Often centrality measures are used in social network analysis. The goal of this presentation is to explain how different centrality works and how they can be compared.
Centrality measures covered: degree, closeness, harmonic, Lin's index, betweenness, eigenvector, seeley's index, pagerank, hits, SALSA
chemical bonding Essentials of Physical Chemistry2.pdf
Network centrality measures and their effectiveness
1. centrality measures
Survey and comparisons
Authors: Antonio Esposito
Emanuele Pesce
Supervisors: Prof. Vincenzo Auletta
Ph.D Diodato Ferraioli
Aprile 2015
University of Salerno, deparment of computer science
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4. centrality of a network
What is a centrality measure?
∙ Given a network, the centrality is a quantitative measure which
aims at reveling the importance of a node
∙ The more a node is centered, the more it is important
∙ Formally, a centrality measure is a real valued function on the
nodes of a graph
What do you mean by center?
∙ There are many intuitive ideas about what a center is, so there are
many different centrality measures
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5. definition of center
The center of a star is at the same time:
∙ the node with largest degree
∙ the node that is closest to the other nodes
∙ the node through which most shortest paths pass
∙ the node with the largest number of incoming paths
∙ the node that maximize the dominant eigenvector of the graph
matrix
Several centrality indices
∙ Different centrality indices capture different properties of a
network
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6. centrality: some applications
Centrality is used often for detecting:
∙ how influential a person is in a social network?
∙ how well used a road is in a transportation network?
∙ how important a web page is?
∙ how important a room is in a building?
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10. geometric measures
The idea
∙ In geometric measures the importance is a function of distances.
∙ A geometric centrality depends on how many nodes exist at every
distance
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11. geometric measures: indegree centrality
∙ Indegree centrality is defined as the number of incoming arcs of a
node x
Cindegree(x) = d−
(x) (1)
∙ The node with the highest degree is the most important
When to use it?
∙ To identify people whom you can talk to
∙ To identify people whom will do favors for you
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13. indegree centrality: examples
Indegree centrality can be deceiving because it is a local measure
Indegree centrality doeas not work well for:
∙ detecting nodes that are broker between two groups
∙ predicting if an information reaches a node
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14. geometric measures: closeness centrality
∙ Closeness centrality of x is defined by:
Ccloseness(x) =
1
∑
d(y,x)<∞
d(y, x)
(2)
∙ Divide it for the max number of nodes (n − 1) to normalize the closeness centrality
∙ Nodes with empty coreachable set have centrality 0
∙ The closer a node is to all others, the more it is important
When to use it?
∙ To identify people whom tend to be very influential person within their local
network
∙ They may often not be public figures, but they are often respected locally
∙ To measure how long it will take to spread information from node x to all other
nodes
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16. geometric measures: harmonic centrality
∙ Harmonic centrality of x, with the convention ∞−1
= 0 is defined
by:
Charmonic(x) =
1
∑
y̸=x
d(y, x)
(3)
∙ It is correlated to closeness centrality in simple networks, but it
also accounts for nodes y that cannot reach x
When to use it?
∙ The same for the closeness but it can be applied to graphs that
are not connected
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18. lin’s index
∙ Lin’s index of x
Clin(x) =
|{y | d(y, x) < ∞}|2
∑
d(y,x)<∞
d(y, x)
(4)
∙ As closeness, but here nodes with a larger coreachable set are
more important
A fact
∙ Surprisingly, Lin’s index was ignored in literature, even though it
seems to provide a reasonable solution for detecting centers in
networks
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19. path-based measures
The idea
∙ Path-based measures exploit not only the existence of shortest
paths but actually take into examination all shortest paths (or all
paths) coming into a node
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20. path-based measures: betweenness centrality
∙ The intuition behind the betweenness centrality is to measure the
probability that a random shortest path passes though a given
node. Betweenness of x is defined as:
Cbetweenness(x) =
∑
y,z̸=x,αyz̸=0
αyz(x)
αyz
(5)
∙ αyz is the number of shortest paths going from y to z
∙ αyz(x) is the number of shortest paths that pass through x
∙ The higher is the fraction of shortest paths which passes through
a node, the more the node is important
When to use it?
