1. Lecture 5: Network centrality Slides are modified from Lada Adamic

2. Measures and Metrics Knowing the structure of a network, we can calculate various useful quantities or measures that capture particular features of the network topology. basis of most of such measures are from social network analysis So far, Degree distribution, Average path length, Density Centrality Degree, Eigenvector, Katz, PageRank, Hubs, Closeness, Betweenness, …. Several other graph metrics Clustering coefficient, Assortativity, Modularity, … 2

3. Characterizing networks: Who is most central? ? ? ? 3

4. network centrality Which nodes are most ‘central’? Definition of ‘central’ varies by context/purpose Local measure: degree Relative to rest of network: closeness, betweenness, eigenvector (Bonacich power centrality), Katz, PageRank, … How evenly is centrality distributed among nodes? Centralization, hubs and authorities, … 4

5. centrality: who’s important based on their network position indegree In each of the following networks, X has higher centrality than Y according to a particular measure outdegreebetweennesscloseness 5

6. Outline Degree centrality Centralization Betweenness centrality Closeness centrality Eigenvector centrality Bonacich power centrality Katz centrality PageRank Hubs and Authorities 6

7. He who has many friends is most important. degree centrality (undirected) When is the number of connections the best centrality measure? o people who will do favors for you o people you can talk to (influence set, information access, …) o influence of an article in terms of citations (using in-degree) 7

8. degree: normalized degree centrality divide by the max. possible, i.e. (N-1) 8

9. Prestige in directed social networks when ‘prestige’ may be the right word admiration influence gift-giving trust directionality especially important in instances where ties may not be reciprocated (e.g. dining partners choice network) when ‘prestige’ may not be the right word gives advice to (can reverse direction) gives orders to (- ” -) lends money to (- ” -) dislikes distrusts 9

10. Extensions of undirected degree centrality - prestige degree centrality indegree centrality a paper that is cited by many others has high prestige a person nominated by many others for a reward has high prestige 10

11. Freeman’s general formula for centralization: (can use other metrics, e.g. gini coefficient or standard deviation) centralization: how equal are the nodes? How much variation is there in the centrality scores among the nodes? maximum value in the network 11

12. degree centralization examples C D = 0.167 C D = 1.0 12

13. degree centralization examples example financial trading networks high centralization: one node trading with many others low centralization: trades are more evenly distributed 13

14. when degree isn’t everything In what ways does degree fail to capture centrality in the following graphs? ability to broker between groups likelihood that information originating anywhere in the network reaches you… 14

15. Outline Degree centrality Centralization Betweenness centrality Closeness centrality Eigenvector centrality Bonacich power centrality Katz centrality PageRank Hubs and Authorities 15

16. betweenness: another centrality measure intuition: how many pairs of individuals would have to go through you in order to reach one another in the minimum number of hops? who has higher betweenness, X or Y? XY 16

17. betweenness on toy networks non-normalized version: ABCED A lies between no two other vertices B lies between A and 3 other vertices: C, D, and E C lies between 4 pairs of vertices (A,D),(A,E),(B,D),(B,E) note that there are no alternate paths for these pairs to take, so C gets full credit 17

18. Where g jk = the number of geodesics connecting j-k, and g jk = the number that actor i is on. Usually normalized by: number of pairs of vertices excluding the vertex itself betweenness centrality: definition 18 betweenness of vertex i paths between j and k that pass through i all paths between j and k directed graph: (N-1)*(N-2)

19. betweenness on toy networks non-normalized version: 19

20. betweenness on toy networks non-normalized version: 20 broker

21. Nodes are sized by degree, and colored by betweenness. example Can you spot nodes with high betweenness but relatively low degree? What about high degree but relatively low betweenness? 21

22. betweenness on toy networks non-normalized version: A B C E D why do C and D each have betweenness 1? They are both on shortest paths for pairs (A,E), and (B,E), and so must share credit: ½+½ = 1 Can you figure out why B has betweenness 3.5 while E has betweenness 0.5? 22

23. Alternative betweenness computations Slight variations in geodesic path computations inclusion of self in the computations Flow betweenness Based on the idea of maximum flow edge-independent path selection effects the results May not include geodesic paths Random-walk betweenness Based on the idea of random walks Usually yields ranking similar to geodesic betweenness Many other alternative definitions exist based on diffusion, transmission or flow along network edges 23

