Reciprocity Reciprocity measures the extent to which a tie from
A to B is reciprocated by a tie from B to A. Obtained for directed
(asymmetric) ties. Networks with lots of reciprocity are often more
balanced, stable, harmonious. Different ways to measure
reciprocity: Dyad based Proportion of dyads (pairs) with
reciprocated ties among all possible dyads: 1/3=.333 (AB/AB,BC,AC)
Proportion of dyads with reciprocated ties among all connected
dyads: 1/2=.5 (AB/AB,BC) Arc (tie) based Proportion of reciprocated
ties among all possible ties: 2/6=.333 (AB,BA/AB,BA,BC,CB,AC,CA)
Proportion of reciprocated ties among all existing ties: 2/3=.667
(AB,BA/AB,BA,BC) In UCINET: Network Cohesion Reciprocity Then
choose either the Dyad-based or Arc-based Method. (You will get the
Proportion of dyads with reciprocated ties among all connected
dyads or the Proportion of reciprocated ties among all existing
ties
Slide 2
Transitivity measures a tendency for a tie from A to C to exist
if a tie from A to B and a tie from B to C exist. If A B & B C
& A C then the three are transitive. Networks with high level
of transitivity are often more stable, balanced, harmonious.
Suppose we have symmetric ties, transitivity then means that If A
is friends with B and B is friends with C (suppose that friendship
is always symmetric) A is friends with C (Fig.1). (In other words:
the triad is closed.) A B, B C, A C In that case, however, it is
also true that if A is friends with C and C is friends with B, A is
friends with B. A C, C B, A B Four other statements also must be
true: B A, A C, B C & B C, C A, B A & C A, A B, C B & C
B, B A, C A This triad is fully transitive: you can take the three
nodes in any configuration, you will get transitivity. Any
particular configuration of three nodes is called a triple. Three
nodes can form triples (3*2*1=) 6 different ways: ABC, ACB, BAC,
BCA, CAB, CBA Suppose we have directed ties, but all happen to be
reciprocal (Fig.2.). We have the same results as with symmetric
ties. Indeed, all triples are transitive. Now C is friend of A but
A is NOT friend of C. How many transitive triples do we have left?
Only 3 B C, C A, B A & C A, A B, C B & C B, B A, C A Now if
you remove B C too you will have only 2 transitive triples
(Fig.4.), but if you remove C B instead, you are down to 1
transitive triple (Fig.5.), and if you remove C A instead, you NO
transitive triple is (Fig.6.). BC A Fig. 2. BC A Fig. 1. BC A Fig.
3. BC A Fig. 4. BC A Fig. 5. BC A Fig. 6.
Slide 3
Transitivity In this network of 4 there is no reciprocal
relationship. But it has one transitive triple ABC: A B, B C, A C.
E.g.: BCA is intransitive: B C, C A, B A. So is BAC : B A, A C, B C
etc. What should we do with ACD? A C, C D, A ? D ACD is not
intransitive, it is called vacuously transitive. A triple that has
fewer than three ties is called vacuously transitive. Therefore
transitive triples are also referred to as non- vacuously
transitive. Reported as: Proportion of transitive triples among all
possible triples: All possible (ordered) triples from 3 nodes is 6.
4 nodes can form 4 triads (leaving out a different one each time.)
All possible triples from 4 nodes is 6*4=24. We find only 1
(non-vacuously) transitive triple: ABC 1/24=.042 Proportion of
transitive triples among triads where one single link could
complete a triad. We have three such triples: ABC, ACD, BCD but the
last two triples have only two ties, so they are vacuously
transitive. 1/3=.333 # transitive triples/(# transitive triples + #
vacuously transitive triples that could be [non-vacuously]
transitive ) In UCINET: Network Cohesion Transitivity Choose
Adjecency for the Type of transitivity
Slide 4
Clustering Clustering measures a tendency towards dense local
neighborhoods neighborhood: other nodes to which ego is connected.
size of the neighborhood: the number of potential connection among
the nodes in the neighborhood. Nodes clustering coefficient:
density of ties between nodes directly adjacent to it, excluding
the ties to the node itself. A has two neighbors B and C. They make
one pair (BC), and have one tie between them. The density of the
network consisting of B and C is 1/1=1. B the same for B C has
three potential pairs in its neighbors: AB,AD, BD. Density of the
network consisting of these nodes is 1/3=.333 For the coefficient
to be calculated, a node has to have at least two ties D has only
one ties, no clustering coefficient can be calculated Note: when
UCINET calculates ties, 1 tie is a symmetric or a reciprocal tie.
