Local Network Analysis Local Network Analysis a b

Local Network Analysis

Local Network Analysis a) b) c) d) Basic Definitions Network Composition Network Structure Social Capital a) Bridging Local & Global? b) Position Generator stuff If time/interest: some software stuff?

Local Network Analysis Basic Definitions Local networks – The network incident to a focal individual. KA BN PP PC DC CM HF CF

Local Network Analysis Basic Definitions Local networks – The network incident to a focal individual.

Local Network Analysis Local network analysis uses data from a simple ego-network survey. These might include information on relations among ego’s contacts, but often not. Questions include: Population Mixing The extent to which one type of person is tied to another type of person (race by race, etc. ) Local Network Composition Peer behavior Cultural milieu Opportunities or Resources in the network Social Support Local Network Structural Network Size Density Holes & Constraint Concurrency Dyadic behavior Frequency of contact Interaction content Specific exchange behaviors

Local Network Analysis Introduction Advantages • Cost: data are easy to collect and can be sampled • Methods are relatively simple extensions of common variable-based methods social scientists are already familiar with • Provides information on the local network context, which is often the primary substantive interest. • Can be used to describe general features of the global network context • Population mixing, concurrency, activity distribution (limited) Disadvantages • Treats each local network as independent, which is false. The poor performance of ‘number of partners’ for predicting STD spread is a clear example. • Impossible to account for how position in a larger context affects local network characteristics. “popular with who” • If “structure matters”, ego-networks are strongly constrained to limit the information you can get on overall structure

Local Network Analysis Network Composition Perhaps the simplest network question is “what types of alters does ego interact with”? Network composition refers to the distribution of types of people in your network. Networks tend to be more homogeneous than the population. Using the GSS, Marsden reports heterogeneity in Age, Education, Race and Gender. He finds that: • Age distribution is fairly wide, almost evenly distributed, though lower than the population at large • Homogenous by education (30% differ by less than a year, on average) • Very homogeneous with respect to race (96% are single race) • Heterogeneous with respect to gender

Local Network Analysis General Questions that you can ask / answer Mixing The extent to which one type of person is tied to another type of person (race by race, etc. ) Aspects of the local context: Peer delinquency Cultural milieu Opportunities Social Support: Extent of resources (and risks) present in a type of network environment. Structural context (next class)

Local Network Analysis Mechanics Calculating local network information. 1) From data, such as the GSS, which has ego-reported information on alter 2) From global network data, such as Add Health, where you have self-reports on alters behaviors.

Local Network Analysis Mechanics Calculating local network information 1: GSS style data. This is the easiest situation. Here you have a separate variable for each alter characteristic, and you can construct density items by summing over the relevant variables. You would, for example, have variables on age of each alter such as: Age_alt 1 age_alt 2 age_alt 3 age_alt 4 age_alt 5 15 35 20 12. You get the mean age, then, with a statement such as: meanage=mean(Age_alt 1, age_alt 2, age_alt 3, age_alt 4, age_alt 5); Be sure you know how the program you use (SAS, SPSS) deals with missing data.

Local Network Analysis Mechanics Calculating local network information 1: GSS style data. In addition to the moments of the distribution (mean, std. dev, etc. ) we are often interested in “mixing” and use a “mixing matrix. ” This is just a cross tab of ego’s characteristics by alters' characteristics: W White Black Hispanic Other B H O

Local Network Analysis Mechanics Calculating local network information 1: GSS style data. Data from Sexual Mixing patterns in China

Local Network Analysis Mechanics Calculating local network information 1: GSS style data. 9 – 12 schools (N of schools: 51) 9 th Graders 10 th Graders 11 th Graders 12 th Graders 9 th Graders 80. 42 10. 23 5. 57 3. 79 10 th Graders 10. 49 69. 49 12. 35 7. 67 11 th Graders 5. 68 12. 52 67. 13 14. 67 12 th Graders 3. 42 7. 76 14. 96 73. 86

Local Network Analysis Mechanics Calculating local network information 2: From a global network. There are multiple options when you have complete network information. Type of tie: Sent, Received, or both? Once you decide on a type of tie, you need to get the information of interest in a form similar to that in the example above.