24. Extending betweenness centrality to directed networks We now consider the fraction of all directed paths between any two vertices that pass through a node Only modification: when normalizing, we have (N-1)*(N-2) instead of (N-1)*(N-2)/2, because we have twice as many ordered pairs as unordered pairs betweenness of vertex i paths between j and k that pass through i all paths between j and k 24

25. Directed geodesics A node does not necessarily lie on a geodesic from j to k if it lies on a geodesic from k to j k j 25

26. Outline Degree centrality Centralization Betweenness centrality Closeness centrality Eigenvector centrality Bonacich power centrality Katz centrality PageRank Hubs and Authorities 26

27. closeness: another centrality measure What if it’s not so important to have many direct friends? Or be “between” others But one still wants to be in the “middle” of things, not too far from the center 27

28. Closeness is based on the length of the average shortest path between a vertex and all vertices in the graph Closeness Centrality: Normalized Closeness Centrality closeness centrality: definition 28 depends on inverse distance to other vertices

29. closeness centrality: toy example ABCED 29

30. closeness centrality: more toy examples 30

31. degree number of connections denoted by size closeness length of shortest path to all others denoted by color how closely do degree and betweenness correspond to closeness? 31

32. Closeness centrality Values tend to span a rather small dynamic range typical distance increases logarithmically with network size In a typical network the closeness centrality C might span a factor of five or less It is difficult to distinguish between central and less central vertices a small change in network might considerably affect the centrality order Alternative computations exist but they have their own problems 32

33. Influence range The influence range of i is the set of vertices who are reachable from the node i 33

34. Extensions of undirected closeness centrality closeness centrality usually implies all paths should lead to you paths should lead from you to everywhere else usually consider only vertices from which the node i in question can be reached 34

35. Outline Degree centrality Centralization Betweenness centrality Closeness centrality Eigenvector centrality Bonacich power centrality Katz centrality PageRank Hubs and Authorities Applications to Information Retrieval LexRank 35

36. Eigenvalues and eigenvectors have their origins in physics, in particular in problems where motion is involved, although their uses extend from solutions to stress and strain problems to differential equations and quantum mechanics. Eigenvectors are vectors that point in directions where there is no rotation. Eigenvalues are the change in length of the eigenvector from the original length. The basic equation in eigenvalue problems is: Eigenvalues and eigenvectors Slides from Fred K. Duennebier

37. In words, this deceptively simple equation says that for the square matrix A, there is a vector x such that the product of Ax is a SCALAR,, that, when multiplied by x, results in the same product. The multiplication of vector x by a scalar constant is the same as stretching or shrinking the coordinates by a constant value. The vector x is called an eigenvector and the scalar is called an eigenvalue. Eigenvalues and eigenvectors

38. Do all matrices have real eigenvalues? No, they must be square and the determinant of A- I must equal zero. This is easy to show: This can only be true if det(A- I )=|A- I |=0 Are eigenvectors unique? No, if x is an eigenvector, then  x is also an eigenvector and  is an eigenvalue. A(  x)=  Ax =  x =  (  x) (E.02) (E.03) (E.04) (E.01)

39. How do you calculate eigenvectors and eigenvalues? Expand equation (E.03): det(A- I )=|A- I |=0 for a 2x2 matrix: For a 2-dimensional problem such as this, the equation above is a simple quadratic equation with two solutions for. In fact, there is generally one eigenvalue for each dimension, but some may be zero, and some complex. (E.05)

40. The solution to E.05 is: (E.06) This “characteristic equation” does not involve x, and the resulting values of can be used to solve for x. Consider the following example: (E.07) Eqn. E.07 doesn’t work here because a 11 a 22 -a 12 a 12 =0, so we use E.06:

41. We see that one solution to this equation is =0, and dividing both sides of the above equation by yields =5. Thus we have our two eigenvalues, and the eigenvectors for the first eigenvalue, =0 are: These equations are multiples of x=-2y, so the smallest whole number values that fit are x=2, y=-1