An asymmetric directed tie counts as half a tie. Here all ties are
symmetric. Average node (overall graph) clustering:
(1+1+.333)/3=.778 Average node clustering weighted by the size of
nodes neighborhood: (1*1+1*1+.333*3)/(1+1+3)=.600 In UCINET:
Network Cohesion Clustering coefficient
Slide 5
Correlation between Two Networks with the Same Actors In
UCINET: Tools Testing Hypotheses Dyadic (QAP) QAP Correlation (old)
Bivariate Statistics 1 2 3 4 5 6 7 Value Signif Avg SD P(Large)
P(Small) NPerm --------- --------- --------- --------- ---------
--------- --------- 1 Pearson Correlation: 0.331 0.194 0.004 0.244
0.194 0.964 2500.000 2 Simple Matching: 0.667 0.194 0.513 0.115
0.194 0.964 2500.000 3 Jaccard Coefficient: 0.412 0.194 0.254 0.115
0.194 0.964 2500.000 4 Goodman-Kruskal Gamma: 0.625 0.194 -0.001
0.460 0.194 0.964 2500.000 5 Hamming Distance: 10.000 0.194 14.598
3.447 0.964 0.194 2500.000 Friendship Network Invitation to a
Birthday Party Network
Slide 6
Measures of Correlation between Two Networks with the Same
Actors The units of analysis or cases here are the dyads. With N
actors you have M=N*(N-1) cases. The data file used here is of the
familiar cases by variables format Which correlation measure to use
depends on how the tie is measured. Binary ties (the two variables
are dichotomous): If the information content of 0 is less than the
information content of 1. E.g., if we both mention X as our best
friend that reveals our similarity. But if neither of us mentions X
as our best friend that does not necessarily mean we are similar.
Jaccard coefficient J=M11/(M01+M10+M00) M11=# of dyads where both
ties are 1, M01=# of dyads where 1 st tie is 0, 2 nd tie is 1 etc.
If the information content of 0 is the same as the information
content of 1. E.g., if we are forced to sort people into friend or
enemy and we both choose X as friend, that is as informative as
both of us choosing X as our enemy. Simple Matching
S=(M11+M00)/(M00+M01+M10+M11) Hamming Distance=(1-S)*M or the
number of mismatched dyads Ordinal ties (the two variables are
ordinal, e.g.: Do you talk often, rarely, never?) Goodman-Kruskall
gamma Interval/Ratio ties (the two variables are interval/ratio,
e.g.: How many times did you talk last week?) Pearsons
correlation
Slide 7
Network Positions and Social Roles Similarity or equivalence of
actors positions can be defined in several ways: Structural
equivalence two nodes have the same relations with the same set of
other nodes Actors A and B each is tied to nodes C,D,E,F,G Actors
C,D,E,F,G each is tied to both A and B Automorphic equivalence
identifies actors in the same configuration of ties. They do not
have to have ties to the same set. But they have the same
centrality, ego density and clique size. Actors A, B A is tied to
C,G,D and B is tied to E,F,D and C,G are like F,E Actors C, G, F, E
Actors C and G are not just automorphically but also structurally
equivalent and so are F and E. Regular equivalence two nodes have
the same profile of ties with members of other sets of actors. It
describes social roles, e.g. mother in a family. Actors A, B --
e.g. mothers Actors C, G, D, F, E e.g. children Actors C and G are
also automorphically and structurally equivalent, so are D,F,E
Slide 8
Network Positions and Social Roles In the figure you find:
Structural equivalence Actors E and F Actors H and I Automorphic
equivalence Actors B, D Actors E, F, H, I Regular equivalence
Actors B, C, D Actors E, F, G, H, I Actors that are structurally
equivalent are also automorphically and regularly equivalent.
