
What is the most important page on the web?
Which protein is most vital in a biological network?
Who is the most respected author in a scientific citation network?
What is the most crucial router in an internet topology network?
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 28 / 74
Centrality
Node centrality: the importance of a node with respect to the othernodes based on the structure of the network
Centrality measure: computes the centrality value of all nodes inthe graph
For all v ∈ V a measure M returns a value CM(v) ∈ [0; 1]
CM(v) > CM(w) means that node v is more important than w
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 29 / 74
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 30 / 74
Degree centrality
Undirected graphs – degree centrality: measure the number ofadjacent nodes
Cd(v) =deg(v)
n − 1
Directed graphs — indegree centrality and outdegree centrality
Local measure
O(1) time to compute
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 31 / 74
Degree distribution
100
101
102
103
104
105
106
107
0 500 1000 1500 2000 2500
fre
qu
en
cy
degree
Not so many distinct values in the lower ranges
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 32 / 74
Degree centrality
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 33 / 74
Degree centrality
Waymar Royce
Will Gared
Eddard Stark
Catelyn StarkRobb StarkBran Stark
Rickon Stark
Sansa Stark
Arya Stark
Jon Snow
Robert Baratheon
Joffrey Baratheon
Cersei Baratheon
Jaime Lannister
Tyrion Lannister
Rodrik Cassel
Jory Cassel
Maester Luwin
Sandor Clegane
Benjen Stark
Theon Greyjoy
Ros
Septa MordaneDaenerys Targaryen
Viserys TargaryenJorah Mormont
Illyrio Mopatis
Drogo
Myrcella
Ilyn Payne
Doreah
Jhiqui
Irri Varys
Petyr Baelish
Pycelle Renly Baratheon
Old Nan
PyparAlistair Thorn Joer Mormont
Grenn
Rast
Lancel Lannister
Barristan Selmy
Yoren
Maester Aemon
Syrio Forel
The Three-Eyed Raven
Hodor
Samwell Tarly
Janos Slynt
Hugh of the Vale
Gendry
Gregor Clegane
Bronn
Willis Wode
KurleketKnight of House Frey
Loras Tyrell
Vardis Egan
Lysa ArrynMord
Robin Arryn
Lyanna Stark
Osha
Beric Dondarrion
Tywin Lannister
Stannis Baratheon
Meryn Trant
ShaggaTimett
Chella
Mirri Maz Duur
Mago
Greatjon Umber
Galbart GloverKevan Lannister
Walder Frey
Shae
Qotho
Hot Pie
Wine Seller
Mycah
Rakharo
Figure: Character co-occurence network. Node size based on degree.Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 34 / 74
Closeness centrality
Closeness centrality: based on the average distance to all othernodes
Cc(v) =
(1
n − 1
∑w∈V
d(v ,w)
)−1Global distance-based measure
O(mn) to compute: one BFS in O(m) for each of the n nodes
Connected component(s). . .
Harmonic centrality: variant of closeness (not normalized)
Ch(v) =∑w∈V
1
d(w , v)
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 35 / 74
Closeness centrality
Closeness centrality: based on the average distance to all othernodes
Cc(v) =
(1
n − 1
∑w∈V
d(v ,w)
)−1Global distance-based measure
O(mn) to compute: one BFS in O(m) for each of the n nodes
Connected component(s). . .
