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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

SAS Data steps

3) Calculating Ego-Network Measures

From Ego-network modules

From 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:

Cohesion

Structural Equivalence

Structural Holes & Weak Ties

Efficiency

Maximize the number of non-redundant contacts

Effectiveness

Draw your primary contacts from different social worlds

Structural Holes & Weak Ties

Maximum

Efficiency

Decreasing Efficiency

Number of Non-Redundant Contacts

Increasing Efficiency

Minimum

Efficiency

Number of Contacts

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 Size

2) Efficiency

3) Constraint

4) 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 Size

Burt’s 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.

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

Structural Holes & Weak Ties

Effective Size:

Piq is the proportion of actor i’s relations that are spent with q.

3

2

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

Structural Holes & Weak Ties

Effective Size:

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.

Structural Holes & Weak Ties

Effective Size:

3

2

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

Sum=1, so

Effective size = 4-1 = 3.

Structural Holes & Weak Ties

Effective Size:

3

2

When you work it out, redundancy reduces to the average degree, not counting ties with ego of ego’s alters.

1

5

4

Node Degree

2 1

3 0

4 1

5 2

Mean: 4/4 = 1

Structural Holes & Weak Ties

Effective Size:

3

2

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

Structural Holes & Weak Ties

Efficiency is the effective size divided by the observed size.

3

2

Effective

Node Size Size: Efficiency

143 .75

2 21 .5

3 111.0

4 21 .5

5 3 1.67 .55

1

5

4

Structural Holes & Weak Ties

Constraint

Conceptually, constraint refers to how much room you have to negotiate or exploit potential structural holes in your network.

3

2

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)

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

Structural Holes & Weak Ties

Constraint

3

2

1

5

4

q

piq

pqj

i

j

pij

Structural Holes & Weak Ties

Constraint

Cij = Direct investment (Pij) + Indirect investment

Structural Holes & Weak Ties

3

2

Constraint

1

5

4

Given the p matrix, you can get indirect constraint (piqpqj) with the 2-step path distance.

P*P

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

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.

Structural Holes & Weak Ties

Constraint

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:

Structural Holes & Weak Ties

Hierarchy

3

2

1

2 3 4 5 C

C: .11 .06 .11 .25 .53

.83 .46 .83 1.9

5

4

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:

PAJEK

UCINET

NEGOPY

STRUCTURE

Playing with data: Using SAS to move data.

Basic Elements:

SAS is a language:

Data Steps = Nouns

Procedures = Verbs

Data needs:

Creation / Read

Organization

Transformation

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/

SAS

The procedure we have been using is IML or the

Interactive Matrix Language.

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