Structural Holes

1
Structural Holes Weak Ties
Overview Granovetter Strength of Weak
Ties What are weak ties? why are they
strong? Burt Structural Holes What are
they? What do they do? How do they
work? Methods Measures 1) Go Over
assignment 1 2) Moving data around SAS Data
steps 3) Calculating Ego-Network
Measures From Ego-network modules From
Global Networks
2
The Strength of Weak Ties
Granovetter argues that, under many
circumstances, strong ties are less useful than
weak ties. Why?
Redundancy
Local Density, Global Fragmentation
3
The Strength of Weak Ties
What are the implications?
For individuals? For Communities?
4
Structural Holes Weak Ties
Burt. Structural Holes
Similar idea to SWT Your ties matter because of
who your connects are not connected to.
What is (for Burt) Social Capital?
Relationships with other players
Why does it matter?
Social capital is as important as competition is
imperfect and investment capital is abundant.
5
Structural Holes Weak Ties
A structural Hole is a buffer a space between
the people you are connected to. 2
ways Cohesion Structural Equivalence
6
Structural Holes Weak Ties
Efficiency Maximize the number of non-redundant
contacts Effectiveness Draw your primary
contacts from different social worlds
7
Structural Holes Weak Ties
Maximum Efficiency
Decreasing Efficiency
Number of Non-Redundant Contacts
Increasing Efficiency
Minimum Efficiency
Number of Contacts
8
Structural Holes Weak Ties
Difference between SWT SH
Burts claim is that he focuses directly on the
causal agent active in Granovetter.
9
Structural Holes Weak Ties
Calculating the measures
Burt discusses 4 related aspects of a
network 1) Effective Size 2) Efficiency 3)
Constraint 4) Hierarchy
10
Structural Holes Weak Ties
Effective Size
Conceptually the effective size is the number of
people ego is connected to, minus the redundancy
in the network, that is, it reduces to the
non-redundant elements of the network. Effective
size Size - Redundancy
11
Structural Holes Weak Ties
Effective Size
Burts measures for effective size is
Where j indexes all of the people that ego i has
contact with, and q is every third person other
than i or j. The quantity (piqmjq) inside the
brackets is the level of redundancy between ego
and a particular alter, j.
12
Structural Holes Weak Ties
Effective Size
Piq is the proportion of actor is relations that
are spent with q.
2
3
Adjacency 1 2 3 4 5 1 0 1 1 1 1 2 1 0 0 0 1 3 1
0 0 0 0 4 1 0 0 0 1 5 1 1 0 1 0
1
5
4
13
Structural Holes Weak Ties
Effective Size
mjq is the marginal strength of contact js
relation with contact q. Which is js interaction
with q divided by js strongest interaction with
anyone. For a binary network, the strongest link
is always 1 and thus mjq reduces to 0 or 1
(whether j is connected to q or not - that is,
the adjacency matrix). The sum of the product
piqmjq measures the portion of is relation with
j that is redundant to is relation with other
primary contacts.
14
Structural Holes Weak Ties
Effective Size
2
3
Working with 1 as ego, we get the following
redundancy levels
1
P 1 2 3 4 5 1 .00 .25 .25 .25 .25 2
.50 .00 .00 .00 .50 3 1.0 .00 .00 .00 .00 4 .50
.00 .00 .00 .50 5 .33 .33 .00 .33 .00
PM1jq 1 2 3 4 5 1 --- --- --- ---
--- 2 --- .00 .00 .00 .25 3 --- .00 .00 .00 .00 4
--- .00 .00 .00 .25 5 --- .25 .00 .25 .00
5
4
Sum1, so Effective size 4-1 3.
15
Structural Holes Weak Ties
Effective Size
2
3
When you work it out, redundancy reduces to the
average degree, not counting ties with ego of
egos alters.
1
5
4
Node Degree 2 1 3 0 4
1 5 2 Mean 4/4 1
16
Structural Holes Weak Ties
Effective Size
2
3
Since the average degree is simply another way
to say density, we can calculate redundancy
as 2t/n where t is the number of ties (not
counting ties to ego) and n is the number of
people in the network (not counting
ego). Meaning that effective size n - 2t/n
1
5
4
17
Structural Holes Weak Ties
Efficiency is the effective size divided by the
observed size.