Actors that are automorphically equivalent are also regularly
equivalent In UCINET: Network Roles & Positions Structural
Profile for stuctural equivalence (for full s.e. you look for a
coefficient of 0.00) Network Roles & Positions Automorphic All
Permutations for automorphic equivalence Network Roles &
Positions Maximal Regular Optimization for regular equivalence
Slide 9
Network Subgroups: Cliques Clique - a sub-set of a network in
which the actors are more closely and intensely tied to one another
than they are to other members of the network. It is a cohesive
subgroup connected with many direct and reciprocated ties.
Formally, a clique is the maximum number of actors but at least
three, who have all possible ties present among themselves Within a
clique the geodesic distance is 1 for everyone (everyone is
directly related) In terms of friendship ties, for example, it is
not unusual for people in human groups to form "cliques" on the
basis of age, gender, race, ethnicity, religion/ideology, and many
other things Cliques tend to indicate stronger relationships,
similarity in information and resources available, more constraint,
but also more support The above definition of the clique is very
strict, so there are many other types of sub-groups you can
identify in a network (N-cliques, N-clans, K-plexes, K-cores,
F-groups) with less restrictive assumptions about in-group and
out-group ties
Slide 10
Cliques 1 cliques found. 1: Ana Jen Liz Pat Actor-by-Actor
Clique Co-Membership Matrix 1 2 3 4 5 6 A J L P N M - - - - - - 1
Ana 1 1 1 1 0 0 2 Jen 1 1 1 1 0 0 3 Liz 1 1 1 1 0 0 4 Pat 1 1 1 1 0
0 5 Nancy 0 0 0 0 0 0 6 Mona 0 0 0 0 0 0 HIERARCHICAL CLUSTERING OF
EQUIVALENCE MATRIX N a M A J L P n o n e i a c n a n z t y a Level
1 2 3 4 5 6 ----- - - - - - - 1.000 XXXXXXX.. 0.000 XXXXXXXXXXX In
UCINET: Network Subgroups Cliques
Slide 11
In UCINET: Tools Scaling/Decomposition Correspondence
Correspondence Analysis for Two-Mode Networks
Slide 12
SINGULAR VALUES FACTOR VALUE PERCENT CUM % RATIO PRE CUM PRE
------- ------ ------- ------- ------- ------- ------- 1: 0.848
47.9 47.9 1.495 0.616 0.616 2: 0.567 32.0 79.9 1.597 0.275 0.892 3:
0.355 20.1 100.0 0.108 1.000 ======= ====== ======= ======= =======
1.770 100.0 Row Scores 1 2 3 ------ ------ ------ 1 Ana 0.678 0.596
0.273 2 Jen 0.545 -0.032 -0.672 3 Pat 0.240 -0.941 0.239 4 Liz
0.240 -0.941 0.239 5 Nancy -0.841 -0.196 0.072 6 Mona -1.921 0.549
-0.096 Column Scores 1 2 3 ------ ------ ------ 1 BDP1 0.203 -0.534
0.085 2 BDP2 0.721 0.497 -0.562 3 BDP3 0.799 1.051 0.768 4 BDP4
-1.628 0.312 -0.034 There are four birthday parties, therefore we
can display every girl in a four dimensional space. Correspondence
analysis (CA) tries to find a simpler space with fewer dimensions,
that still describes the relative positions of the six girls fairly
accurately. It is always possible to derive K-1 dimensions (or
factors) from K dimensions, if you are willing to take the sum of
the four dimensions as given. The last dimension then can be
obtained from the sum by subtraction. CA derived 3 factors from the
4 birthday parties. The first factor explains 47.9% of the
connections among the 6 girls. The second explains 32%, the third
20.1%. The factors are always ordered from the highest to the
lowest explanatory power. The hope is that one can derive a few
(say, two) factors that explains a large percent of the
connections. The plot takes the first two dimensions and places
each girl and birthday party according to their scores on those two
factors. E.g. Anas position on the 1 st factor is 0.678 and on the
2 nd 0.596. Pat and Liz have identical values (0.240, -0.941) and
they are occupying the exact same spot. The question is: what do
these factors mean? What explains the pattern of association? If we
know something about the parties we can speculate. E.g. Suppose
Factor 1 is the size of the party, Factor 2 is the amount of
dancing at the party. Then BDP3 was the largest, BDP2 was almost as
large, and BDP4 was the smallest. BDP3 was the danciest and BDP1
was the least dancy. Ana is invited to larger and dancier parties,
Mona to small and dancy parties etc.This may tell you something
about the relationships among the girls.