structural holes & weak ties
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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 - PowerPoint PPT PresentationTRANSCRIPT
Structural Holes & Weak TiesOverview
Granovetter: Strength of Weak TiesWhat are ‘weak ties’?why are they ‘strong’?
Burt: Structural HolesWhat are they?What do they do?How do they work?
Methods & Measures:1) Go Over assignment 12) Moving data around
SAS Data steps3) Calculating Ego-Network Measures
From Ego-network modulesFrom Global Networks
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
The Strength of Weak Ties
What are the implications?
For individuals?
For Communities?
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.”
Structural Holes & Weak Ties
A structural Hole is a buffer: a space between the people you are connected to.
2 ways:CohesionStructural Equivalence
Structural Holes & Weak Ties
EfficiencyMaximize the number of non-redundant contacts
EffectivenessDraw your primary contacts from different social worlds
Structural Holes & Weak Ties
Number of Contacts
Num
ber
of N
on-R
edun
dant
Con
tact
s
Maximum Efficiency
Minimum Efficiency
Decreasing Efficiency
Increasing Efficiency
Structural Holes & Weak Ties
Difference between SWT & SH:
Burt’s claim is that he focuses directly on the causal agent active in Granovetter.
Structural Holes & Weak Ties
Calculating the measures
Burt discusses 4 related aspects of a network:1) Effective Size2) Efficiency3) Constraint4) Hierarchy
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
Structural Holes & Weak Ties
Effective SizeBurt’s measures for effective size is:
j qjqiqmp1
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.
Structural Holes & Weak Ties
Effective Size:
j qjqiqmp1
Piq is the proportion of actor i’s relations that are spent with q.
1
2
4 5
3 Adjacency 1 2 3 4 51 0 1 1 1 12 1 0 0 0 13 1 0 0 0 04 1 0 0 0 15 1 1 0 1 0
P 1 2 3 4 51 .00 .25 .25 .25 .252 .50 .00 .00 .00 .503 1.0 .00 .00 .00 .004 .50 .00 .00 .00 .505 .33 .33 .00 .33 .00
Effective Size:
j qjqiqmp1
Structural Holes & Weak Ties
mjq is the marginal strength of contact j’s relation with contact q. Which is j’s interaction with q divided by j’s 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 i’s relation with j that is redundant to i’s relation with other primary contacts.
Effective Size:
j qjqiqmp1
Structural Holes & Weak Ties
1
2
4 5
3
P 1 2 3 4 51 .00 .25 .25 .25 .252 .50 .00 .00 .00 .503 1.0 .00 .00 .00 .004 .50 .00 .00 .00 .505 .33 .33 .00 .33 .00
Working with 1 as ego, we get the following redundancy levels:
PM1jq
1 2 3 4 51 --- --- --- --- ---2 --- .00 .00 .00 .253 --- .00 .00 .00 .004 --- .00 .00 .00 .255 --- .25 .00 .25 .00
Sum=1, so Effective size = 4-1 = 3.
Effective Size:
j qjqiqmp1
Structural Holes & Weak Ties
1
2
4 5
3 When you work it out, redundancy reduces to the average degree, not counting ties with ego of ego’s alters.
Node Degree 2 1 3 0 4 1 5 2Mean: 4/4 = 1
Effective Size:
j qjqiqmp1
Structural Holes & Weak Ties
1
2
4 5
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
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
Structural Holes & Weak Ties
1
2
4 5
3
Efficiency is the effective size divided by the observed size.
Structural Holes & Weak Ties
1
2
4 5
3
Constraint
Conceptually, constraint refers to how much room you have to negotiate or exploit potential structural holes in your network.
“..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)
Structural Holes & Weak Ties
1
2
4 5
3
Constraint
P 1 2 3 4 51 .00 .25 .25 .25 .252 .50 .00 .00 .00 .503 1.0 .00 .00 .00 .004 .50 .00 .00 .00 .505 .33 .33 .00 .33 .00
2
qqjiqijij pppC
Structural Holes & Weak Ties
Constraintq
i jpij
piq pqj
Cij = Direct investment (Pij) + Indirect investment
2
qqjiqijij pppC
Structural Holes & Weak Ties
Constraint2
qqjiqijij pppC
Given the p matrix, you can get indirect constraint (piqpqj) with the 2-step path distance.
P 1 2 3 4 51 .00 .25 .25 .25 .252 .50 .00 .00 .00 .503 1.0 .00 .00 .00 .004 .50 .00 .00 .00 .505 .33 .33 .00 .33 .00
P*P 1 2 3 4 51 ... .083 .000 .083 .2502 .165 ... .125 .290 .1253 .000 .250 ... .250 .2504 .165 .290 .125 ... .1255 .330 .083 .083 .083 ...
1
2
4 5
3
Structural Holes & Weak Ties
Constraint2
qqjiqijij pppC
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.
Structural Holes & Weak Ties
Constraint2
qqjiqijij pppC
P+P2 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
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:
)ln(
ln
NN
NC
C
NC
C
H j
ijij
Structural Holes & Weak Ties
Hierarchy
)ln(
ln
NN
NC
C
NC
C
H j
ijij
2 3 4 5 CC: .11 .06 .11 .25 .53
.83 .46 .83 1.9
1
2
4 5
3
NC
Cij
H=.514
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/s884/data.htm
Common things people did:Typos in the original data matrix. Wrong data in, wrong answer out.
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.
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:PAJEKUCINETNEGOPYSTRUCTURE
Playing with data: Using SAS to move data.
Back-up: 1) How does SAS store & move data?
2) How do you store & use programs over again?
Basic Elements:SAS is a language:
Data Steps = Nouns
Procedures = Verbs
Data needs:Creation / Read OrganizationTransformationManipulation
Procedures:SummarizeAnalyzeCommunicateManipulate
http://wks.uts.ohio-state.edu/sasdoc/
SAS
The procedure we have been using is IML or the Interactive Matrix Language.
Data
Libraries: Links to where data are storedDatasets: the actual data
You refer to a data set by a two-level name:library.data