Measures of similarity and structural equivalence

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Measures of similarity and
structural equivalence
Jing Zhou
Contents
• Measuring similarity/dissimilarity
• Valued relations
• Binary relations
• Visualizing similarity and distance
• Clustering tools
• Multi-dimensional scaling tools
• Describing structural equivalence sets
• Clustering similarities or distance profiles
• Concor
• Optimization by Tabu search
• Recommendation
Measuring similarity and dissimilarity
• Actors 2,5,7 might be
structurally similar in that they
seem to have reciprocal ties
with each other and almost
everyone else.
• Actors 6,8,10 are “regularly”
similar in that they are rather
isolated, but they are not
structurally similar.
Two actors may be said to be structurally equivalent to if they have the same
patterns of ties with other actors.
CO COM
MAY
NEW UWA
EDUC INDU
WRO
WELF WEST
UN M
R
S
Y
1
1
0
0
1
0
1
0
1
0
1
1
1
0
1
1
1
0
1
1
1
1
0
0
1
1
0
1
0
0
0
0
1
1
1
1
1
0
1
0
0
0
0
1
0
2
1
3
0
1
4
1
1
0
5
1
1
1
1
6
0
0
1
0
0
7
0
1
0
1
1
0
8
1
1
0
1
1
0
1
9
0
1
0
0
1
0
1
0
10 1
1
1
0
1
0
1
0
0
0
COUN COMM EDUC INDU MAYR WRO NEWS UWAY WELF WEST
-1
0
0
1
0
1
0
1
0
1
-1
1
1
0
1
1
1
0
0
1
-1
1
1
1
0
0
1
1
1
0
-1
0
1
0
0
0
1
1
1
1
-0
1
1
1
1
0
0
1
0
0
-1
0
1
0
0
1
0
1
1
0
-0
0
0
1
1
0
1
1
0
1
-1
0
0
1
0
0
1
0
1
0
-0
1
1
1
0
1
0
1
0
0
--1
0
1
1
0
0
1
0
1
-1
1
1
1
0
1
1
1
1
-0
1
0
1
1
0
0
0
1
-0
1
1
1
0
1
1
0
0
-1
1
1
1
0
1
1
1
1
-0
0
1
0
0
0
0
0
0
-1
1
1
1
1
1
1
1
1
-0
1
0
0
1
0
0
0
0
-1
1
0
0
1
1
0
1
0
-0
0
1
0
1
0
0
0
0
Valued relations: Pearson correlation coefficients
• -1: two actors have exactly the opposite ties to each other actor
• 0 : knowing one actor’s tie to a third party doesn’t help us at all in guessing what
the other actor’s tie to the third party might be.
Actor 1 and 9 have
identical pattern of ties
• 1 : the two actors always have exactly the same tie to other actors
Actor 6 and 7 have different pattern of ties. For
actor 6 to have ties to actor that actor 7 does not
Euclidean distance
COUN COM EDUC INDU MAYR WRO NEWS UWAY WELF WEST
M
1
-1
0
0
1
0
1
0
1
0
2
1 -1
1
1
0
1
1
1
0
3
0
1 -1
1
1
1
0
0
1
4
1
1
0 -1
0
1
0
0
0
5
1
1
1
1 -0
1
1
1
1
6
0
0
1
0
0 -1
0
1
0
7
0
1
0
1
1
0 -0
0
0
8
1
1
0
1
1
0
1 -1
0
9
0
1
0
0
1
0
1
0 -0
10
1
1
1
0
1
0
1
0
0 --
What is Euclidean distance?
1,2 =
(1 − 0)2 +(1 − 0)2 +(1 − 1)2 +(0 − 0)2 +(1 − 1)2 +(1 − 0)2 +(1 − 1)2 +(0 − 0)2 = 3=1.732
Binary relations: matches, Jaccard and Hamming
CO COM
UN M
1
1
2
1
3
0
1
4
1
1
5
1
1
6
0
0
7
0
1
8
1
1
9
0
1
10 1
1
• Matches: 5/8=0.625
• In comparing actor 1 and 2, they have the same tie to other
actors 62.5% of the time.