∙ To identify nodes which have a large influence on the transfer of
items through the network
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23. betweenness and closeness
∙ Betweenness and closeness measures applied to the same
network
∙ The nodes are sized by degree and colored by betweenness
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24. spectral measures
The idea
∙ In spectral measures the importance is related to the iterated
computation of the left dominant eigenvector of the adjacency
matrix.
∙ In the spectral centrality the importance of a node is given by the
importance of the neighbourhood
∙ The more important are the nodes pointing at you, the more
important you are
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25. spectral measures
How many of them?
∙ The dominant eigenvector
∙ Seeley’s index
∙ Katz’s index
∙ PageRank
∙ HITS
∙ SALSA
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26. spectral measures: some useful notation
Given the adjacency matrix A we can compute:
∙ The ℓ1 norm of the matrix ¯A
∙ Each element of the row i is divided by the sum of its elements
∙ The symmetric graph G′
of the given graph G
∙ The transpose of AT
of the adjacency matrix A
∙ The number of k−lenght path from a node i to another node j
∙ Ak
: in such a matrix, each element aij will be the number of paths with
lenght = k from the node i to the node j
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27. spectral measures: the left dominant eigenvector
Dominant eigenvector
∙ Taking in consideration the left dominant eigenvector means to consider the
incoming edges of a node.
∙ To find out the node’s importance, we perform an iterated computation of:
xt+1
i
=
1
λ
n∑
i=0
A
(t)
ij
(6)
where:
∙ x0
i = 1 ∀ i at step 0
∙ xt
is the score after t iterations
∙ λ is the dominant eigenvalue of the adjacency matrix A
∙ After that, the vector x is normalized and the process iterated until convergence
∙ Each node starts with the same score. Then, in iteration, it receives the sum of the
connected neighbor’s score
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28. eigenvector centrality: example
In figure 1 there are applications on the same graph of degree and
eigenvector centrality
Figure 1: Degree and eigenvector centrality
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29. spectral measures: seeley’s index
∙ Why give away all of our importance?
∙ It would have more sense to equally divide our importance among our successors
∙ The process will remains the same, but from an algebric point of view that means
normalizing each row of the adjacency matrix:
xt+1
i
=
1
λ
n∑
i=0
¯A
(t)
ij
(7)
where:
∙ x0
i = 1 ∀ i at step 0
∙ xt
is the score after t iterations
∙ λ is the dominant eigenvalue of the adjacency matrix A
∙ ¯A is the normalized form of the adjacency matrix
∙ Isolated nodes of a non strongly connected graph will have null score over
iterations
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30. spectral measures: katz’s index
Katz’s index weighs all incoming paths to a node and then compute:
x = 1
∞∑
i=0
βi
Ai
(8)
where:
∙ x is the output’s scores vector
∙ 1 is the weight’s vector (for example all 1)
∙ βi
is an attenuation factor (β < 1
λ )
∙ Ai
contains in the generic element aij the number of i-lenght path
from i to j
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31. spectral measures: pagerank
PageRank - a little overview
∙ It’s supposed to be how the Google’s search engine works
∙ It is the unique vector p satisfying
p = (1 − α)v(1 − α¯A)−1
∙ where:
∙ α ∈ [0, 1) is a dumping factor
∙ v is a preference vector (a distribution)
∙ ¯A is the ℓ1 normalized adjacency matrix
∙ As shown, PageRank and Katz’s index differ by a constant factor
and the ℓ1 normalization of the adjacency matrix A
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32. spectral measures: eigenvector and pagerank
In figure 2 there are applications of the same graph of eigenvector
PageRank centrality
Figure 2: Degree and eigenvector centrality
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33. spectral measures: hits
HITS - a little overview by Kleinberg
∙ The key here is the mutual reinforcement
∙ A node ( such as a page ) is authoritative if it is pointed by many
good hubs
∙ Hubs: pages containing good list of authoritative pages
∙ Then an Hub is good if it points to many authoritative pages
∙ We iteratively compute the:
∙ ai: authoritativeness score ( where a0 = 1)
∙ hi: hubbiness score
as the following:
hi+1 = aiAT
ai+1 = hi+1A
∙ This process converges to the left dominant eigenvector of the
matrix AT
A giving the final score of authoritativeness, called ”HITS”
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34. spectral measures: salsa
SALSA was ideated by Lempel and Moran
∙ Based on the same mutual reinforcement between
authoritativeness and hubbiness, but ℓ1normalizing the matrices A
and AT
.