Harmonic centrality: variant of closeness (not normalized)
Ch(v) =∑w∈V
1
d(w , v)
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 35 / 74
Closeness centrality
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 36 / 74
Degree vs. closeness centrality
Waymar Royce
Will Gared
Eddard Stark
Catelyn StarkRobb StarkBran Stark
Rickon Stark
Sansa Stark
Arya Stark
Jon Snow
Robert Baratheon
Joffrey Baratheon
Cersei Baratheon
Jaime Lannister
Tyrion Lannister
Rodrik Cassel
Jory Cassel
Maester Luwin
Sandor Clegane
Benjen Stark
Theon Greyjoy
Ros
Septa MordaneDaenerys Targaryen
Viserys TargaryenJorah Mormont
Illyrio Mopatis
Drogo
Myrcella
Ilyn Payne
Doreah
Jhiqui
Irri Varys
Petyr Baelish
Pycelle Renly Baratheon
Old Nan
PyparAlistair Thorn Joer Mormont
Grenn
Rast
Lancel Lannister
Barristan Selmy
Yoren
Maester Aemon
Syrio Forel
The Three-Eyed Raven
Hodor
Samwell Tarly
Janos Slynt
Hugh of the Vale
Gendry
Gregor Clegane
Bronn
Willis Wode
KurleketKnight of House Frey
Loras Tyrell
Vardis Egan
Lysa ArrynMord
Robin Arryn
Lyanna Stark
Osha
Beric Dondarrion
Tywin Lannister
Stannis Baratheon
Meryn Trant
ShaggaTimett
Chella
Mirri Maz Duur
Mago
Greatjon Umber
Galbart GloverKevan Lannister
Walder Frey
Shae
Qotho
Hot Pie
Wine Seller
Mycah
Rakharo
Figure: Node size based on degree, color based on closeness centrality.Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 37 / 74
Betweenness centrality
Betweenness centrality: measure the number of shortest paths thatrun through a node
Cb(u) =∑
v ,w∈Vv 6=w ,u 6=v ,u 6=w
σu(v ,w)
σ(v ,w)
σ(v ,w) is the number of shortest paths from v to w
σu(v ,w) is the number of such shortest paths that run through u
Divide by largest value to normalize to [0; 1]
Global path-based measure
O(2mn) time to compute (two “BFSes” for each node)
U. Brandes, ”A faster algorithm for betweenness centrality”, Journal of Mathematical Sociology 25(2): 163–177, 2001
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 38 / 74
Betweenness centrality
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 39 / 74
Degree vs. betweeness centrality
Waymar Royce
Will Gared
Eddard Stark
Catelyn StarkRobb StarkBran Stark
Rickon Stark
Sansa Stark
Arya Stark
Jon Snow
Robert Baratheon
Joffrey Baratheon
Cersei Baratheon
Jaime Lannister
Tyrion Lannister
Rodrik Cassel
Jory Cassel
Maester Luwin
Sandor Clegane
Benjen Stark
Theon Greyjoy
Ros
Septa MordaneDaenerys Targaryen
Viserys TargaryenJorah Mormont
Illyrio Mopatis
Drogo
Myrcella
Ilyn Payne
Doreah
Jhiqui
Irri Varys
Petyr Baelish
Pycelle Renly Baratheon
Old Nan
PyparAlistair Thorn Joer Mormont
Grenn
Rast
Lancel Lannister
Barristan Selmy
Yoren
Maester Aemon
Syrio Forel
The Three-Eyed Raven
Hodor
Samwell Tarly
Janos Slynt
Hugh of the Vale
Gendry
Gregor Clegane
Bronn
Willis Wode
KurleketKnight of House Frey
Loras Tyrell
Vardis Egan
Lysa ArrynMord
Robin Arryn
Lyanna Stark
Osha
Beric Dondarrion
Tywin Lannister
Stannis Baratheon
Meryn Trant
ShaggaTimett
Chella
Mirri Maz Duur
Mago
Greatjon Umber
Galbart GloverKevan Lannister
Walder Frey
Shae
Qotho
Hot Pie
Wine Seller
Mycah
Rakharo
Figure: Node size based on degree, color based on betweenness centrality.Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 40 / 74
Computing betweenness centrality
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 41 / 74
Counting shortest paths
Bellman criterion: v lies on a shortest path from u to w ifd(u, v) + d(v ,w) = d(u,w).