Jaccard coefficients (I can’t get the results as the output said)
• The number of times that both actors
report a tie to the same third actors as
percentage of the total number of ties
reported.
4/11=36.4%???
CO COM
UN M
1
1
2
1
3
0
1
4
1
1
5
1
1
6
0
0
7
0
1
8
1
1
9
0
1
10 1
1
Hamming distance: The number of entries in the vector for one actor
that would need to be changed in order to make it identical to the
vector of the other actor
CO COM
MAY
NEW UWA
EDUC INDU
WRO
WELF WEST
UN M
R
S
Y
1
1
0
0
1
0
1
0
1
0
1
1
1
0
1
1
1
0
1
1
1
1
0
0
1
1
0
1
0
0
0
0
1
1
1
1
1
0
1
0
0
0
0
1
0
2
1
3
0
1
4
1
1
0
5
1
1
1
1
6
0
0
1
0
0
7
0
1
0
1
1
0
8
1
1
0
1
1
0
1
9
0
1
0
0
1
0
1
0
10 1
1
1
0
1
0
1
0
0
0
Visualizing similarity and distance
Clustering tool
• E-I index is often most helpful. It measures the ratio of the numbers
of ties within the clusters to ties between clusters.
• But how can negative number exit???
Information on the relative size of
the clusters at each stage
Multi-dimensional scaling tools
• MDS represents the patterns of similarity or dissimilarity associated with multiple
underlying dimensions.
Measure of badness of fit
Coordinates of
the case in two
dimensions
• Graph the nodes according
to their coordinates. In this
case, maybe case 1 and
case 2 are similar.
Describing structural equivalence sets
• Two actors that are structurally equivalence have the same ties to all
other actors. They are perfectly substitutable. But in real data maybe
approximate equivalence is meaningful.
• Clustering analysis
• Concor
• Numerical optimization by tabu search
Clustering similarities or distances profiles
We measure dissimilarity in this case
Rough character-mapped
graphic of the clustering and
dendogram of structural
equivalence dta
CONCOR
• Correlate each pair of actors
• Generate the row of this actor-by-actor correlation matrix
• Then correlate with each other row
• The process is repeated until eventually the elements in this “iterated
correlation matrix” converge on value of either 1 or -1
• Concor divides the data into two sets on the basis of correlation
• Then within each set, the process is repeated
• It continues until all actors are separated
5
1
2
3
22% of the variance in the ties in
the concor model can be
accounted for by a perfect
structural block model
4
Optimization by Tabu search
• Search for sets of actors who, if placed into a block, produce the
smallest sum of within-block variances in the tie profiles.
• That is, if actors in a block have similar ties, their variance around the
block mean profile will be small
• This analysis can help us answer three answers:
• How many equivalence classes or approximate equivalence classes are there?
• How good is the fit of this simplification into equivalence classes in
summarizing the information about all the nodes?
• What is the position of each class as defined by it’s relations to the other
classes?
Answer
question 1
Answer
question 2
Answer
question 3
Recommendation
• Structural equivalence in a journal network
• Present the description of the structure of the journal network at
three time periods by considering the block structures of it.
• Two way of grouping journals
• Journal roles: aims and objectives of the journals in the network
• Journal position: patterns of citation activity
• Construct an image structure of the network and this reveals a simple
core-periphery structure of the network.
• Each of the periphery sociological journals are tied to the core but not to each
other.
Citation network
Grouping journals
• Journal roles
• Disciplinary
• Comprehensive sociology: AJS ASR SF
• Social structural
HR SSPQ
• Methodology/Models
SM SSR SMR JMS QQ
• Interdisciplinary
•
•
•
•
Behavioral science
Public opinion
Structural analysis
Social conflict
BS
POQ
SN
JCR
Journal positions
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