centrality measuresSurvey and comparisons

Authors: Antonio EspositoEmanuele Pesce

Supervisors: Prof. Vincenzo Auletta

Ph.D Diodato Ferraioli

Aprile 2015

University of Salerno, deparment of computer science

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outline

Introduction

Centrality measures

Geometric measures

Path-based measures

Spectral measures

Effectiveness of centrality measures

Axioms for centrality

Information retrieval

Conclusions

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introduction

centrality of a network

What is a centrality measure?

∙ Given a network, the centrality is a quantitative measure whichaims 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 thenodes of a graph

What do you mean by center?

∙ There are many intuitive ideas about what a center is, so there aremany different centrality measures

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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 graphmatrix

Several centrality indices

∙ Different centrality indices capture different properties of anetwork

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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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centrality measures

centrality measures

Geometric measures∙ Indegree

∙ Closeness

∙ Harmonic

∙ Lin’s Index

Path-based measures∙ Betweeness

Spectral measures∙ The left dominant eigenvector

∙ Seeley’s index

∙ Katz’s index

∙ PageRank

∙ HITS

∙ SALSA7

different centrality measures

Example of different centrality measures applied to the samenetwork

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geometric measures

The idea

∙ In geometric measures the importance is a function of distances.∙ A geometric centrality depends on how many nodes exist at everydistance

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geometric measures: indegree centrality

∙ Indegree centrality is defined as the number of incoming arcs of anode 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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indegree centrality: examples

Indegree measure applied on different networks

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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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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 localnetwork

∙ 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 othernodes

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closeness centrality: example

Closeness measure applied to different networks

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geometric measures: harmonic centrality

∙ Harmonic centrality of x, with the convention∞−1 = 0 is definedby:

Charmonic(x) =1∑

y=xd(y, x) (3)

∙ It is correlated to closeness centrality in simple networks, but italso accounts for nodes y that cannot reach x

When to use it?

∙ The same for the closeness but it can be applied to graphs thatare not connected

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harmonic centrality: examples

Harmonic and indegree measures applied to the same network(Zachary’s karate club)

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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 aremore important

A fact

∙ Surprisingly, Lin’s index was ignored in literature, even though itseems to provide a reasonable solution for detecting centers innetworks

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path-based measures

The idea

∙ Path-based measures exploit not only the existence of shortestpaths but actually take into examination all shortest paths (or allpaths) coming into a node

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path-based measures: betweenness centrality

∙ The intuition behind the betweenness centrality is to measure theprobability that a random shortest path passes though a givennode. Betweenness of x is defined as:

Cbetweenness(x) =∑

y,z=x,αyz =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 througha node, the more the node is important

When to use it?

∙ To identify nodes which have a large influence on the transfer ofitems through the network

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betweenness centrality: examples

Betweenness applied to different networks

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betweenness and indegree

Betweenness and indegree measures applied to the same network(Zachary’s karate club)

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betweenness and closeness

∙ Betweenness and closeness measures applied to the samenetwork

∙ The nodes are sized by degree and colored by betweenness

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spectral measures

The idea

∙ In spectral measures the importance is related to the iteratedcomputation of the left dominant eigenvector of the adjacencymatrix.

∙ In the spectral centrality the importance of a node is given by theimportance of the neighbourhood

∙ The more important are the nodes pointing at you, the moreimportant you are

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spectral measures

How many of them?

∙ The dominant eigenvector∙ Seeley’s index∙ Katz’s index∙ PageRank∙ HITS∙ SALSA

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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 withlenght = k from the node i to the node j

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spectral measures: the left dominant eigenvector

Dominant eigenvector

∙ Taking in consideration the left dominant eigenvector means to consider theincoming edges of a node.