Predecessors: Pu(w) is the set of predecessors of node w on ashortest path from u, formally:
Pu(w) = {v ∈ V : (v ,w) ∈ E , d(u,w) = d(u, v) + 1}
Counting the number of shortest paths σ(u,w) for u 6= w :
σ(u,w) =∑
v∈Pu(w)
σ(u, v)
with σ(u, u) = 1
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 42 / 74
Counting shortest paths
Task: compute σ(u,w), the number of shortest paths from u to w. . .
b
a
u e
cf
d
h w
i
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 43 / 74
Counting shortest paths
Task: compute σ(u,w), the number of shortest paths from u to wFirst a BFS from u to find w
b
a
u
0
e
cf
d
h w
i
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 44 / 74
Counting shortest paths
Task: compute σ(u,w), the number of shortest paths from u to wFirst a BFS from u to find w
b
a
1
u
0
e
c
1
f
d
h w
i
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 45 / 74
Counting shortest paths
Task: compute σ(u,w), the number of shortest paths from u to wFirst a BFS from u to find w
b
2a
1
u
0
e
c
1
f
d
h w
i
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 46 / 74
Counting shortest paths
Task: compute σ(u,w), the number of shortest paths from u to wFirst a BFS from u to find w
b
2a
1
u
0
e
3
c
1
f
3
d
3
h w
i
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 47 / 74
Counting shortest paths
Task: compute σ(u,w), the number of shortest paths from u to wFirst a BFS from u to find w
b
2a
1
u
0
e
3
c
1
f
3
d
3
h
4
w
i
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 48 / 74
Counting shortest paths
Task: compute σ(u,w), the number of shortest paths from u to wσ(u,w) ? Ask predecessors Pu(w) = {h} for its value
b
2a
1
u
0
e
3
c
1
f
3
d
3
h
4
w
5
i
5
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 49 / 74
Counting shortest paths
Task: compute σ(u,w), the number of shortest paths from u to wσ(u, h) ? Ask predecessors Pu(h) = {d , e, f } for its value
b
2a
1
u
0
e
3
c
1
f
3
d
3
h
4
w
5
i
5
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 50 / 74
Counting shortest paths
Task: compute σ(u,w), the number of shortest paths from u to wAsk predecessors . . .
b
2a
1
u
0
e
3
c
1
f
3
d
3
h
4
w
5
i
5
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 51 / 74
Counting shortest paths
Task: compute σ(u,w), the number of shortest paths from u to wAsk predecessors . . .
b
2a
1
u
0
e
3
c
1
f
3
d
3
h
4
w
5
i
5
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 52 / 74
Counting shortest paths
Task: compute σ(u,w), the number of shortest paths from u to wAsk predecessors . . .
b
2a
1
u
0
e
3
c
1
f
3
d
3
h
4
w
5
i
5
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 53 / 74
Counting shortest paths
Task: compute σ(u,w), the number of shortest paths from u to wσ(u, u) = 1, now propagate back...
b
2a
1
u
0
1
e
3
c
1
f
3
d
3
h
4
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5
i
5
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 54 / 74
Counting shortest paths
Task: compute σ(u,w), the number of shortest paths from u to wσ(u, a) = σ(u, c) = 1
b
2a
1
1u
0
1c
1
1
e
3
f
3
d
3
h
4
w
5
i
5
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 55 / 74
Counting shortest paths
Task: compute σ(u,w), the number of shortest paths from u to wσ(u, b) = σ(u, a) + σ(u, c) = 2
b
2
2
a
1
1u
0
1c
1
1
e
3
f
3
d
3
h
4
w
5
i
5
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 56 / 74
Counting shortest paths
Task: compute σ(u,w), the number of shortest paths from u to wσ(u, d) = σ(u, e) = σ(u, f ) = σ(u, c) = 2
b
2
2
a
1
1u
0
1c
1
1
e
3
2
f
3
2
d
3
2
h
4
w
5
i
5
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 57 / 74
Counting shortest paths
Task: compute σ(u,w), the number of shortest paths from u to wσ(u, h) = σ(u, d) + σ(u, e) + σ(u, f ) = 6
b
2
2
a
1
1u
0
1c
1
1
e
3
2
f
3
2
d
3
2
h
4
6
w
5
i
5
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 58 / 74
Counting shortest paths
Done!σ(u,w) = σ(u, h) = 6
b
2
2
a
1
1u
0
1c
1
1
e
3
2
f
3
2
d
3
2
h
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6
w
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i
5
6
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 59 / 74
Centrality measures compared
Figure: Degree, closeness and betweenness centrality
Source: ”Centrality”’ by Claudio Rocchini, Wikipedia File:Centrality.svg
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 60 / 74
Eccentricity centrality
Node eccentricity: length of a longest shortest path (distance to anode furthest away)
e(v) = maxw∈V
d(v ,w)
Eccentricity centrality:
Ce(v) =1
e(v)
Worst-case variant of closeness centrality
O(mn) to compute: one BFS in O(m) for each of the n nodes
Large optimizations possible using lower and upper bounds, seeF.W. Takes and W.A. Kosters, Computing the Eccentricity Distributionof Large Graphs, Algorithms, vol. 6, nr. 1, pp. 100-118, 2013.
