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.

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 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 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 1 4 3 .75 2 2 1 .5 3 1 1 1.0 4 2 1 .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)

Structural Holes & Weak Ties Constraint 3 2 1 5 4 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

q i j Structural Holes & Weak Ties Constraint pij piq pqj 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