Reciprocity

1
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
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 (321) 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.).
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 6424. 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
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 1/11.
• 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
  (11.333)/3.778
• Average node clustering weighted by the size of
  nodes neighborhood (1111.3333)/(113).600

In UCINET Network ? Cohesion? Clustering
coefficient
5
Correlation between Two Networks with the Same
Actors
Friendship Network
Invitation to a Birthday Party Network
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
In UCINET Tools ? Testing Hypotheses ? Dyadic
(QAP) ? QAP Correlation (old)
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 MN(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
• JM11/(M01M10M00) M11 of dyads
  where both ties are 1, M01 of dyads where 1st
  tie is 0, 2nd 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(M11M00)/(M00M01M10M11)
• 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?)

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

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
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

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
11
Correspondence Analysis for Two-Mode Networks

• In UCINET Tools ? Scaling/Decomposition ?
  Correspondence

12
Correspondence Analysis for Two-Mode Networks

• 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 1st factor is 0.678 and on the
  2nd 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.

• 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