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Transcript of Lecture 5: Network centrality Slides are modified from Lada Adamic.
Lecture 5:
Network centrality
Slides are modified from Lada Adamic
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
Characterizing networks:Who is most central?
?
?
?
3
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
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
outdegree betweenness closeness
5
Outline
Degree centrality Centralization
Betweenness centrality Closeness centrality
Eigenvector centrality Bonacich power centrality
Katz centrality PageRank Hubs and Authorities
6
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 youo people you can talk to (influence set, information access, …)o influence of an article in terms of citations (using in-degree)
7
degree: normalized degree centrality
divide by the max. possible, i.e. (N-1)
8
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
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
Freeman’s general formula for centralization: (can use other metrics, e.g. gini coefficient or standard deviation)
CD CD (n*) CD (i)
i1
g[(N 1)(N 2)]
centralization: how equal are the nodes?
How much variation is there in the centrality scores among the nodes?
maximum value in the network
11
degree centralization examples
CD = 0.167
CD = 0.167CD = 1.0
12
degree centralization examples
example financial trading networks
high centralization: one node trading with many others
low centralization: trades are more evenly distributed
13
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
Outline
Degree centrality Centralization
Betweenness centrality Closeness centrality
Eigenvector centrality Bonacich power centrality
Katz centrality PageRank Hubs and Authorities
15
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
betweenness on toy networks
non-normalized version:
A B C ED
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
CB (i) g jk (i) /g jkjk
Where gjk = the number of geodesics connecting j-k, and
gjk = the number that actor i is on.
Usually normalized by:
CB' (i) CB (i ) /[(n 1)(n 2) /2]
number of pairs of vertices excluding the vertex itself
betweenness centrality: definition
18
betweenness of vertex ipaths between j and k that pass through i
all paths between j and k
directed graph: (N-1)*(N-2)
betweenness on toy networks
non-normalized version:
19
betweenness on toy networks
non-normalized version:
20
broker
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
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
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
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
CB (i) g jkj ,k
(i) /g jk
betweenness of vertex ipaths between j and k that pass through i
all paths between j and k
CB
' (i) CB(i) /[(N 1)(N 2)]
24
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
Outline
Degree centrality Centralization
Betweenness centrality Closeness centrality
Eigenvector centrality Bonacich power centrality
Katz centrality PageRank Hubs and Authorities
26
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
Closeness is based on the length of the average shortest path between a vertex and all vertices in the graph
Cc (i) d(i, j)j1
N
1
)1)).((()(' NiCiC CC
Closeness Centrality:
Normalized Closeness Centrality
closeness centrality: definition
28
depends on inverse distance to other vertices
Cc' (A)
d(A, j)j1
N
N 1
1
1 2 3 4
4
1
10
4
1
0.4
closeness centrality: toy example
A B C ED
29
closeness centrality: more toy examples
30
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
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
Influence range
The influence range of i is the set of vertices who are reachable from the node i
33
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
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
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:
Axx
Eigenvalues and eigenvectors
Slides from Fred K. Duennebier
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.
Axx
Eigenvalues and eigenvectors
AxxDo 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.
Ax x0 x A I 0
A(x)= Ax = x = (x)
(E.02)
(E.03)
(E.04)
(E.01)
How do you calculate eigenvectors and eigenvalues?
Expand equation (E.03): det(A- I )=|A- I |=0 for a 2x2 matrix:
A I a11 a12
a21 a22
1 0
0 1
a11 a12
a21 a22
det A I a11 a12
a21 a22 a11 a22 a12a21 0
0 a11a22 a12a21 a11 a22 2
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)
The solution to E.05 is:
0 a11a22 a12a21 a11 a22 2
a11 a22 a11 a22 2
4 a11a22 a12a21
(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:
A1 2
2 4
(E.07)
Eqn. E.07 doesn’t work here because a11a22-a12a12=0, so we use E.06:
0 a11a22 a12a21 a11 a22 2
0 14 22 (14) 2
(14) 2
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:
Axx, A I x0
1 2
2 4
0
0
x
y
1 2
2 4
x
y
1x2y
2x4y
0
0
These equations are multiples of x=-2y, so the smallest whole number values that fit are x=2, y=-1
For the other eigenvalue, =5:
1 2
2 4
5 0
0 5
x
y
4 2
2 1
x
y
4x2y
2x 1y
0
0
-4x + 2y = 0, and 2x y0, so, x1, y2
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
A2 .70 .45
.30 .55
A
3 .65 .525
.35 .475
A
100 .600 .600
.400 .400
Or we could find the eigenvalues of A and obtain A100 very quickly using eigenvalues.
What is A100 ?
We can get A100 by multiplying matrices many many times:
What good are such things?
Consider the matrix:
A.8 .3
.2 .7
For now, I’ll just know that there are two eigenvectors for A:
x1 .6
.4
and Ax1
.8 .3
.7 .2
.6
.4
x1 (1 = 1)
x2 1
1
and Ax2
.8 .3
.7 .2
1
1
.5
.5
(2 = 0.5)
The eigenvectors are x1=[.6 ; .4] and x2=[1 ; -1], and
the eigenvalues are 1=1 and 2=0.5.