42. For the other eigenvalue, =5: This example is rather special; A -1 does not exist, the two rows of A- I are dependent and thus one of the eigenvalues is zero. (Zero is a legitimate eigenvalue!) EXAMPLE: A more common case is A =[1.05.05 ;.05 1] used in the strain exercise. Find the eigenvectors and eigenvalues for this A, and then calculate [V,D]=eig[A]. The procedure is: 1)Compute the determinant of A- I 2)Find the roots of the polynomial given by | A- I|=0 3)Solve the system of equations (A- I)x=0

43. Or we could find the eigenvalues of A and obtain A 100 very quickly using eigenvalues. What is A 100 ? We can get A 100 by multiplying matrices many many times: What good are such things? Consider the matrix:

44. For now, I’ll just know that there are two eigenvectors for A: The eigenvectors are x 1 =[.6 ;.4] and x 2 =[1 ; -1], and the eigenvalues are 1 =1 and 2 =0.5. Note that, if we multiply x 1 by A, we get x 1. If we multiply x 1 by A again, we STILL get x 1. Thus x 1 doesn’t change as we mulitiply it by A n.

45. What about x 2 ? When we multiply A by x 2, we get x 2 /2, and if we multiply x 2 by A 2, we get x 2 /4. This number gets very small fast. Note that when A is squared the eigenvectors stay the same, but the eigenvalues are squared! Back to our original problem; we note that for A 100, the eigenvectors will be the same, the eigenvalues 1 =1 and 2 =(0.5) 100, which is effectively zero. Each eigenvector is multiplied by its eigenvalue whenever A is applied,

46. Outline Degree centrality Centralization Betweenness centrality Closeness centrality Eigenvector centrality Bonacich power centrality Katz centrality PageRank Hubs and Authorities Applications to Information Retrieval LexRank 46

47. Eigenvector Centrality 47

48. Eigenvector Centrality 48

49. Eigenvector Centrality Can be calculated for directed graphs as well We need to decide between incoming or outgoing edges A has no incoming edges, hence a centrality of 0 B has only an incoming edge from A hence its centrality is also 0 Only vertices that are in a strongly connected component of two or more vertices or the out-component of such a component have non-zero centrality 49 B A C D E

50. Katz centrality 50

51. Katz Centrality:   The magnitude of  reflects the radius of power Small values of  weight local structure Larger values weight global structure  If  > 0, ego has higher centrality when tied to people who are central  If  < 0, then ego has higher centrality when tied to people who are not central  With  = 0, you get degree centrality 51

52.  =.25 Katz Centrality: examples  =-.25 Why does the middle node have lower centrality than its neighbors when  is negative? 52

53. PageRank: bringing order to the web It’s in the links: links to URLs can be interpreted as endorsements or recommendations the more links a URL receives, the more likely it is to be a good/entertaining/provocative/authoritative/interesting information source but not all link sources are created equal a link from a respected information source a link from a page created by a spammer Many webpages scattered across the web an important page, e.g. slashdot if a web page is slashdotted, it gains attention

54. PageRank 54

55. Ranking pages by tracking a drunk A random walker following edges in a network for a very long time will spend a proportion of time at each node which can be used as a measure of importance

56. Trapping a drunk Problem with pure random walk metric: Drunk can be “trapped” and end up going in circles

57. Ingenuity of the PageRank algorithm Allow drunk to teleport with some probability e.g. random websurfer follows links for a while, but with some probability teleports to a “random” page bookmarked page or uses a search engine to start anew

58. PageRank algorithm where p 1,p 2,...,p N are the pages under consideration, M(p i ) is the set of pages that link to p i, L(p j ) is the number of outbound links on page p j, and N is the total number of pages. d is the random jumping probability (d = 0.85 for google)

59. GUESS PageRank demo Exercise: PageRank What happens to the relative PageRank scores of the nodes as you increase the teleportation probability? Can you construct a network such that a node with low indegree has the highest PageRank? http://projects.si.umich.edu/netlearn/GUESS/pagerank.html

60. example: probable location of random walker after 1 step 1 2 3 4 5 7 68 t=0 t=1 20% teleportation probability

61. 1 2 3 4 5 7 68 example: location probability after 10 steps t=0 t=1 t=10

62. Matrix-based Centrality measures 62 Divide by out-degree No division with constant termwithout constant term PageRank Degree centrality Eigenvector centrality Katz centrality

63. Outline Degree centrality Centralization Betweenness centrality Closeness centrality Eigenvector centrality Bonacich power centrality Katz centrality PageRank Hubs and Authorities Applications to Information Retrieval LexRank 63