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 61 / 74
Eccentricity centrality
Node eccentricity: length of a longest shortest path (distance to anode furthest away)
e(v) = maxw∈V
d(v ,w)
Eccentricity centrality:
Ce(v) =1
e(v)
Worst-case variant of closeness centrality
O(mn) to compute: one BFS in O(m) for each of the n nodes
Large optimizations possible using lower and upper bounds, seeF.W. Takes and W.A. Kosters, Computing the Eccentricity Distributionof Large Graphs, Algorithms, vol. 6, nr. 1, pp. 100-118, 2013.
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 61 / 74
Eccentricity centrality
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 62 / 74
Degree vs. eccentricity centrality
Waymar Royce
Will Gared
Eddard Stark
Catelyn StarkRobb StarkBran Stark
Rickon Stark
Sansa Stark
Arya Stark
Jon Snow
Robert Baratheon
Joffrey Baratheon
Cersei Baratheon
Jaime Lannister
Tyrion Lannister
Rodrik Cassel
Jory Cassel
Maester Luwin
Sandor Clegane
Benjen Stark
Theon Greyjoy
Ros
Septa MordaneDaenerys Targaryen
Viserys TargaryenJorah Mormont
Illyrio Mopatis
Drogo
Myrcella
Ilyn Payne
Doreah
Jhiqui
Irri Varys
Petyr Baelish
Pycelle Renly Baratheon
Old Nan
PyparAlistair Thorn Joer Mormont
Grenn
Rast
Lancel Lannister
Barristan Selmy
Yoren
Maester Aemon
Syrio Forel
The Three-Eyed Raven
Hodor
Samwell Tarly
Janos Slynt
Hugh of the Vale
Gendry
Gregor Clegane
Bronn
Willis Wode
KurleketKnight of House Frey
Loras Tyrell
Vardis Egan
Lysa ArrynMord
Robin Arryn
Lyanna Stark
Osha
Beric Dondarrion
Tywin Lannister
Stannis Baratheon
Meryn Trant
ShaggaTimett
Chella
Mirri Maz Duur
Mago
Greatjon Umber
Galbart GloverKevan Lannister
Walder Frey
Shae
Qotho
Hot Pie
Wine Seller
Mycah
Rakharo
Figure: Node size based on degree, color based on eccentricity centrality.Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 63 / 74
Centrality measures
Distance/path-based measures:
Degree centrality O(n)Closeness centrality O(mn)Betweenness centrality O(mn)Eccentricity centrality O(mn)
(complexity is for computing centralities of all n nodes)
Many more: Eigenvector centrality, Katz centrality, . . .
