Structural Holes & Weak Ties

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.”


A structural Hole is a buffer: a space between the people you are connected to.

2 ways:CohesionStructural Equivalence


EfficiencyMaximize the number of non-redundant contacts

EffectivenessDraw your primary contacts from different social worlds


Number of Contacts

Num

ber

of N

on-R

edun

dant

Con

tact

s

Maximum Efficiency

Minimum Efficiency

Decreasing Efficiency

Increasing Efficiency


Difference between SWT & SH:

Burt’s claim is that he focuses directly on the causal agent active in Granovetter.


Calculating the measures

Burt discusses 4 related aspects of a network:1) Effective Size2) Efficiency3) Constraint4) Hierarchy


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


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.


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


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


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


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


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


1

2

4 5

3

Efficiency is the effective size divided by the observed size.


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)


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


Constraintq

i jpij

piq pqj

Cij = Direct investment (Pij) + Indirect investment

2

qqjiqijij pppC


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


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.


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


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


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

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Structural Holes & Weak Ties

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