Effective Node Size
Size Efficiency 1 4 3 .75 2
2 1 .5 3 1 1 1.0 4 2 1 .5
5 3 1.67 .55
2
3
1
5
4
18
Structural Holes Weak Ties
Constraint
Conceptually, constraint refers to how much room
you have to negotiate or exploit potential
structural holes in your network.
2
3
1
5
4
..opportunities are constrained to the extent
that (a) another of your contacts q, in whom you
have invested a large portion of your network
time and energy, has (b) invested heavily in a
relationship with contact j. (p.54)
19
Structural Holes Weak Ties
Constraint
2
3
1
5
4
20
Structural Holes Weak Ties
Constraint
Cij Direct investment (Pij) Indirect
investment
21
Structural Holes Weak Ties
2
3
Constraint
1
5
4
Given the p matrix, you can get indirect
constraint (piqpqj) with the 2-step path distance.
PP 1 2 3 4 5 1 ... .083
.000 .083 .250 2 .165 ... .125 .290 .125 3 .000
.250 ... .250 .250 4 .165 .290 .125 ... .125 5
.330 .083 .083 .083 ...
P 1 2 3 4 5 1 .00 .25 .25 .25 .25 2
.50 .00 .00 .00 .50 3 1.0 .00 .00 .00 .00 4 .50
.00 .00 .00 .50 5 .33 .33 .00 .33 .00
22
Structural Holes Weak Ties
Constraint
Total constraint between any two people then is
C (P P2)2
Where P is the normalized adjacency matrix, and
means to square the elements of the matrix.
23
Structural Holes Weak Ties
Constraint
PP2 Cij C .00 .33 .25 .33 .50
.00 .11 .06 .11 .25 .53 .67 .00 .13 .29 .63
.44 .00 .02 .08 .39 1.0 .25 .00 .25 .25
1.0 .06 .00 .06 .06 .67 .29 .13 .00 .63 .44
.08 .02 .00 .39 .66 .41 .08 .41 .00 .44 .17
.01 .17 .00
24
Structural Holes Weak Ties
Hierarchy
Conceptually, hierarchy (for Burt) is really the
extent to which constraint is concentrated in a
single actor. It is calculated as
25
Structural Holes Weak Ties
Hierarchy
2
3
1
2 3 4 5 C C .11 .06 .11 .25
.53 .83 .46 .83 1.9
5
4
H.514
26
Homework
The solution program for assignment 1 can be
found on the course data programs page, called
solutions1.sas Look at this for the
answers. http//www.soc.sbs.ohio-state.edu/jwm/s8
84/data.htm
Common things people did Typos in the original
data matrix. Wrong data in, wrong answer out.
27
Homework
Common things people did Typos in the original
data matrix. Wrong data in, wrong answer
out. Adjacency lists should include a row for
every node, even if they do not send any ties in
the network What is the longest possible path in
a network? How would you write a program to stop
automatically? Many of you were able to identify
the symmetric / asymmetric relations. But you
left them as 2 in the matrix. Usually you
would add one more line (or use a slightly
different syntax) to change them to 1 as well.
28
Playing with data Getting information from one
program to another
If our data are in one format (SAS, for example)
how do we get it into a program like PAJEK or
UCINET?
1) Type it in by hand. Too slow, error prone,
impossible for very large networks 2) Write a
program that moves data around for you
automatically SPAN contains programs that
write to PAJEK UCINET NEGOPY STRUCTURE
29
Playing with data Using SAS to move data.
Basic Elements SAS is a language Data Steps
Nouns Procedures Verbs Data
needs Creation / Read Organization Transfo
rmation Manipulation Procedures Summarize
Analyze Communicate Manipulate
Back-up 1) How does SAS store move data? 2)
How do you store use programs over again?
http//wks.uts.ohio-state.edu/sasdoc/
30
SAS
The procedure we have been using is IML or the
Interactive Matrix Language.
31
Data
Libraries Links to where data are
stored Datasets the actual data You refer to a
data set by a two-level name library.data