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

0 outline

Introduction

Centrality measures Geometric measures Path-based measures Spectral measures

Effectiveness of centrality measures Axioms for centrality Information retrieval

Conclusions

1 introduction 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

3 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

4 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?

5 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 ∙ SALSA 7 different centrality measures

Example of different centrality measures applied to the same network

8 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

9 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

10 indegree centrality: examples

Indegree measure applied on different networks

11 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

12 geometric measures: closeness centrality

∙ Closeness centrality of x is defined by:

1 C (x) = ∑ (2) closeness d(y, x) d(y,x)<∞

∙ 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

13 closeness centrality: example

Closeness measure applied to different networks

14 geometric measures: harmonic centrality

∙ Harmonic centrality of x, with the convention ∞−1 = 0 is defined by: 1 C (x) = ∑ (3) harmonic d(y, x) y≠ x

∙ 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

15 harmonic centrality: examples

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

16 lin’s index

∙ Lin’s index of x |{y | d(y, x) < ∞}|2 C (x) = ∑ (4) lin d(y, x) d(y,x)<∞

∙ 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

17 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

18 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: ∑ αyz(x) Cbetweenness(x) = (5) αyz y,z≠ x,αyz≠ 0

∙ 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 19 betweenness centrality: examples

Betweenness applied to different networks

20 betweenness and indegree

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

21 betweenness and closeness

∙ Betweenness and closeness measures applied to the same network ∙ The nodes are sized by degree and colored by betweenness

22 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

23 spectral measures

How many of them?

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

24 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 k ∙ A : in such a matrix, each element aij will be the number of paths with lenght = k from the node i to the node j

25 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:

1 ∑n xt+1 = A(t) (6) i λ ij i=0 where: 0 ∀ ∙ xi = 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

26 eigenvector centrality: example

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

Figure 1: Degree and eigenvector centrality 27 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:

1 ∑n xt+1 = A¯(t) (7) i λ ij i=0

where: 0 ∀ ∙ xi = 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

28 spectral measures: katz’s index

Katz’s index weighs all incoming paths to a node and then compute: ∑∞ x = 1 βiAi (8) i=0

where:

∙ x is the output’s scores vector ∙ 1 is the weight’s vector (for example all 1) i 1 ∙ β is an attenuation factor (β < λ ) i ∙ A contains in the generic element aij the number of i-lenght path from i to j

29 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 factor

and the ℓ1 normalization of the adjacency matrix A

30 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

31 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: T hi+1 = aiA ai+1 = hi+1A ∙ This process converges to the left dominant eigenvector of the matrix ATA giving the final score of authoritativeness, called ”HITS” 32 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 ¯T ∙ hi+1 = aiA ¯ ∙ ai+1 = aiA ∙ Contrarily to HITS there is no need of a large number of iteration with SALSA

33 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

34 effectiveness of centrality mea- sures 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

36 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 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

38 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.

39 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.

40 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

41 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

42 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 conclusions 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

45 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

46 Thank you for your attention!

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