∙ To find out the node’s importance, we perform an iterated computation of:

xt+1i =1λ

n∑i=0

A(t)ij (6)

where:

∙ x0i = 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 theconnected neighbor’s score

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eigenvector centrality: example

In figure 1 there are applications on the same graph of degree andeigenvector centrality

Figure 1: Degree and eigenvector centrality27

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 meansnormalizing each row of the adjacency matrix:

xt+1i =1λ

n∑i=0

A(t)ij (7)

where:

∙ x0i = 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 overiterations

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spectral measures: katz’s index

Katz’s index weighs all incoming paths to a node and then compute:

x = 1∞∑i=0

βiAi (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 pathfrom i to j

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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− αA)−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 factorand the ℓ1 normalization of the adjacency matrix A

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spectral measures: eigenvector and pagerank

In figure 2 there are applications of the same graph of eigenvectorPageRank centrality

Figure 2: Degree and eigenvector centrality

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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 manygood 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 = aiATai+1 = hi+1A

∙ This process converges to the left dominant eigenvector of thematrix ATA giving the final score of authoritativeness, called ”HITS”

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spectral measures: salsa

SALSA was ideated by Lempel and Moran

∙ Based on the same mutual reinforcement betweenauthoritativeness and hubbiness, but ℓ1normalizing the matrices Aand AT.∙ Starting value: a0 = 1∙ hi+1 = aiAT

∙ ai+1 = aiA

∙ Contrarily to HITS there is no need of a large number of iterationwith SALSA

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spectral measures: some applications

∙ Left dominant eigenvector: the idea on which networks structureanalysisis is based

∙ Seeley’s index: feedback’s network∙ Katz’s index: citations networks∙ expecially good with direct acyclic graphs (where the basic dominanteigenvector don’t perform well)

∙ HITS: web page’s citations∙ Pagerank: Google’s search engine∙ SALSA: link structure analysis

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effectiveness of centrality mea-sures

axioms for centrality

∙ Boldi and Vigna in 2013 tried to provide a method to evaluate andcompare different centrality measures

∙ They defined three axioms that an index should satisfy to behavepredictably∙ Size axiom∙ Density axiom∙ The score-monotonicity axiom

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axioms for centrality: size axiom

Given a graph Sk,p (figure 3), made by a k− clique and a directedp− 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 moreimportant

∙ 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

axioms for centrality: density axiom

∙ Given a graph Dk,p(figure 4), made by a k− clique and a directedp− cycle connected by a bidirectional bridge x ↔ y, where x is anode of the clique and y a node of the cycle.

∙ A centrality measure satisfies the density axiom for k = p, if thecentrality of x is strictly larger than the centrality of y.

Figure 4: Graph Gk,p

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axioms for centrality: the score-monotonicity axiom

∙ A centrality measure satisfies the score-monotonicity axiom if forevery graph G and every pair of node x, y such that x ↛ y, when weadd x → y to G the centrality of y increases.

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axioms for centrality: centrality axioms: comparisons

Figure 5: For each centrality and each axiom, the report whether it issatisfied

The harmonic centrality satisfies all axioms.

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information retrieval: sanity check

∙ Boldi and Vigna have applied centrality measures on standarddatasets in order to find out the behavior of different indices

∙ There are standard datasets with associated queries and groundtruth about which documents are relevant for every query

∙ Those collections are typically used to compare the merits and thedemerits about retrieval methods

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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 hostare excluted from the graph

Measures of effectiveness chosen:

∙ P@10: precision at 10, fraction of relevant documents retrievedamong the first ten

∙ NDCG@10: discounted cumulative gain at 10, measure theusefulness, or gain, of a document based on its position in theresult list

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information retrieval: results

For each centrality measure the discounted cumulative and precision at 10, on GOV2dataset 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

conclusions

conclusions

∙ A very simple measure as harmonic centrality, turned out to be agood 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 itdepends on the specific application domain

∙ So, the best approach is to understand which measure fits theproblem better

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references and useful resources

Paolo Boldi and Sebastiano VignaAxioms for centrality.

Nicola Perra and Santo FortunatoSpectral centrality measures in complex networks.

M. E. J. NewmanNetworks: an introduction

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Thank you for your attention!

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