64. Hubs and Authorities In directed networks, vertices that point to important resources should also get a high centrality e.g. review articles, web indexes recursive definition: hubs are nodes that links to good authorities authorities are nodes that are linked to by good hubs

65. Hyperlink-Induced Topic Search HITS algorithm start with a set of pages matching a query expand the set by following forward and back links take transition matrix E, where the i,j th entry E ij =1/n i where i links to j, and n i is the number of links from i then one can compute the authority scores a, and hub scores h through an iterative approach:

66. Outline Degree centrality Centralization Betweenness centrality Closeness centrality Eigenvector centrality Bonacich power centrality Katz centrality PageRank Hubs and Authorities Applications to Information Retrieval LexRank 66

67. Applications to Information Retrieval Can we use the notion of centrality to pick the best summary sentence? Can we use the subgraph of query results to infer something about the query? Can we use a graph of word translations to expand dictionaries? disambiguate word meanings? How might one use the HITS algorithm for document summarization? Consider a bipartite graph of sentences and words

68. Centrality in summarization Extractive summarization pick k sentences that are most representative of a collection of n sentences Motivation: capture the most central words in a document or cluster Centroid score [Radev & al. 2000, 2004a] Alternative methods for computing centrality?

69. Sample multidocument cluster 1 (d1s1) Iraqi Vice President Taha Yassin Ramadan announced today, Sunday, that Iraq refuses to back down from its decision to stop cooperating with disarmament inspectors before its demands are met. 2 (d2s1) Iraqi Vice president Taha Yassin Ramadan announced today, Thursday, that Iraq rejects cooperating with the United Nations except on the issue of lifting the blockade imposed upon it since the year 1990. 3 (d2s2) Ramadan told reporters in Baghdad that "Iraq cannot deal positively with whoever represents the Security Council unless there was a clear stance on the issue of lifting the blockade off of it. 4 (d2s3) Baghdad had decided late last October to completely cease cooperating with the inspectors of the United Nations Special Commission (UNSCOM), in charge of disarming Iraq's weapons, and whose work became very limited since the fifth of August, and announced it will not resume its cooperation with the Commission even if it were subjected to a military operation. 5 (d3s1) The Russian Foreign Minister, Igor Ivanov, warned today, Wednesday against using force against Iraq, which will destroy, according to him, seven years of difficult diplomatic work and will complicate the regional situation in the area. 6 (d3s2) Ivanov contended that carrying out air strikes against Iraq, who refuses to cooperate with the United Nations inspectors, ``will end the tremendous work achieved by the international group during the past seven years and will complicate the situation in the region.'' 7 (d3s3) Nevertheless, Ivanov stressed that Baghdad must resume working with the Special Commission in charge of disarming the Iraqi weapons of mass destruction (UNSCOM). 8 (d4s1) The Special Representative of the United Nations Secretary-General in Baghdad, Prakash Shah, announced today, Wednesday, after meeting with the Iraqi Deputy Prime Minister Tariq Aziz, that Iraq refuses to back down from its decision to cut off cooperation with the disarmament inspectors. 9 (d5s1) British Prime Minister Tony Blair said today, Sunday, that the crisis between the international community and Iraq ``did not end'' and that Britain is still ``ready, prepared, and able to strike Iraq.'' 10 (d5s2) In a gathering with the press held at the Prime Minister's office, Blair contended that the crisis with Iraq ``will not end until Iraq has absolutely and unconditionally respected its commitments'' towards the United Nations. 11 (d5s3) A spokesman for Tony Blair had indicated that the British Prime Minister gave permission to British Air Force Tornado planes stationed in Kuwait to join the aerial bombardment against Iraq. (DUC cluster d1003t)

70. Cosine between sentences Let s 1 and s 2 be two sentences. Let x and y be their representations in an n-dimensional vector space The cosine between is then computed based on the inner product of the two. The cosine ranges from 0 to 1.