Local Network Analysis Mechanics Calculating local network information from a global network. An example network: All senior males from a small (n~350) public HS.

Local Network Analysis Mechanics Suppose you want to identify ego’s friends, calculate what proportion of ego’s female friends are older than ego, and how many male friends they have (this example came up in a model of fertility behavior). You need to: • Construct a dataset with (a) ego's id. This allows you to link each person in the network. (b) age of each person, (c) the friendship nominations variables. • Then you need to: a) Identify ego's friends b) Identify their age c) compare it to ego's age d) count it if it is greater than ego's. There is a SAS program described in the exercise that shows you how to do this kind of work, using the graduate student network data.

Local Network Analysis Composition - homophily Highly cited: Google Scholar 6258 as of 1. 29. 15

Local Network Analysis Composition - homophily Key bit here is a duality between settings and relations

Local Network Analysis Composition - homophily “Status homophily includes the major socio-demographic dimensions that stratify society—ascribed characteristics like race, ethnicity, sex, or age, and acquired characteristics like religion, education, occupation, or behavior patterns. Value homophily includes the wide variety of internal states presumed to shape our orientation toward future behavior. ”

Local Network Analysis Composition - homophily This is the proportion different. So 4. 7% of pairs in 1985 were different race, compared to 9. 8% in 2004. Smith et al “Social Distance in the U. S. A: Sex, Race, Religion, Age, and Education: Note this is *raw* rates, not adjusted for chance.

Local Network Analysis Composition - homophily Use matched sample logit model & predict tie as a function of social distance. Smith et al “Social Distance in the U. S. A: Sex, Race, Religion, Age, and Education”

Local Network Analysis Composition - homophily Smith et al “Social Distance in the U. S. A: Sex, Race, Religion, Age, and Education”

Local Network Analysis Composition - homophily

Local Network Analysis Composition - homophily Coefficient for logit model of any cross sex ties, by setting: highly contextual

Local Network Analysis Composition - homophily Di. Prete, Thomas A. , Andrew Gelman Tyler Mc. Cormic Julien Teitler & Tian Zheng. “Segregation in Social Networks based on Acquaintanceship and Trust” American Journal of Sociology 116: 1234 -1283

Local Network Analysis Composition - homophily Network Scaleup is an old technique, here I provide a calculator for doing it: http: //www. soc. duke. edu/~jmoody 77/Hydra/scaleupcalc. htm

Local Network Analysis Composition - homophily Key issue for homophily is “choice” versus AJS Volume 115 Number 2 (September 2009): 405– 50

Local Network Analysis Network structure Homophily / diversity are functions related to the composition of the network, we are often interested as well in the structure of the local network. -Size -Density -Tie patterns (reciprocity, transitivity) - local connectivity (number of components, structural holes)

Local Networks GSS Networks Network Size X 1985: 2. 9 X 2004: 2. 1 Increase in Social Isolation From time to time, most people discuss important matters with other people. Looking back over the last six months—who are the people with whom you discussed matters important to you? Just tell me their first names or initials. IF LESS THAN 5 NAMES MENTIONED, PROBE: Anyone else?

Local Networks GSS Networks Size by: Age: Drops with age at an increasing rate. Elderly have few close ties. Education: Increases with education. College degree ~ 1. 8 times larger Sex (Female): No gender differences on network size. Race: African Americans networks are smaller (2. 25) than White Networks (3. 1).