Approximating these measures is also possible
Also: propagation-based centrality measures like PageRank
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 64 / 74
Periodic table of centrality
8000 1979
DC
Degree
224 1971
BC
Betweenness
942 1966
CC
Closeness
1279 1972
EC
Eigenvector
1306 1953
KS
Katz Status
8053 1999
PR
Page Rank
484 2005
SC
Subgraph
239 2008
EBC
Endpoint BC
239 2008
PBC
Proxy BC
239 2008
LSBC
LscaledBC
239 2008
DBBC
DBounded BC
239 2008
DSBC
DScaled BC
613 1991
FBC
Flow BC
224 1971
EBC
Edge BC
979 2005
RWBC
RWalk BC
291 1953
σ
Stress
14 2012
RLBC
RLimited BC
53 2009
CBC
Commun. BC
477 1991
TEC
Total Effects
477 1991
IEC
Immediate Eff.
477 1991
MEC
Mediative Eff.
236 2007
∆C
Delta Cent.
42 2009
LI
Lobby Index
1 2014
DM
Degree Mass
69 2010
LEVC
Leverage Cent.
5 2010
MDC
MD Cent.
11 2008
MC
Mod Cent.
10 2012
LAPC
Laplacian C.
35 2010
TC
Topological C.
0 2015
EYC
Entropy C.
0 2014
COMCC
Community C.
0 2012
ABC
Attentive BC
X X
SDC
Sphere Degree
2 2013
CAC
Comm. Ability
45 2012
ECCoef
ECCoef
1699 2001
STRC
Straightness C
15 2010
ZC
Zonal Cent.
56 2007
EPTC
Entropy PC
0 2015
SMD
Super Mediat.
0 2015
SNR
Silent Node R.
14 2013
CI
Collab. Index
281 1971
CCoef
Clust. Coef.
1 2014
UCC
United Comp.
15 2011
HPC
Harm. Prot.
11 2013
CoEWC
CoEWC
42 2012
PeC
PeC
4 2012
WDC
WDC
26 2011
LAC
Local Average
45 2012
NC
NC
427 2007
BN
Bottleneck
119 2008
MNC
MNC
119 2008
DMNC
DMNC
108 2010
MLC
Moduland C.
43 2009
EI
Essentiality I.
9068 1999
HITS
Hubs/Authority
26 1989
kPC
kPath C.
43 2009
KL
Clique Level
3 2013
LR
Lurker Rank
X X
RSC
Resolvent SC
275 2002
EGO
Ego
573 2006
g-kPC
geodesic kPath
573 2006
e-kPC
e-disjoint kPC
179 2005
BIP
Bipartivity
2457 1987
β-C
β Cent.
1 2014
SWIPD
SWIPD
51 2004
HYPER
Hypergraphs
296 1999
GROUP
Groups/Classes
573 2006
v-kPC
v-disjoint kPC
426 1988
GPI
GPI Power
X X
HYP
Hyperbolic C.
36 2009
XXXX
LinComb
279 1997
AFF
Affiliation C.
80 2006
HYPSC
Hyperg. SC
505 2010
WEIGHT
Weighted C.
116 1991
kRPC
Reachability
27 2012
kEPC
k-edge PC
0 2014
BCPR
BCPR
399 2 001
α-C
α-Cent.
34 2010
t-SC
t-Subgraph
17 2013
TCom
Total Comm.
58 2007
SCodd
odd Subgraph
13 2007
FC
Functional C.
0 2014
TPC
Tunable PC
178 1995
ECC
Eccentricity
518 1989
IC
Information C
116 1998
RAD
Radiality
116 1998
INT
Integration
586 2004
RWCC
RWalk CC
0 2014
HCC
Hierar. CC
0 2015
EDCC
Effective Dist.
1
2
3
4
5
6
7
1 IA
2 IIA
3 IIIA 4 IVB 5 VB 6 VIB 7 VIIB 8 VIIIB 9 VIIIB 10 VIIIB 11 IB 12 IIB
13 IIIA 14 IVA 15 VA 16 VIA 17 VIIA
18 VIIIA
8000 1979
Freeman
Conceptual
942 1966
Sabidussi
Axiomatic
573 2006
Borgatti/Everett
Conceptual
1130 2005
Borgatti
Conceptual
24 2014
Boldi/Vigna
Axiomatic
252 1974
Nieminen
Axiomatic
6 1981
Kishi
Axiomatic
3 2012
Kitti
Axiomatic
3 2009
Garg
Axiomatic
2065 1934
Moreno
Historic
1546 1950
Bavelas
Historic
780 1948
Bavelas
Historic
1475 1951
Leavitt
Historic
297 1992
Borgatti/Everett
Conceptual
3649 2001
Jeong et al.