Note that, if we multiply x1 by A, we get x1.
If we multiply x1 by A again, we STILL get x1.
Thus x1 doesn’t change as we mulitiply it by An.
What about x2?
When we multiply A by x2, we get x2/2,
and if we multiply x2 by A2, we get x2/4 .
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 A100,
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,
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
Eigenvector Centrality
47
Eigenvector Centrality
48
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
CD
E
Katz centrality
50
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
=.25
Katz Centrality: examples
=-.25
Why does the middle node have lower centrality than itsneighbors when is negative?
52
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 scatteredacross the web
an important page, e.g. slashdot
if a web page isslashdotted, it gains attention
PageRank
54
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 nodewhich can be used as a measure ofimportance
Trapping a drunk
Problem with pure random walk metric: Drunk can be “trapped” and end up going in circles
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
PageRank algorithm
where p1,p2,...,pN are the pages under consideration,
M(pi) is the set of pages that link to pi,
L(pj) is the number of outbound links on page pj, and
N is the total number of pages.
d is the random jumping probability (d = 0.85 for google)
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
example: probable location of random walker after 1 step
1
2
34
5
7
6 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 2 3 4 5 6 7 8
Pag
eRan
k
t=0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 2 3 4 5 6 7 8
Pag
eRan
k
t=1
20% teleportation probability
1
2
34
5
7
6 8
example: location probability after 10 steps
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 2 3 4 5 6 7 8
Pag
eRan
k
t=0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 2 3 4 5 6 7 8
Pag
eRan
k
t=1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 2 3 4 5 6 7 8
Pag
eRan
k
t=10
Matrix-based Centrality measures
62
Divide by out-degree
No division
with constant term without constant term
PageRank Degree centrality
Eigenvector centralityKatz centrality
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
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
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,jth entry Eij =1/ni
where i links to j, and ni is the number of links from i
then one can compute the authority scores a, and hub scores h through an iterative approach:
hEa T' Eah '
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
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
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?
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)
Cosine between sentences
Let s1 and s2 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.
yx
yx
yx niii
,1),cos(
The cosine ranges from 0 to 1.
LexRank (Cosine centrality)
1 2 3 4 5 6 7 8 9 10 11
1 1.00 0.45 0.02 0.17 0.03 0.22 0.03 0.28 0.06 0.06 0.00
2 0.45 1.00 0.16 0.27 0.03 0.19 0.03 0.21 0.03 0.15 0.00
3 0.02 0.16 1.00 0.03 0.00 0.01 0.03 0.04 0.00 0.01 0.00
4 0.17 0.27 0.03 1.00 0.01 0.16 0.28 0.17 0.00 0.09 0.01
5 0.03 0.03 0.00 0.01 1.00 0.29 0.05 0.15 0.20 0.04 0.18
6 0.22 0.19 0.01 0.16 0.29 1.00 0.05 0.29 0.04 0.20 0.03
7 0.03 0.03 0.03 0.28 0.05 0.05 1.00 0.06 0.00 0.00 0.01
8 0.28 0.21 0.04 0.17 0.15 0.29 0.06 1.00 0.25 0.20 0.17
9 0.06 0.03 0.00 0.00 0.20 0.04 0.00 0.25 1.00 0.26 0.38
10 0.06 0.15 0.01 0.09 0.04 0.20 0.00 0.20 0.26 1.00 0.12
11 0.00 0.00 0.00 0.01 0.18 0.03 0.01 0.17 0.38 0.12 1.00
d4s1
d1s1
d3s2
d3s1
d2s3
d2s1
d2s2
d5s2d5s3
d5s1
d3s3
Lexical centrality (t=0.3)
d4s1
d1s1
d3s2
d3s1
d2s3
d2s1
d2s2
d5s2d5s3
d5s1
d3s3
Lexical centrality (t=0.2)
d4s1
d1s1
d3s2
d3s1
d2s3d3s3
d2s1
d2s2
d5s2d5s3
d5s1
Lexical centrality (t=0.1)
Sentences vote for the most central sentence…
d4s1
d3s2
d2s1
N
dTiTETp
Tc
dTiTETp
Tc
dTip nn
n
1)()(
)(...)()(
)()( ,,11
1
LexRank
T1…Tn are pages that link to A,
c(Ti) is the outdegree of pageTi, 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
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)
Menczer, Filippo (2004) The evolution of document networks.
Content similarity distributions forweb pages (DMOZ) and scientific articles (PNAS)
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?
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?
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 …)
AB
C
D E
bibliographic coupling: both A and B are cite many of the same articles (F,G,H …)
FG
H
AB
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….
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.pl
Source: undetermined
Using network algorithms (specifically proximity) to improve movie recommendations can pay off
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 or or
final IR application: machine translation
spring English
printemps French
primavera Spanish
ربيعArabic
koanga Maori
udaherri Basque
1
vzmet Slovenian пружина
Russian
ressort French
2
veer Dutch
рысора Belarusian
……
……
3
11 1
1
3
3
…
…
…3
3
…
444
2
22
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 ’
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
ربيعArabic
koanga Maori
udaherri Basque
1…
…
3
11 1
1
3
3
…
…3
33
1
пружина Russian…
22
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