Local Networks GSS Networks Network Density Recall that density is the average value of the relation among all pairs of ties. Here, density is only calculated over the alters in the network. 2 1 R 5 1 2 3 1 3 4 5 4 D=0. 5 1 2 3 4 5 1 1 1

Local Networks GSS Networks 45 40 35 30 25 20 15 10 5 0 1985 2004 <. 25 -. 49 . 50 -. 74 Density >. 74

Local Networks GSS Networks Network Structure Summary (Marsden, based on GSS 1985)

Local Networks GSS Networks Non-Kin 1985 Non-Kin 2004 Kin 1985 Kin 2004

Local Networks To Dwell Among Friends One of the best-known books on the stuff of local social networks; framed as an attempt to test the idea that cities are socially isolating. Network Size

Local Networks To Dwell Among Friends Network Composition One of the best-known books on the stuff of local social networks; framed as an attempt to test the idea that cities are socially isolating.

Local Networks To Dwell Among Friends Network Composition (non-kin) One of the best-known books on the stuff of local social networks; framed as an attempt to test the idea that cities are socially isolating.

Local Networks To Dwell Among Friends One of the best-known books on the stuff of local social networks; framed as an attempt to test the idea that cities are socially isolating.

Fischer’s Work. What does Fischer have to say about the size of local nets (by context)?

Fischer’s Work. What does Fischer have to say about the density of local nets (by context)?

Social Network Mechanisms Strength of Weak Ties & Structural Holes Granovetter argues that, under many circumstances, strong ties are less useful than weak ties. Why?

Social Network Mechanisms Strength of Weak Ties & Structural Holes Key element is the correlation between tie strength and structure: The argument rests on the association, the mechanism is diffusion of information /opportunity through nonredundant ties

Social Network Mechanisms Strength of Weak Ties & Structural Holes 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. ”

Social Network Mechanisms Strength of Weak Ties & Structural Holes The basic notion of bridging a structural hole involves connecting people who are not otherwise connected. These two networks, for example, the focal nodes (red) have the same number of ties, but the node on the left spans more structural holes than the node on the right. As such, we would expect the node to have access to unique information or be able to play the one side of his network against the other. The key to power & effectiveness, according to Burt, often depends on having ties that are non-redundant.

Social Network Mechanisms Strength of Weak Ties & Structural Holes Intuitively, a tie is redundant if it connects you to somebody you are already connected to. The most efficient networks will be those where each tie takes you to an entirely new social world. The limit, of course, is when every person you are connected to is unconnected to anyone else. Number of Non-Redundant Contacts Maximum Efficiency Decreasing Efficiency Increasing Efficiency Number of Contacts Minimum Efficiency

Social Network Mechanisms Strength of Weak Ties & Structural Holes Intuitively, a tie is redundant if it connects you to somebody you are already connected to. The most efficient networks will be those where each tie takes you to an entirely new social world. Redundant & constrained Locally Redundant Unconstrained Low High Constraint

Social Network Mechanisms Strength of Weak Ties & Structural Holes Burt’s idea discussion network

Social Network Mechanisms Strength of Weak Ties & Structural Holes The results show a strong effect of network constraint on salary, evaluation and promotion, independent of the job/age characteristics related to human capital explanations.

Social Network Mechanisms Strength of Weak Ties & Structural Holes

Social Network Mechanisms Strength of Weak Ties & Structural Holes Why does this work? 2. 5 mechanisms 1) Social Balance & Homophily: A Friend of a Friend is a Friend • We are most similar to those we are closest to, so strong ties tend to be redundant & carry no information 2) Information Arbitrage & Control: Simmel’s Tertius Gaudens • Power comes from the ability to control a resource others need • Brokers can exploit their holes (up until a limit of legitimacy!) 2. 5) Social Skills Training • Brokers have to navigate multiple worlds, which is hard. • Doing this for long hones a particular kind of “people skill” that leads to success. This is Burt’s most recent argument.