Empirical
4167 1998
Tsai/Ghoshal
Empirical
961 1993
Ibarra
Empirical
71 2008
Valente
Empirical
“Traditional”Betweenness-likeFriedkin MeasuresMiscellaneousPath-basedSpecific Network TypeSpectral-basedCloseness-like
citations year
C
Name
c©David Schoch (University of Konstanz)
Periodic Table of Network Centrality
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 65 / 74
Course project
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 66 / 74
Course project
Project on a specific SNA subtopic makes up 60% of your coursegrade
Teams of exactly 2 students
Deliverables:Presentation on a topic-related paper. 30 minutes for your talk, 15minutes for questions and discussion (duration subject to change)Paper with a contribution to SNA that goes beyond what is done inthe paper you study, e.g.:
Comparing similar algorithms from different papersTesting algorithms on larger datasetsValidating algorithms using different metricsAddressing future work posed in the paper
Short peer review documentRelevant project code and supplementary materialBonus for open-source or open science contributions
Topics divided over teams based on first come, first serve
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 67 / 74
Presentation
Present one paper on the topic of your project
Convey the main message of the paper in an understandable way
Show a nice demo, pictures, movies or visualization
Have a clearly structured presentation
Briefly discuss your project plans
Demo presentation will follow
Discussion with other students is expected (from both presenters andattendees)
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 68 / 74
Course project
Read your paper, understand the main problem
Do a bit of research on related literature
Determine which algorithms/techniques/parameters/datasets/subproblems you are going to compare (i.e., your contribution)
Program (or obtain code of) the different algorithms and techniques
Obtain and describe applicable datasets for comparing the algorithms
Perform and report on experiments to compare the algorithms
Determine and discuss results
Write a sensible conclusion
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 69 / 74
Course project paper
Scientific paper
LATEX
6 to 10 pages, two columns
Images, figures, graphs, diagrams, tables, references, . . .
Between 5 and 9 sections
Peer review after first few sections
Option for “intermediary paper check” before final hand-in
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 70 / 74
Common pitfalls/excuses
Starting to read your project paper after Assignment 2 (you are verylate)
Starting just before the first paper deadline (you are likely too late)
Starting too late with writing the paper (your paper will likely be toomeager)
Starting too late with writing code (your algorithms will be slow andyou will not have enough time to run your experiments becauseeveryone is using the servers)
Copying from the internet
A presentation without pictures
Literally reading out every sentence on your slides
Too much text on your slides
Not writing the paper in LATEX
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 71 / 74
Course project schedule
Sep 24: deadline for registering as a group in Brightspace for thecourse project topic
Sep 25: remaining students/topics matched
From mid October onwards: presentations by students
Nov. 11: deadline for the first three sections (for peer review)
Nov. 23: optional deadline for “intermediary paper check”
Nov. 27: deadline for having decent code ready for code review
Dec. 9: deadline for full course project paper
Dec. 18: all grades submitted to student administration
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 72 / 74
Homework for next week
Make serious progress with Assignment 1
Make choice of participation in course explicit, before September 14:
In Brightspace, choose whether you are in the “On-site” or “On-line”group via “Course Tools”, “Groups”Un-enroll no later than September 14 should you want to do so
Next week: consult the list of project topics on course website,and think of what you may want to work on
Ask questions
Today: stick around at the end of the lab session if you are alreadycertain that you will take the course, and need a teammate
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 73 / 74
Lab session this week
From 10:30 to 12:15 in Gorlaeus room 03
Instructions on course website
Hands-on introduction to NetworkX
Continue working on Assignment 1
Frank Takes — SNACS — Lecture 2 — Advanced network concepts and centrality 74 / 74
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