Working with Node Attributes

Transcription

1 Working with Node Attributes Steve Borgatti 2009 LINKS Center Workshop ADVANCED Session

2 Egonet Composition Concept Characterize the alters that an ego has ties a given type of ties with What proportion of friends are men, women, gay, etc. Average income of a person s friends Can define ego network in terms of outgoing ties or incoming ties What kinds of actors are nominating a given node What kinds of actors is a given actor nominating? Categorical & continuous alter attributes Gender, department, religion, etc. Age, income, centrality, structural holes, etc. of the alter

3 Egonet Strength Concept Average value of an alter attribute, such as wealth or power Social capital ala social resource theory (Lin) Measure Given adjacency matrix A and attribute vector v, the matrix product Av gives the sum of attribute values for the alters of each node Can also compute average, maximum, minimum etc s = j i a j ij a v ij j

4 Egonet Heterogeneity Concept Measure the diversity of an actor s contacts For categorical attributes, e.g., gender use Blau Herfindahl index = 2 H p k k where pk gives the proportion of alters that fall into category k For continuous attributes, e.g., wealth of alters Calculate standard deviation

5 Homophily Concept Extent to which actors tend to have ties with actors who aresimilar to themselves E.g., girls confide mostly to girls, boys to boys Dozens of Measures Pct homophilous matches E I Correlation

6 converting attributes to matrices Problem Given vector v representing a node attribute, construct a matrix X Categorical attributes x ij = 1 if v i = v j, and x ij = 0 otherwise Continuous attributes x ij = v i v j x ij = (v i v j ) 2

7 converting attributes to matrices Categorical example V x ij = 1 if v i = v j, and x ij = 0 otherwise

8 Set up for homophily measures Given A social ilrelation R A categorical attribute vector a Construct Similarity il i relation S in which h s ij = 1 if a i = a j, and s ij = 0 otherwise f f f f f f f f mmmmmmmmmm f f f f f f f f m m mmmmmmmm f f f f f f f f f f f f f f f f m m m m m m m m m m m m m m m m m m m m R a S

9 Homophily set up cont Construct relational contingency table S 1 0 R 1 a b 0 c d R = the data the social relation S = similarity is 1 if same attrib value Campnet dataset: S 1 0 R

10 Pct homophilous matches (H%) H% = a/(a+b) S 1 0 R 1 a b 0 c d Campnet dataset S 1 0 R H% =

11 E I index Krackhardt & Stern Number external ties minus number of internal ties as a proportion of allties S 1 0 Negative values d R 1 a b b a EI = indicated greater 0 c d b + a homophily Campnet S 1 0 R EI = 0.667

12 Point bi serial correlation (pbsc) approach Take into account non choices as well: S 1 0 R 1 a b 0 c d r(r,s) = Campnet dataset: S 1 0 R S 1 0 R r(r,s) = 0.33 r(r,s) = 0.00 H% = 0.83 H% = 0.83

13 Correlation The pbsc measure is the same as a QAP correlation of the two dyadic variables R and S r(r,s) = f f f f f f f fmmmmmmmmmm f f f f f f f fmmmmmmmmmm f f f f f f f f f f f f f f f f m m m m m m m m m m m m m m m m m m m m R S

14 Concept Egonet Homophily To what extent an ego s alters are like ego on a given attribute Approach Construct relational contingency table for each node Measures Pct homophilous (%H) = 0.67 E I index = PBSC = 0.24 HOLLY S 1 0 R

15 Density Tables Concept Number of ties within and between groups Called density when expressed as a function of the number possible BHS CCG DCL ES HEW IS MS SRG STAT TAS N BHS CCG DCL ES HEW IS MS SRG STAT TAS BHS BHS CCG CCG DCL DCL ES ES HEW HEW IS IS MS SRG STAT TAS N MS SRG STAT TAS Tie Frequencies Densities

16 BILL Adding Attributes to DON HARRY CENTRALITY MICHAEL HOLLY LEE STEVE GERY PAM PAT JENNI BRAZEY RUSS JOHN PAULINE CAROL BERT ANN

17 Individual Level E I Index Degree centrality Intern Extern Total E-I HOLLY BRAZEY CAROL PAM PAT JENNIE We can partition degree centrality into inward and 7 PAULINE outward 8 ANN components 9 MICHAEL BILL Can we similarly 11 LEE DON partition other 13 JOHN centrality 14 HARRY measures the 15 GERY STEVE same way? 17 BERT RUSS

18 EI Degree Centrality Degree Intern Extern centrality Total E-I HOLLY BRAZEY CAROL PAM PAT We can partition degree centrality into inward and 6 JENNIE outward 7 PAULINE ANN components 9 MICHAEL Can we similarly 10 BILL LEE partition other 12 DON centrality 13 JOHN measures the 14 HARRY GERY same way? 16 STEVE BERT RUSS

19 EI Eigenvector Internal External Total HOLLY Av = λv 2 BRAZEY CAROL PAM PAT JENNIE PAULINE v i = λ 1 So i s score is sum 8 ANN of scores of those 9 MICHAEL BILL adjacent to him 11 LEE DON Easily partitioned into groups 13 JOHN HARRY GERY STEVE v = ig λ aij v j j g 17 BERT RUSS j a ij v j

20 Closeness Centrality Internal External Total HOLLY Separately sum 2 BRAZEY CAROL distances to group 4 PAM insiders and 5 PAT JENNIE outsiders 7 PAULINE ANN Call this EI Closeness 9 MICHAEL BILL LEE DON JOHN HARRY GERY STEVE BERT RUSS

21 EI Betweenness Same Different Total HOLLY BRAZEY CAROL PAM PAT JENNIE PAULINE ANN MICHAEL BILL LEE DON JOHN HARRY GERY STEVE BERT RUSS A node can be along the shortest path between nodes that belong to the samegroup aseach other (broker A), or to different groups (broker B) Broker A Broker B

22 Extending betweenness ala Gould & Fernandez brokerage Coordinator Liaison Gatekeeper Representative Consultant

23 Brokerage Roles a b c Broker Gould & Fernandez Broker is middle node of directed triad What if nodes belong to different organizations? 2006 Steve Borgatti

24 B Gould & Fernandez Brokerage Roles A C Coordinator B B A C A C Representative Gatekeeper B B A C A C Liaison Consultant 2006 Steve Borgatti

25 Example Coord Gate Rep Cons Liaisi Ttl Total JB TB MC CC BD TD PD JF KG SM BS AS JT PW CW TW Total Steve Borgatti

26 Role Profiles Observed 50 JB BD TD TB MC CC BD TD PD 25 JF PW KG JB KG SM BS AS 5 0 TB CW Coordinator Gatekeeper Representative Consultant Liaison 2006 Steve Borgatti JT PW CW TW