Structural Holes & Weak Ties Calculations Calculating the measures Burt discusses 4 related aspects of a network: 1) Effective Size 2) Efficiency 3) Constraint 4) Hierarchy

Structural Holes & Weak Ties Calculations 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 Calculations 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 Calculations Effective Size: Piq is the proportion of actor i’s relations that are spent with q. 3 2 1 4 5 Adjacency 1 2 3 4 5 1 0 1 1 2 1 0 0 0 1 3 1 0 0 4 1 0 0 0 1 5 1 1 0 P 1 2 3 4 5 1. 00. 50 1. 0. 50. 33 2. 25. 00. 00. 33 3. 25. 00. 00 4. 25. 00. 00. 33 5. 25. 50. 00

Structural Holes & Weak Ties Calculations 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 Calculations Effective Size: 3 2 Working with 1 as ego, we get the following redundancy levels: 1 4 P 5 1 2 3 4 5 1. 00. 50 1. 0. 50. 33 2. 25. 00. 00. 33 3. 25. 00. 00 4. 25. 00. 00. 33 5. 25. 50. 00 1 2 3 4 5 1 ------ PM 1 jq 2 3 --- --. 00. 00. 00. 25. 00 4 --. 00. 00. 25 5 --. 25. 00 Sum=1, so Effective size = 4 -1 = 3.

Structural Holes & Weak Ties Calculations Effective Size: 3 2 1 4 5 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 2 Mean: 4/4 = 1

Structural Holes & Weak Ties Calculations Effective Size: 3 2 1 4 5 Since the average degree is simply another way to say density, we can calculate redundancy as: 2 t/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 - 2 t/n

Structural Holes & Weak Ties Calculations Efficiency is the effective size divided by the observed size. 3 2 1 4 5 Node 1 2 3 4 5 Size 4 2 1 2 3 Effective Size: 3 1 1. 67 Efficiency. 75. 5 1. 0. 5. 55

Structural Holes & Weak Ties Calculations Constraint 3 2 1 4 5 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 Calculations Constraint 3 2 1 4 5 P 1 2 3 4 5 1. 00. 50 1. 0. 50. 33 2. 25. 00. 00. 33 3. 25. 00. 00 4. 25. 00. 00. 33 5. 25. 50. 00

Structural Holes & Weak Ties Calculations Constraint q piq i pqj pij Cij = Direct investment (Pij) + Indirect investment j

Structural Holes & Weak Ties Calculations 3 2 Constraint 1 5 4 Given the p matrix, you can get indirect constraint (piqpqj) with the 2 -step path distance. P 1 2 3 4 5 1. 00. 50 1. 0. 50. 33 2. 25. 00. 00. 33 3. 25. 00. 00 4. 25. 00. 00. 33 5. 25. 50. 00 1 1 2 3 4 5 . . 165. 000. 165. 330 2. 083. . 250. 290. 083 P*P 3. 000. 125. 083 4. 083. 290. 250. . 083 5. 250. 125. . .

Structural Holes & Weak Ties Calculations Constraint Total constraint between any two people then is: C = (P + P 2)##2 Where P is the normalized adjacency matrix, and ## means to square the elements of the matrix.

Structural Holes & Weak Ties Calculations Constraint . 00. 67 1. 0. 67. 66 P+P 2. 33. 25. 00. 13. 25. 00. 29. 13. 41. 08 . 33. 29. 25. 00. 41 . 50. 63. 25. 63. 00. 44 1. 0. 44 Cij. 11. 06. 00. 02. 06. 00. 08. 02. 17. 01 . 11. 08. 06. 00. 17 . 25. 39. 06. 39. 00 C. 53

Structural Holes & Weak Ties Calculations 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 Calculations Hierarchy 3 2 1 4 5 C: 2 3 4 5. 11. 06. 11. 25. 83. 46. 83 1. 9 H=. 514 C. 53

Local Networks Social Capital (soc journals, as of 2001)

Local Networks Social Capital Relational capital – ability to use social ties to access resources. Position generator:

Local Networks Summary Advantages: • Can collect as random sample • Informative for most homophily/segregation measures • (though comes with reporting bias) • Structural information for local patterns can be collected • …and with statistical models can generalize to larger structure • Position generators & scale-up methods are probably most robust to particular survey design issues Disadvantages: • Ego-network module is complex to implement in a survey • Fully dependent on ego self-report • No path-level information