Presentation on theme: "Social Network Analysis"— Presentation transcript:
1 Social Network Analysis Introduction toSocial Network AnalysisColumbia UniversityNovember 2004James MoodyOhio State University
2 IntroductionWe live in a connected world:“To speak of social life is to speak of the association between people – their associating in work and in play, in love and in war, to trade or to worship, to help or to hinder. It is in the social relations men establish that their interests find expression and their desires become realized.”Peter M. BlauExchange and Power in Social Life, 1964"If we ever get to the point of charting a whole city or a whole nation, we would have … a picture of a vast solar system of intangible structures, powerfully influencing conduct, as gravitation does in space. Such an invisible structure underlies society and has its influence in determining the conduct of society as a whole."J.L. Moreno, New York Times, April 13, 1933These patterns of connection form a social space, that can be seen in multiple contexts:
3 IntroductionSource: Linton Freeman “See you in the funny pages” Connections, 23, 2000,
4 High Schools as Networks IntroductionHigh Schools as Networks
7 IntroductionAnd yet, standard social science analysis methods do not take this space into account.“For the last thirty years, empirical social research has been dominated by the sample survey. But as usually practiced, …, the survey is a sociological meat grinder, tearing the individual from his social context and guaranteeing that nobody in the study interacts with anyone else in it.”Allen Barton, 1968 (Quoted in Freeman 2004)Moreover, the complexity of the relational world makes it impossible to identify social connectivity using only our intuition.Social Network Analysis (SNA) provides a set of tools to empirically extend our theoretical intuition of the patterns that compose social structure.
8 Introduction Why do Networks Matter? Local vision Consider the following example. Here we have sampled respondents (red dots) reporting on their interaction with romantic partners. A classic local network module would ask about their characteristics and behaviors, then attempt to relate those characteristics to ego’s behavior. All of these sampled nodes have the exact same number of partners.
9 Introduction Why do Networks Matter? Local vision But these nodes are situated in dramatically different parts of the real underlying global network. Here some of them (lower left) are truly local isolates, but most are embedded in a larger network structure.These are real data from Add Health, on romantic involvement.
10 Social network analysis is: IntroductionSocial network analysis is:a set of relational methods for systematically understanding and identifying connections among actors. SNAis motivated by a structural intuition based on ties linking social actorsis grounded in systematic empirical datadraws heavily on graphic imageryrelies on the use of mathematical and/or computational models.Social Network Analysis embodies a range of theories relating types of observable social spaces and their relation to individual and group behavior.
12 “Networks as Variables” approaches IntroductionKey QuestionsSocial Network analysis lets us answer questions about social interdependence. These include:“Networks as Variables” approachesAre kids with smoking peers more likely to smoke themselves?Do unpopular kids get in more trouble than popular kids?Are people with many weak ties more likely to find a job?Do central actors control resources?“Networks as Structures” approachesWhat generates hierarchy in social relations?What network patterns spread diseases most quickly?How do role sets evolve out of consistent relational activity?We don’t want to draw this line too sharply: emergent role positions can affect individual outcomes in a ‘variable’ way, and variable approaches constrain relational activity.
13 We represent actors with points and relations with lines. Social Network DataThe unit of interest in a network are the combined sets of actors and their relations.We represent actors with points and relations with lines.Actors are referred to variously as:Nodes, vertices, actors or pointsRelations are referred to variously as:Edges, Arcs, Lines, TiesExample:bdace
14 Social Network data consists of two linked classes of data: Basic Data ElementsSocial Network data consists of two linked classes of data:Nodes: Information on the individuals (actors, nodes, points, vertices)Network nodes are most often people, but can be any other unit capable of being linked to another (schools, countries, organizations, personalities, etc.)The information about nodes is what we usually collect in standard social science research: demographics, attitudes, behaviors, etc.Often includes dynamic information about when the node is activeb) Edges: Information on the relations among individuals (lines, edges, arcs)Records a connection between the nodes in the networkCan be valued, directed (arcs), binary or undirected (edges)One-mode (direct ties between actors) or two-mode (actors share membership in an organization)Includes the times when the relation is activeGraph theory notation: G(V,E)
15 Social Network DataBasic Data ElementsIn general, a relation can be: (1) Binary or Valued (2) Directed or UndirectedabcedUndirected, binaryDirected, binaryUndirected, ValuedDirected, Valued1342The social process of interest will often determine what form your data take. Almost all of the techniques and measures we describe can be generalized across data format.