71. LexRank (Cosine centrality) 1234567891011 11.000.450.020.170.030.220.030.280.06 0.00 20.451.000.160.270.030.190.030.210.030.150.00 30.020.161.000.030.000.010.030.040.000.010.00 40.170.270.031.000.010.160.280.170.000.090.01 50.03 0.000.011.000.290.050.150.200.040.18 60.220.190.010.160.291.000.050.290.040.200.03 7 0.280.05 1.000.060.00 0.01 80.280.210.040.170.150.290.061.000.250.200.17 90.060.030.00 0.200.040.000.251.000.260.38 100.060.150.010.090.040.200.000.200.261.000.12 110.00 0.010.180.030.010.170.380.121.00

72. d4s1 d1s1 d3s2 d3s1 d2s3 d2s1 d2s2 d5s2 d5s3 d5s1 d3s3 Lexical centrality (t=0.3)

73. d4s1 d1s1 d3s2 d3s1 d2s3 d2s1 d2s2 d5s2 d5s3 d5s1 d3s3 Lexical centrality (t=0.2)

74. d4s1 d1s1 d3s2 d3s1 d2s3d3s3 d2s1 d2s2 d5s2 d5s3 d5s1 Lexical centrality (t=0.1) Sentences vote for the most central sentence… d4s1 d3s2 d2s1

75. LexRank T 1 …T n are pages that link to A, c(T i ) is the outdegree of pageT i, and N is the total number of pages. d is the “damping factor”, or the probability that we “jump” to a far-away node during the random walk. It accounts for disconnected components or periodic graphs. When d = 0, we have a strict uniform distribution. When d = 1, the method is not guaranteed to converge to a unique solution. Typical value for d is between [0.1,0.2] (Brin and Page, 1998). Güneş Erkan and Dragomir R. Radev, LexRank: Graph-based Lexical Centrality as Salience in Text Summarization

76. lab: Lexrank demo http://tangra.si.umich.edu/demos/lexrank/ how does the summary change as you: increase the cosine similarity threshold for an edge how similar two sentences have to be? increase the salience threshold (minimum degree of a node)

77. Menczer, Filippo (2004) The evolution of document networks. Content similarity distributions for web pages (DMOZ) and scientific articles (PNAS)

78. what is that good for? How could you take advantage of the fact that pages that are similar in content tend to link to one another?

79. What can networks of query results tell us about the query? Jure Leskovec, Susan Dumais: Web Projections: Learning from Contextual Subgraphs of the Web If query results are highly interlinked, is this a narrow or broad query? How could you use query connection graphs to predict whether a query will be reformulated?

80. How can bipartite citation graphs be used to find related articles? co-citation: both A and B are cited by many other papers (C, D, E …) A B C D E bibliographic coupling: both A and B are cite many of the same articles (F,G,H …) F G H A B

81. which of these pairs is more proximate according to cycle free effective conductance: the probability that you reach the other node before cycling back on yourself, while doing a random walk….

82. Proximity as cycle free effective conductance Measuring and Extracting Proximity in Networks by Yehuda Koren, Stephen C. North, Chris Volinsky, KDD 2006 demo: http://public.research.att.com/~volinsky/cgi-bin/prox/prox.plhttp://public.research.att.com/~volinsky/cgi-bin/prox/prox.pl

83. Source: undetermined Using network algorithms (specifically proximity) to improve movie recommendations can pay off

84. final IR application: machine translation not all pairwise translations are available e.g. between rare languages in some applications, e.g. image search, a word may have multiple meanings “spring” is an example in english But in other languages, the word may be unambiguous. automated translation could be the key or

85. final IR application: machine translation spring English printemps French primavera Spanish ربيع Arabic koanga Maori udaherri Basque 1 vzmet Slovenian пружина Russian ressort French 2 veer Dutch рысора Belarusian … … … … 3 1 11 1 3 3 … … … 3 3 … 4 4 4 2 2 2 4 2 2 4 3 4 1 if we combine all known word pairs, can we construct additional dictionaries between rare languages? source: Reiter et al., ‘Lexical Translation with Application to Image Search on the Web ’Lexical Translation with Application to Image Search on the Web

86. Automatic translation & network structure Two words more likely to have same meaning if there are multiple indirect paths of length 2 through other languages spring English printemps French primavera Spanish ربيع Arabic koanga Maori udaherri Basque 1 … … 3 1 11 1 3 3 … … 3 3 3 1 пружина Russian … 2 2

87. summary the web can be studied as a network this is useful for retrieving relevant content network concepts can be used in other IR tasks summarization query prediction machine translation