∙ Starting value: a0 = 1
∙ hi+1 = ai
¯AT
∙ ai+1 = ai
¯A
∙ Contrarily to HITS there is no need of a large number of iteration
with SALSA
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35. spectral measures: some applications
∙ Left dominant eigenvector: the idea on which networks structure
analysisis is based
∙ Seeley’s index: feedback’s network
∙ Katz’s index: citations networks
∙ expecially good with direct acyclic graphs (where the basic dominant
eigenvector don’t perform well)
∙ HITS: web page’s citations
∙ Pagerank: Google’s search engine
∙ SALSA: link structure analysis
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37. axioms for centrality
∙ Boldi and Vigna in 2013 tried to provide a method to evaluate and
compare different centrality measures
∙ They defined three axioms that an index should satisfy to behave
predictably
∙ Size axiom
∙ Density axiom
∙ The score-monotonicity axiom
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38. axioms for centrality: size axiom
Given a graph Sk,p (figure 3), made by a k − clique and a directed
p − cycle, the size axioms is satisfied if there are threshold values,
of p and k such that:
∙ p > k (if the cycle is very large) the nodes of the cycle are more
important
∙ k > p the nodes of clique are more important
∙ intuitively, for p = k, the nodes of the clique are more important
Figure 3: Graph Sk,p 37
39. axioms for centrality: density axiom
∙ Given a graph Dk,p(figure 4), made by a k − clique and a directed
p − cycle connected by a bidirectional bridge x ↔ y, where x is a
node of the clique and y a node of the cycle.
∙ A centrality measure satisfies the density axiom for k = p, if the
centrality of x is strictly larger than the centrality of y.
Figure 4: Graph Gk,p
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40. axioms for centrality: the score-monotonicity axiom
∙ A centrality measure satisfies the score-monotonicity axiom if for
every graph G and every pair of node x, y such that x ↛ y, when we
add x → y to G the centrality of y increases.
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41. axioms for centrality: centrality axioms: comparisons
Figure 5: For each centrality and each axiom, the report whether it is
satisfied
The harmonic centrality satisfies all axioms.
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42. information retrieval: sanity check
∙ Boldi and Vigna have applied centrality measures on standard
datasets in order to find out the behavior of different indices
∙ There are standard datasets with associated queries and ground
truth about which documents are relevant for every query
∙ Those collections are typically used to compare the merits and the
demerits about retrieval methods
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43. information retrieval: datasets
Dataset GOV2, tested in two different ways:
∙ with all links: complete dataset
∙ with inter-host link only: links between pages of the same host
are excluted from the graph
Measures of effectiveness chosen:
∙ P@10: precision at 10, fraction of relevant documents retrieved
among the first ten
∙ NDCG@10: discounted cumulative gain at 10, measure the
usefulness, or gain, of a document based on its position in the
result list
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44. information retrieval: results
For each centrality measure the discounted cumulative and precision at 10, on GOV2
dataset using all links (on the left) and using only inter-host links (on the right).
Figure 6: All links Figure 7: Inter-host links 43
46. conclusions
∙ A very simple measure as harmonic centrality, turned out to be a
good notion of centrality.
∙ it satisfies all centrality axioms proposed
∙ it works well to retrieve information
Choose the right measure
∙ No centrality measure is better than the others in every situation
∙ Some are better than others to reach a particular goal, but it
depends on the specific application domain
∙ So, the best approach is to understand which measure fits the
problem better
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47. references and useful resources
Paolo Boldi and Sebastiano Vigna
Axioms for centrality.
Nicola Perra and Santo Fortunato
Spectral centrality measures in complex networks.
M. E. J. Newman
Networks: an introduction
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