16 Social Network Data Global-Net Basic Data Elements: Levels of analysis 2-stepPartial networkPrimaryGroupGlobal-NetEgo-NetBest FriendDyad
17 We can examine networks across multiple levels: Social Network DataBasic Data Elements: Levels of analysisWe can examine networks across multiple levels:1) Ego-network- Have data on a respondent (ego) and the people they are connected to (alters). Example: 1985 GSS module- May include estimates of connections among alters2) Partial network- Ego networks plus some amount of tracing to reach contacts of contacts- Something less than full account of connections among all pairs of actors in the relevant population- Example: CDC Contact tracing data for STDs
18 We can examine networks across multiple levels: Social Network DataBasic Data Elements: Levels of analysisWe can examine networks across multiple levels:3) Complete or “Global” data- Data on all actors within a particular (relevant) boundary- Never exactly complete (due to missing data), but boundaries are setExample: Coauthorship data among all writers in the social sciences, friendships among all students in a classroom
19 Social Network DataBasic Data StructuresWorking with pictures.No standard way to draw a sociogram: each of these are equal:
20 Social Network Data Basic Data Structures In general, graphs are cumbersome to work with analytically, though there is a great deal of good work to be done on using visualization to build network intuition.I recommend using layouts that optimize on the feature you are most interested in. The two I use most are a hierarchical layout or a force-directed layout are best.Start w. simply opening CNAT.Then look at ZOOM, Refresh, etc.Then re-draw the network based on DegreeThen redraw from a particular ego.
21 a b c d e 1 a b c d e 1 Social Network Data Basic Data Structures a b From pictures to matricesabcedabcedUndirected, binaryDirected, binaryabcde1abcde1
22 a b c d e 1 Arc List Adjacency List a b b a b c c b c d c e d c d e Social Network DataBasic Data StructuresFrom matrices to listsabcde1Arc ListAdjacency Lista bb ab cc bc dc ed cd ee ce da bb a cc b d ed c ee c d
23 Communication, friendship, giving orders, sending email. Social Network DataBasic Data Elements: ModesSocial network data are substantively divided by the number of modes in the data.1-mode data represents edges based on direct contact between actors in the network. All the nodes are of the same type (people, organization, ideas, etc). Examples:Communication, friendship, giving orders, sending .There are no constraints on connections between classes of nodes.1-mode data are usually singly reported (each person reports on their friends), but you can use multiple-informant data, which is more common in child development research (Cairns and Cairns).
24 People as members of groups People as authors on papers Social Network DataBasic Data Elements: ModesSocial network data are substantively divided by the number of modes in the data.2-mode data represents nodes from two separate classes, where all relations cross classes. Examples:People as members of groupsPeople as authors on papersWords used often by peopleEvents in the life history of peopleThe two modes of the data represent a duality: you can project the data as people connected to people through joint membership in a group, or groups to each other through common membershipN-mode data generalizes the constraint on ties between classes to N groups
25 Breiger: 1974 - Duality of Persons and Groups Social Network DataBasic Data Elements: ModesBreiger: Duality of Persons and GroupsArgument:Metaphor: people intersect through their associations, which defines (in part) their individuality.The Duality argument is that relations among groups imply relations among individuals
26 Social Network Data Basic Data Elements: Modes Bipartite networks imply a constraint on the mixing, such that ties only cross classes.Here we see a tie connecting each woman with the party she attended (Davis data)
27 Social Network Data Basic Data Elements: Modes Bipartite networks imply a constraint on the mixing, such that ties only cross classes.Here we see a tie connecting each woman with the party she attended (Davis data)
28 Social Network Data Basic Data Elements: Modes By projecting the data, one can look at the shared between people or the common memberships in groups: this is the person-to-person projection of the 2-mode data.
29 Social Network Data Basic Data Elements: Modes By projecting the data, one can look at the shared between people or the common memberships in groups: this is the group-to-group projection of the 2-mode data.
30 A = Social Network Data Working with two-mode data Basic Data Elements: ModesWorking with two-mode dataA person-to-group adjacency matrix is rectangular, with persons down rows and groups across columnsEach column is a group, each row a person, and the cell = 1 if the person in that row belongs to that group.You can tell how many groups two people both belong to by comparing the rows: Identify every place that both rows = 1, sum them, and you have the overlap.ABCDEFA =
31 A = Social Network Data Working with two-mode data Basic Data Elements: ModesWorking with two-mode dataCompare persons A and F:Person A is in 1 group, Person F is in two groups, and they are in no groups together.ABCDEFA =SA = 1F = 2AF = 0Or persons D and F:Person D is in 4 groups, Person F is in two groups, and they are in 2 groups together.SD = 4F = 4DF = 2
32 Similarly for Groups: A = Social Network Data Basic Data Elements: ModesWorking with two-mode dataSimilarly for Groups:ABCDEFA =1 2 1•2ABCDEFGroup 1 has 2 members, group 2 has 2 members and they overlap by 1 members (C).
33 Persons-to-Persons Groups-to-Groups Social Network Data Basic Data Elements: ModesWorking with two-mode dataIn general, you can get the overlap for any pair of groups / persons by summing the multiplied elements of the corresponding rows/columns of the persons-to-groups adjacency matrix. That is:Persons-to-PersonsGroups-to-Groups
34 AT = A = Social Network Data Working with two-mode data Basic Data Elements: ModesWorking with two-mode dataOne can get these easily with a little matrix multiplication. First define AT as the transpose of A (simply reverse the rows and columns). If A is of size P x G, then AT will be of size G x P.ABCDEFA =A B C D E FAT =
35 P = A(AT) G = AT(A) (5x6) (6x5) A * AT = P AT * A = P (6x5)(5x6) (6x6) Social Network DataBasic Data Elements: ModesABCDEFA B C D E FP = A(AT)G = AT(A)A =AT =(5x6)(6x5)A * AT = P(6x5)(5x6) (6x6)PA B C D E FABCDEFAT * A = P(5x6) 6x5) (5x5)G
36 The resulting network: 1) Is always symmetric Social Network DataBasic Data Elements: ModesTheoretically, these two equations define what Breiger means by duality:“With respect to the membership network,…, persons who are actors in one picture (the P matrix) are with equal legitimacy viewed as connections in the dual picture (the G matrix), and conversely for groups.” (p.87)The resulting network:1) Is always symmetric2) the diagonal tells you how many groups (persons) a person (group) belongs to (has)In practice, most network software (UCINET, PAJEK) will do all of these operations. It is also simple to do the matrix multiplication in programs like SAS or SPSS
37 Social Network Data Existing Sources of Social Network Data: Network Data Sources: Existing data sourcesExisting Sources of Social Network Data:There are lots of network data archived. Check INSNA for a listing. The PAJEK data page includes a number of exemplars for large-scale networks.2-Mode DataOne can construct networks from many different data sources if you want to work with 2-mode data. Any list can be so transformed.Director interlocksProtest event participation1-Mode DataLocal Network data:Fairly common, because it is easy to collect from sample surveys.GSS, NHSL, Urban Inequality Surveys, etc.Pay attention to the question askedKey features are (a) number of people named and (b) whether alters are able to nominate each other.
38 Existing Sources of Social Network Data: 1-Mode Data Network Data Sources: Existing data sourcesExisting Sources of Social Network Data:1-Mode DataPartial network data:Much less common, because cost goes up significantly once you start tracing to contacts.Snowball data: start with focal nodes and trace to contactsCDC style data on sexual contact tracingLimited snowball samples:Colorado Springs drug users dataGeneology dataSmall-world network samplesLimited Boundary data: select data within a limited boundCross-national trade dataFriendships within a classroomFamily support ties
39 Existing Sources of Social Network Data: 1-Mode Data Network Data Sources: Existing data sourcesExisting Sources of Social Network Data:1-Mode DataComplete network data:Significantly less common and never perfect.Start by defining a theoretically relevant boundaryThen identify all relations among nodes within that boundaryCo-sponsorship patterns among legislatorsFriendships within strongly bounded settings (sororities, schools)Examples:Add Health on adolescent friendshipsHallinan data on within-school friendshipsMcFarland’s data on verbal interactionElectronic data on citations or coauthorship (see Pajek data page)See INSNA home page for many small-scale networks
40 Boundary Specification Problem Social Network DataNetwork Data Sources: Collecting network dataBoundary Specification ProblemNetwork methods describe positions in relevant social fields, where flows of particular goods are of interest. As such, boundaries are a fundamentally theoretical question about what you think matters in the setting of interest.See Marsden (19xx) for a good review of the boundary specification problemIn general, there are usually relevant social foci that bound the relevant social field. We expect that social relations will be very clumpy. Consider the example of friendship ties within and between a high-school and a Jr. high:
41 Social Network DataNetwork Data Sources: Collecting network dataNetwork data collection can be time consuming. It is better (I think) to have breadth over depth. Having detailed information on <50% of the sample will make it very difficult to draw conclusions about the general network structure.Question format:If you ask people to recall names (an open list format), fatigue will result in under-reportingIf you ask people to check off names from a full list, you can often get over-reportingc) It is common to limit people to a small number if nominations (~5). This will bias network measures, but is sometimes the best choice to avoid fatigue.d) Concrete relational indicators are best (who did you talk to?) over attitudes that are harder to define (who do you like?)
42 Boundary Specification Problem Social Network DataNetwork Data Sources: Collecting network dataBoundary Specification ProblemWhile students were given the option to name friends in the other school, they rarely do. As such, the school likely serves as a strong substantive boundary
43 When using a survey, common to use an “ego-network module.” Social Network DataNetwork Data Sources: Collecting network dataLocal Network data:When using a survey, common to use an “ego-network module.”First part: “Name Generator” question to elicit a list of namesSecond part: Working through the list of names to get information about each person namedThird part: asking about relations among each person named.GSS Name Generator:“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.”Why this question?Only time for one questionNormative pressure and influence likely travels through strong tiesSimilar to ‘best friend’ or other strong tie generatorsNote there are significant substantive problems with this name generator
44 Electronic Small World name generator: Social Network DataNetwork Data Sources: Collecting network dataElectronic Small World name generator:
45 The second part usually asks a series of questions about each person Social Network DataNetwork Data Sources: Collecting network dataLocal Network data:The second part usually asks a series of questions about each personGSS Example:“Is (NAME) Asian, Black, Hispanic, White or something else?”ESWP example:Will generate N x (number of attributes) questions to the survey
46 Social Network Data Local Network data: Network Data Sources: Collecting network dataLocal Network data:The third part usually asks about relations among the alters. Do this by looping over all possible combinations. If you are asking about a symmetric relation, then you can limit your questions to the n(n-1)/2 cells of one triangle of the adjacency matrix:12345GSS: Please think about the relations between the people you just mentioned. Some of them may be total strangers in the sense that they wouldn't recognize each other if they bumped into each other on the street. Others may be especially close, as close or closer to each other as they are to you. First, think about NAME 1 and NAME 2. A. Are NAME 1 and NAME 2 total strangers? B. ARe they especially close? PROBE: As close or closer to eahc other as they are to you?
47 Social Network Data Local Network data: Network Data Sources: Collecting network dataLocal Network data:The third part usually asks about relations among the alters. Do this by looping over all possible combinations. If you are asking about a symmetric relation, then you can limit your questions to the n(n-1)/2 cells of one triangle of the adjacency matrix:
48 Random Walk designs (Klovdahl) Strong tie designs All names designs Social Network DataNetwork Data Sources: Collecting network dataSnowball Samples:Snowball samples work much the same as ego-network modules, and if time allows I recommend asking at least some of the basic ego-network questions, even if you plan to sample (some of) the people your respondent names.Start with a name generator, then any demographic or relational questions.Have a sample strategyRandom Walk designs (Klovdahl)Strong tie designsAll names designsGet contact information from the people namedSnowball samples are very effective at providing network context around focal nodes. Detailed treatments of snowball sampling estimates are given in Frank ().
49 Social Network Data Snowball Samples: Network Data Sources: Collecting network dataSnowball Samples:
50 Recall surveys (“Name all of your best friends”) Social Network DataNetwork Data Sources: Collecting network dataComplete Network dataData collection is concerned with all relations within a specified boundary.Requires sampling every actor in the population of interest (all kids in the class, all nations in the alliance system, etc.)The network survey itself can be much shorter, because you are getting information from each person (so ego does not report on alters).Two general formats:Recall surveys (“Name all of your best friends”)Check-list formats: Give people a list of names, have them check off those with whom they have relations.
51 Social Network DataNetwork Data Sources:Collecting network dataComplete network surveys require a process that lets you link answers to respondents.You cannot have anonymous surveys.Recall:Need Id numbers & a roster to link, or hand-code names to find matchesChecklistsNeed a roster for people to check through
52 Social Network DataNetwork Data Sources: Missing DataWhatever method is used, data will always be incomplete. What are the implications for analysis?Example 1. Ego is a matchable person in the SchoolUnOutOutOutUnUnMEgoMEgoMMMMMMTrue NetworkObserved Network
53 Social Network Data Network Data Sources: Missing Data Example 2. Ego is not on the school rosterMMUnMUnMMMMMMMUnUnUnTrue NetworkObserved Network
54 Social Network Data Example 3: Network Data Sources: Missing DataExample 3:Node population: 2-step neighborhood of Actor XRelational population: Any connection among all nodesFFullFull (0)Full (0)F126.96.36.199.188.8.131.52.184.108.40.206.220.127.116.11.3FFullFullFull (0)1-stepF(0)FullFullUK2-step3-stepF(0)Full (0)UnknownUK
55 Social Network Data Example 4 Network Data Sources: Missing DataExample 4Node population: 2-step neighborhood of Actor XRelational population: Trace, plus All connections among 1-step contactsF1-step2-step3-stepFullFull (0)Full (0)F18.104.22.168.22.214.171.124.126.96.36.199.188.8.131.52.3FFullFullFull (0)F(0)FullUnknownUKF(0)Full (0)UnknownUK
56 Social Network Data Example 5. Network Data Sources: Missing DataExample 5.Node population: 2-step neighborhood of Actor XRelational population: Only tracing contactsFFullFull (0)Full (0)F184.108.40.206.220.127.116.11.18.104.22.168.22.214.171.124.3FUnknownFullFull (0)1-stepF(0)FullUnknownUK2-step3-stepF(0)Full (0)UnknownUK
57 Social Network Data Example 6 Network Data Sources: Missing DataExample 6Node population: 2-step neighborhood from 3 focal actorsRelational population: All relations among actorsFocal1-Step2-Step3-StepFocalFullFullFull (0)Full (0)1-StepFullFullFullFull (0)Full(0)FullFullUK2-Step3-StepFull(0)Full (0)UnknownUK
58 Social Network Data Example 7. Network Data Sources: Missing DataExample 7.Node population: 1-step neighborhood from 3 focal actorsRelational population: Only relations from focal nodesFocal1-Step2-Step3-StepFocalFullFullFull (0)Full (0)1-StepFullUnknownUnknownFull (0)Full(0)UnknownUnknownUK2-Step3-StepFull(0)Full (0)UnknownUK
59 Local Network Analysis IntroductionLocal 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 MixingThe extent to which one type of person is tied to another type of person (race by race, etc.)Local Network CompositionPeer behaviorCultural milieuOpportunities or Resources in the networkSocial SupportLocal Network StructuralNetwork SizeDensityHoles & ConstraintConcurrencyDyadic behaviorFrequency of contactInteraction contentSpecific exchange behaviors
60 Local Network Analysis IntroductionAdvantagesCost: data are easy to collect and can be sampledMethods are relatively simple extensions of common variable-based methods social scientists are already familiar withProvides information on the local network context, which is often the primary substantive interest.Can be used to describe general features of the global network contextPopulation mixing, concurrency, activity distribution (limited)DisadvantagesTreats 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
61 Local Network Analysis Network CompositionPerhaps 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 largeHomogenous 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
62 Local Network Analysis Network CompositionClaude Fischer’s book “To Dwell Among Friends” is a classic study of urbanism that makes good use of local network data.Age heterogeneity varies by ego’s age and across urban settings.
63 Local Network Analysis Network CompositionClaude Fischer’s book “To Dwell Among Friends” is a classic study of urbanism that makes good use of local network data.Marital composition similarly varies across respondents and settings
64 Local Network Analysis Network CompositionCalculating network composition using “GSS style” data.Generally you have a separate variable for each alter characteristic, and you can construct items by summing over the relevant variables.You would, for example, have variables on age of each alter such as:Age_alt1 age_alt2 age_alt3 age_alt4 age_alt5You get the mean age, then, with a statement such as:meanage=mean(Age_alt1, age_alt2, age_alt3, age_alt4, age_alt5);Be sure you know how the program you use (SAS, SPSS) deals with missing data.
65 Local Network Analysis Network CompositionCalculating local network information from global network dataDefine the local neighborhood:Distance (1-step, 2-steps, what?)Direction of tieSent, Received, or both?Pull the relevant altersMatch the alters to the variables of interestOnce 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.
66 Local Network Analysis Network CompositionAn example network:All senior males from a small (n~350) public HS.SPAN will do this for you
67 Local Network Analysis Network CompositionCommon composition measures:Level measures:Mean of a given attribute (average income of alters)Proportion with a particular attribute (proportion who smoke)Counts (number of peers who have had sex)Dispersion measures:Heterogeneity index (Racial heterogeneity)Index of dissimilarityStandard DeviationAbsolute value of the differencesVariable range of valuesComposition measures for multiple variables simultaneouslyAverage correlation across all altersEuclidean / Mahalanobis distance measures
68 Local Network Analysis Network MixingA common interest in network research is identifying how likely persons of one category are to interact with people of another category.Examples:Race mixing: how likely are people of one race to interact with people of another?Sexual activity mixing: Are people with many partners likely to associate with each other?Neighborhood / location mixing: Are people likely to name friends from the same neighborhood.These questions can be answered by cross classifying the category of the nominator with the category of the nominated in a “mixing matrix”.
69 Local Network Analysis Network MixingRace mixing in one of the Add Health schools
70 Local Network Analysis Network MixingWhite Black Hispan Asian Mix/OtherWhiteBlackHispanicAsianMix/Other
71 Local Network Analysis Network MixingWorking with mixing matrices:Group segregation index (Freeman 1972)Associations between rows and columns (valued relations)“Assortative mixing”Correlations or QLog-linear modelsAssessing chance levels depends on the data available. If you have full network data you can look at density between groups, without you can only focus on the sheer volume of ties (without information on the size of the “target” groups)
72 Local Network Analysis Network StructureWhile network structure data are limited, there are a number of features that can be of interest, assuming you have data on the relations among ego’s contacts.Basic arguments:“structural amplification:” that some feature of the arrangement of ties amplifies any peer effect of network composition (see Haynie’s paper)“Network range effects:” that being connected to a diverse set of alters -- who are not connected to each other – provides profitable returns. Granovetter’s “Strength of Weak Ties”, Burt’s “Structural Holes”Familiar to students of social theory as the Tertius Gaudens argument from SimmelIn both cases, we use the pattern of ties surrounding ego to characterize the local structure. We start with volume measures, then move on to more complex pattern measures.
73 Distribution of total network size, GSS 1985 Local Network AnalysisNetwork Structure: volumeNetwork SizeDistribution of total network size, GSS 1985
74 Local Network Analysis Network Structure: volumeNetwork Size by:Age:Drops with age at an increasing rate. Elderly have few close ties.Education:Increases with education. College degree ~ 1.8 times largerSex (Female):No gender differences on network size.Race:African Americans networks are smaller (2.25) than White Networks (3.1).
75 Local Network Analysis Network Structure: volumeWhat does Fischer have to say about the size of local nets (by context)?
76 D= 5 / ((5*4)/2) = 5 / 10 = 0.5 Local Network Analysis Network Structure: volumeDensity is the average value of the relation among all pairs of ties.= T / ((N*N-1)/2)Density is usually calculated over the alters in the network.212345231451R345D= 5 / ((5*4)/2)= 5 / 10= 0.5
77 Local Network Analysis Network Structure: volumeWhat does Fischer have to say about the density of local nets (by context)?
78 Local Network Analysis Network Structure: volumeIn general, dense networks should be more cohesive and we would expect that “goods” will flow through the network more efficientlySocial support & peer influence, for example, should be stronger in dense networksDensity is a volume measure, however, and can mask significant structural differences:These two networks, for example, have the same density but very different structures.Most network analysis programs will calculate ego-network density directly.
79 Local Network Analysis Network Structure: Weak Ties & Structural Holes“The Strength of Weak Ties”In a classic article, Granovetter (1972) argues that for many purposes (such as getting a job), the most useful network contacts are through “weak ties.”This is because weak ties connect you to a more diverse set of alters, increasing the ‘range’ of your network.Your strong ties tend to be tied to each other, making them redundant for the purposes of bringing information.Essentially this argument works on a spurious relation. The key value of weak ties is not in the weak affective bond, but in the structural location of the ties. We can measure this directly, and Ron Burt provides a series of measures for doing so.
80 Local Network Analysis Network Structure: Weak Ties & Structural HolesMaximumEfficiencyDecreasing EfficiencyNumber of Non-Redundant ContactsIncreasing EfficiencyMinimumEfficiencyNumber of Contacts
81 Local Network Analysis Network Structure: Weak Ties & Structural HolesEffective SizeConceptually 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 - RedundancyWhere 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.
82 3 2 1 5 4 Local Network Analysis Effective Size: Network Structure: Weak Ties & Structural HolesEffective Size:Piq is the proportion of actor i’s relations that are spent with q.32AdjacencyP154
83 Local Network Analysis Network Structure: Weak Ties & Structural HolesEffective 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)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.
84 Effective Size: 3 2 1 5 4 Redundancy = 1 Effective size = 4-1 = 3 Local Network AnalysisNetwork Structure: Weak Ties & Structural HolesEffective Size:32Working with 1 as ego, we get the following redundancy levels:1PPM1jq54Redundancy = 1Effective size = 4-1 = 3
85 Effective Size: 3 2 1 5 4 Local Network Analysis Network Structure: Weak Ties & Structural HolesEffective Size:32When you work it out, in a binary network, redundancy reduces to the average degree, not counting ties with ego of ego’s alters. Since the average degree is simply another way to say density, we can calculate redundancy as:2t/nwhere 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/n154UCINET, STRUCTURE, SPAN and PAJEK all calculate effective size
86 Ego Size Size: Efficiency 1 4 3 .75 2 2 1 .50 3 1 1 1.00 4 2 1 .50 Local Network AnalysisNetwork Structure: Weak Ties & Structural HolesEfficiency is simply effective size divided by observed size. Taken from each ego’s point of view, efficiency in this network would be:EffectiveEgo Size Size: Efficiency32154
87 3 2 1 5 4 Local Network Analysis Constraint Network Structure: Weak Ties & Structural HolesConstraintConceptually, constraint refers to how much room you have to negotiate or exploit potential structural holes in your network.32“..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)154P
88 q i j Constraint pij piq pqj Local Network AnalysisNetwork Structure: Weak Ties & Structural HolesConstraintqijpijpiqpqjCij = Direct investment (Pij) + Indirect investment (PiqPqj)
89 Local Network Analysis Network Structure: Weak Ties & Structural Holes32Constraint154Given the p matrix, you can get indirect constraint (piqpqj) by simply squaring the matrix:P*PP
90 Total constraint between any two people then is: Local Network AnalysisNetwork Structure: Weak Ties & Structural HolesConstraintTotal constraint between any two people then is:C = (P + P2)##2Where P is the normalized adjacency matrix, and ## means to square the elements of the matrix.
91 Local Network Analysis Network Structure: Weak Ties & Structural HolesHierarchyConceptually, hierarchy (for Burt) is really the extent to which constraint is concentrated in a single actor. It is calculated as:
92 Hierarchy 3 2 1 5 4 H=.514 Local Network Analysis Network Structure: Weak Ties & Structural HolesHierarchy321CC:54H=.514
93 Local Network Analysis Network Structure: Weak Ties & Structural HolesBurt (2004) AJS 110:
94 Local Network Analysis Network Structure: Weak Ties & Structural HolesBurt (2004) AJS 110:
95 Local Network Analysis Network Structure: Weak Ties & Structural HolesBurt (2004) AJS 110:
96 Local Network Analysis Local Network Models: Modeling IssuesLocal Network modeling issuesCase independenceIn very clustered settings, the alters that each person names will overlap. This will lead to non-independence among the cases.If you have enough cases or over time data, you can use random or fixed effect modelsIf you know the names of alters, you can link them to build in a direct network autocorrelation effect.Small network effectsBe aware of the size of your networks. Substantively, having 50% white networks means something different in a net of size 2 vs a net of size 10. I often suggest interactions to check for these kinds of effectsDealing with isolatesIsolated nodes have no network alters, so none of these measures apply. Depending on the context, you can either leave them out of the analysis, or use interaction terms to selectively apply the measures of interest.
97 Local Network Analysis Local Network Models: Modeling IssuesSelectionThat some unobserved factor, z, creates both friendships and the outcome of interest.EndogeneityThat the causal order of peer relations and outcomes is reversed. Peers do not cause Y, but Y causes friendship relations
98 Local Network Analysis Local Network Models: Modeling IssuesSelectionWhat do we know about how friendships form?Opportunity / focal factors- Being members of the same group- In the same class- On the same team- Members of the same churchStructural Relationship factors- Reciprocity- Social BalanceBehavior Homophily- Smoking- Drinking
99 Local Network Analysis Local Network Models: Modeling IssuesSelectionHow to correct this problem?Essentially, this is an omitted variable problem, and the obvious “solution” is been to identify as many potentially relevant alternative variables as you can find.Sensitivity measures (see Ken Frank’s work here)Propensity score matchingIndividual-level fixed effect modelsSubstantively you only look at change in Y as a function of change in X, holding constant (because dummied out) any individual level effect.This works, but it’s drastic. Any endogenous effect of networks on the self are essentially removed
100 where P = some peer function. But the actual model may really be: Local Network AnalysisLocal Network Models: Modeling IssuesEndogeneityEstimated:Y = b0 + b1(P) + ewhere P = some peer function.But the actual model may really be:P = b’0 + b’1(Y) + e
101 Local Network Analysis Local Network Models: Modeling IssuesEndogeneityDoes it matter?Algebraically the relation between y and p should be direct translation of the coefficients since:The statistical problem of endogeneity is that when you estimateb’1, it does not equal 1/b1, because of our assumptions about x, and hence e. There are other models that make different assumptions, where this direction is irrelevant. But they are uncommon and hard to work with in the multivariate context.(see Joel H. Levine, Exceptions are the Rule, for a full discussion of this)
102 Local Network Analysis Local Network Models: Modeling IssuesPossible solutions:Theory: Given what we know about how friendships form, is it reasonable to assume a bi-directional cause? That is, work through the meeting, socializing, etc. process and ask whether it makes sense that Y is a cause of P.Models:Time Order. We are on somewhat firmer ground if P precedes Y in time.- Simultaneous Equation Models. Model both the friendship pattern and the outcome of interest simultaneously. Difficult to identify “instruments” or to specify orders that do not logically make the model inestimable.
103 Local Network Analysis Local Network Models: Peer influence exampleHaynie asks whether peers matter for delinquent behavior, focusing on:a) the distinction between selection and influenceb) the effect of friendship structure on peer influenceTwo basic theories underlie her work:a) Hirchi’s Social Control TheorySocial bonds constrain otherwise criminal behaviorThe theory itself is largely ambivalent toward direction of network effectsb) Sutherland’s Differential AssociationBehavior is the result of internalized definitions of the situationThe effect of peers is through communication of the appropriateness of particular behaviorsHaynie adds to these the idea that the structural context of the network can “boost” the effect of peers: (a) so transmission is more effective in locally dense networks and (b) the effect of peers is stronger on central actors.
104 Local Network Analysis Local Network Models: Peer influence example
105 Local Network Analysis Local Network Models: Peer influence example