签订房屋租赁合同的十大注意事项
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broker
20
example Nodes are sized by degree, and colored by betweenness.
Can you spot nodes with high betweenness but relatively low degree?
What about high degree but relatively low betweenness?
太原房产网 http://www.52youju.com
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.
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
go through you in order to reach one another in the minimum number of hops? who has higher betweenness, X or Y?
YX
16
betweenness centrality: definition
Centralization
Betweenness centrality Closeness centrality
Eigenvector centrality
Bonacich power centrality
Katz centrality PageRank Hubs and Authorities
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, ….
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
Definition of ‘central’ varies by context/purpose
Local measure:
degree
Relative to rest of network:
closeness, betweenness, eigenvector (Bonacich power centrality), Katz, PageRank, …
21
betweenness on toy networks non-normalized version:
why do C and D each have
C
betweenness 1?
They are both on shortest
paths for pairs (A,E), and (B,E),
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
Random-walk betweenness
Based on the idea of random walks Usually yields ranking similar to geodesic betweenness
betweenness of vertex i
paths between j and k that pass through i
CB (i) g jk (i) /g jk
jk
all paths between j and k
Where gjk = the number of geodesics connecting j-k, and gjk = the number that actor i is on.
note that there are no alternate paths for these pairs to take, so C gets full credit
18
betweenness on toy networks non-normalized version:
19
betweenness on toy networks non-normalized version:
A
B
C
D
E
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)
when ‘prestige’ may not be the right word
gives advice to (can reverse direction) gives orders to (- ” -) lends money to (- ” -) dislikes distrusts
6
degree centrality (undirected) He who has many friends is most important.
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)
(can useห้องสมุดไป่ตู้other metrics, e.g. gini coefficient or standard deviation)
maximum value in the network
CD
g i1
CD (n*) CD (i)
[(N 1)(N 2)]
11
degree centralization examples
In each of the following networks, X has higher centrality than Y according to a particular measure
indegree
outdegree
betweenness
closeness
5
Outline Degree centrality
and so must share credit:
A
B
E ½+½ = 1
Can you figure out why B has betweenness 3.5 while E has betweenness 0.5?
D
22
Alternative betweenness computations
Several other graph metrics
Clustering coefficient, Assortativity, Modularity, …
2
Characterizing networks: Who is most central?
? ?
?
3
network centrality Which nodes are most ‘central’?
How evenly is centrality distributed among nodes?
Centralization, hubs and autthorities, …
4
centrality: who’s important based on their network position
network reaches you…
14
Degree centrality
Centralization
Outline
Betweenness centrality
Closeness centrality
15
betweenness: another centrality measure intuition: how many pairs of individuals would have to
CD = 0.167
CD = 1.0
CD = 0.167
12
degree centralization examples
example financial trading networks
high centralization: one node trading with many others
low centralization: trades are more evenly distributed
10
centralization: how equal are the nodes?
How much variation is there in the centrality scores among the nodes?
Freeman’s general formula for centralization:
admiration influence gift-giving trust
directionality especially important in instances where ties may not be reciprocated (e.g. dining partners choice network)
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
Usually normalized by:
CB' (i) CB (i ) /[(n 1)(n 2) /2]
number of pairs of vertices excluding the vertex itself
directed graph: (N-1)*(N-2)
17
betweenness on toy networks non-normalized version:
20
example Nodes are sized by degree, and colored by betweenness.
Can you spot nodes with high betweenness but relatively low degree?
What about high degree but relatively low betweenness?
太原房产网 http://www.52youju.com
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.
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
go through you in order to reach one another in the minimum number of hops? who has higher betweenness, X or Y?
YX
16
betweenness centrality: definition
Centralization
Betweenness centrality Closeness centrality
Eigenvector centrality
Bonacich power centrality
Katz centrality PageRank Hubs and Authorities
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, ….
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
Definition of ‘central’ varies by context/purpose
Local measure:
degree
Relative to rest of network:
closeness, betweenness, eigenvector (Bonacich power centrality), Katz, PageRank, …
21
betweenness on toy networks non-normalized version:
why do C and D each have
C
betweenness 1?
They are both on shortest
paths for pairs (A,E), and (B,E),
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
Random-walk betweenness
Based on the idea of random walks Usually yields ranking similar to geodesic betweenness
betweenness of vertex i
paths between j and k that pass through i
CB (i) g jk (i) /g jk
jk
all paths between j and k
Where gjk = the number of geodesics connecting j-k, and gjk = the number that actor i is on.
note that there are no alternate paths for these pairs to take, so C gets full credit
18
betweenness on toy networks non-normalized version:
19
betweenness on toy networks non-normalized version:
A
B
C
D
E
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)
when ‘prestige’ may not be the right word
gives advice to (can reverse direction) gives orders to (- ” -) lends money to (- ” -) dislikes distrusts
6
degree centrality (undirected) He who has many friends is most important.
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)
(can useห้องสมุดไป่ตู้other metrics, e.g. gini coefficient or standard deviation)
maximum value in the network
CD
g i1
CD (n*) CD (i)
[(N 1)(N 2)]
11
degree centralization examples
In each of the following networks, X has higher centrality than Y according to a particular measure
indegree
outdegree
betweenness
closeness
5
Outline Degree centrality
and so must share credit:
A
B
E ½+½ = 1
Can you figure out why B has betweenness 3.5 while E has betweenness 0.5?
D
22
Alternative betweenness computations
Several other graph metrics
Clustering coefficient, Assortativity, Modularity, …
2
Characterizing networks: Who is most central?
? ?
?
3
network centrality Which nodes are most ‘central’?
How evenly is centrality distributed among nodes?
Centralization, hubs and autthorities, …
4
centrality: who’s important based on their network position
network reaches you…
14
Degree centrality
Centralization
Outline
Betweenness centrality
Closeness centrality
15
betweenness: another centrality measure intuition: how many pairs of individuals would have to
CD = 0.167
CD = 1.0
CD = 0.167
12
degree centralization examples
example financial trading networks
high centralization: one node trading with many others
low centralization: trades are more evenly distributed
10
centralization: how equal are the nodes?
How much variation is there in the centrality scores among the nodes?
Freeman’s general formula for centralization:
admiration influence gift-giving trust
directionality especially important in instances where ties may not be reciprocated (e.g. dining partners choice network)
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
Usually normalized by:
CB' (i) CB (i ) /[(n 1)(n 2) /2]
number of pairs of vertices excluding the vertex itself
directed graph: (N-1)*(N-2)
17
betweenness on toy networks non-normalized version: