Social Network Analysis

Transcription

1 Social Network Analysis Basic Concepts, Methods & Theory University of Cologne Johannes Putzke

2 Agenda Introduction Basic Concepts Mathematical Notation Network Statistics 2

3 Textbooks Hanneman & Riddle (2005) Introduction to Social Network Methods, available at Wasserman & Faust (994): Social Network Analysis Methods and Applications, Cambridge: Cambridge University Press. 3

4 Introduction 4

5 Basic Concepts What is a network? University of Cologne Johannes Putzke 5

6 What is a Network? Actors / nodes / vertices / points Ties / edges / arcs / lines / links 6

7 What is a Network? Actors / nodes / vertices / points Computers / Telephones Persons / Employees Companies / Business Units Articles / Books Can have properties (attributes) Ties / edges / arcs / lines / links 7

8 What is a Network? Actors / nodes / vertices / points Ties / edges / arcs / lines / links connect pair of actors types of social relations friendship acquaintance kinship advice hindrance sex allow different kind of flows messages money diseases

9 What is a Social Network? - Relations among People Rob Paul Steve Mike John Kai Stanley Lee Kim Peter Patrick George Ken Juan Johannes Homer 9

10 What is a Network? - Relations among Institutions 23% as institutions 6% 22% 5% owned by, have partnership / joint venture 2% 0% 5% 8% 72% 00% purchases from, sells to competes with, supports 7% 32% 9% 6% 27% 00% 6% 4% 7% 9% 3% through stakeholders board interlocks Previously worked for 42% 5% Image by MIT OpenCourseWare. 0

11 Why study social networks?

12 Example 2) Homophily Theory I I Male Female Male Female I Birds of a feather flock together See McPherson, Smith-Lovin & Cook (200) > > age / gender network 2

13 Managerial Relevance Social Network Shaneeka Stock Juan Jose Mitch Nichole Burke Callie Andy Ashley Bill Ashton Sean Cody Alisha Ben Brandy Pamela Jody Ewelina 3 Image by MIT OpenCourseWare.

14 vs. Organigram Exploration & Production Exploration Cody Senior Vice President Burke Drilling Ben Production Stock Shaneeka Stock G&G Petrophysical Ashley Production Shaneeka Reservoir Juan Juan Jose Mitch Andy Ewelina Jose Bill Brandy Mitch Nichole Burke Ashton Pamela Jody Sean Ashley Bill Callie Ashton Andy Sean Cody Nichole Alisha Callie Brandy Pamela Alisha Jody Ben Ewelina Image by MIT OpenCourseWare. 4 Source:

15 SNA A Recent Trend in Social Sciences Research Keyword search for social + network in 4 literature databases Abstracts Titles YEAR Source: Knoke, David (2007) Introduction to Social Network Analysis 5

16 SNA A Recent Trend in IS Research Artikelanzahl EBSCO ACM ScienceDirect Jahr 6

17 How to analyze Social Networks? 7

18 Example: Centrality Measures Who is the most prominent? Who knows the most actors? (Degree Centrality) Who has the shortest distance to the other actors? Who controls knowledge flows?... 8

19 Example: Centrality Measures Who is the most prominent? Who knows the most actors? Who has the shortest distance to the other actors? (Closesness Centrality) Who controls knowledge flows?... 9

20 Example: Centrality Measures Who is the most prominent? Who knows the most actors Who has the shortest distance to the other actors? Who controls knowledge flows?... (Betweenness Centrality) 20

21 Basic Concepts 2

22 Dyads, Triads and Relations actor dyad triad friendship kinship relation: collection of specific ties among members of a group 22

23 Strength of a Tie Ken, male, 34 Anna, female, Social network finite set of actors and relation(s) defined on them depicted in graph/ sociogram labeled graph Strength of a Tie dichotomous vs. valued depicted in valued graph or signed graph (+/-) 23

24 Strength of a Tie adjacent node to/from incident node to + - Strength of a Tie nondirectional vs. directional depicted in directed graphs (digraphs) nodes connected by arcs 3 isomorphism classes null dyad mutual / reciprocal / symmetrical dyad asymmetric / antisymmetric dyad converse of a digraph reverse direction of all arcs 24

25 Walks, Trails, Paths (Directed) Walk (W) sequence of nodes and lines starting and ending with (different) nodes (called origin and terminus) Nodes and lines can be included more than once Inverse of a (directed) walk (W - ) Walk in opposite order Length of a walk How many lines occur in the walk? (same line counts double, in weighted graphs add line weights) (Directed) Trail Is a walk in which all lines are distinct (Directed) Path Walk in which all nodes and all lines are distinct Every path is a trail and every trail is a walk 25

26 Walks, Trails and Paths - Repetition l3 l l2 l4 l6 l5 l7 W = n l n2 l2 n3 l4 n5 l6 n6 n n3 W = n l n2 l2 n3 l4 n5 l4 n3 W = n l n2 l2 n3 l4 n5 l5 n4 l3 n3 Path Walk Trail origin terminus University of Cologne Johannes Putzke 26

27 Reachability, Distances and Diameter Reachability If there is a path between nodes n i and n j Geodesic Shortest path between two nodes (Geodesic) Distance d(i,j) Length of Geodesic (also called degrees of separation ) 27

28 Mathematical Notation and Fundamentals 28

29 Three different notational schemes. Graph theoretic 2. Sociometric 3. Algebraic 29

30 . Graph Theoretic Notation N Actors {n, n 2,, n g } n n j there is a tie between the ordered pair <n i, n j > n n j there is no tie (n i, n j ) nondirectional relation <n i, n j > directional relation g(g-) number of ordered pairs in <n i, n j > directional network g(g-)/2 number of ordered pairs in nondirectional network L collection of ordered pairs with ties {l, l 2,, l g } G graph descriped by sets (N, L) Simple graph has no reflexive ties, loops 30

31 2. Sociometric Notation - From Graphs to (Adjacency/Socio)-Matrices I II III V IV VI I 2 3 II 4 III 4 V IV 4 VI Binary, undirected I II III IV V VI I - II - symmetrical III - IV - V - VI - Valued, directed I II III IV V VI I II III IV V VI

32 2. Sociometric Notation X g g sociomatrix on a single relation g g R super-sociomatrix on R relations X R sociomatrix on relation R X ij(r) value of tie from n i to n i (on relation χ r ) where i j 32

33 2. Sociometric Notation From Matrices to Adjacency Lists and Arc Lists Arc List I II III IV V VI I - II - III - IV - V - VI - Adjacency List I II II I III III II IV V IV III V VI V III V VI VI IV V I II II I II III III II III IV III V IV III IV V IV VI V III V IV V VI VI IV VI IV University of Cologne Johannes Putzke 33

34 Network Statistics 34

35 Different Levels of Analysis Actor-Level Dyad-Level Triad-Level Subset-level (cliques / subgraphs) Group (i.e. global) level 35

36 Measures at the Actor-Level: Measures of Prominence: Centrality and Prestige 36

37 Degree Centrality Who knows the most actors? (Degree Centrality) Who has the shortest distance to the other actors? Who controls knowledge flows?... 37

38 Degree Centrality I I I II III V IV VI II III IV V VI I II III IV V Indegree d I (n i ) Popularity, status, deference, degree prestige Outdegree d O (n i ) Expansiveness g CDO ni do( ni ) xij xi Total degree 2 x number of edges VI Marginals of adjacency matrix ( ) g C n d n x x DI i I i ji i j j 38

39 Degree Centrality II Interpretation: opportunity to (be) influence(d) Classification of Nodes Isolates d I (n i ) = d O (n i ) = 0 Transmitters d I (n i ) = 0 and d O (n i ) > 0 Receivers d I (n i ) > 0 and d O (n i ) = 0 Carriers / Ordinaries d I (n i ) > 0 and d O (n i ) > 0 Standardization of C D to allow comparison across networks of different sizes: divide by ist maximum value VI V II I C III ' D VII ( n) i IV d n i g 39

40 Closeness Centrality Who knows the most actors? Who has the shortest distance to the other actors? (Closesness Centrality) Who controls knowledge flows?... 40

41 Closeness Centrality I I II III IV V VI I II II III V III IV IV VI V VI Index of expected arrival time C ( n) C i g j Reciprocal of marginals of geodesic distance matrix Standardize by multiplying (g-) Problem: Only defined for connected graphs d n, n i j 4

42 Proximity Prestige I i/ (g-) P number of actors in the influence domain of n i normed by maximum possible number of actors in influence domain Σd(n j,n i )/ I i ( n) P i g j / ( g) average distance these actors are to n i I i d n, n / I j i i

43 Eccentricity / Association Number Largest geodesic distance between a node and any other node max j d(i,j)

44 Betweenness Centrality Who knows the most actors? 2 Who has the shortest distance to the other actors? Who controls knowledge flows? (Betweenness Centrality)

45 Betweenness Centrality How many geodesic linkings between two actors j and k contain actor i? g jk (n i )/g jk probability that distinct actor n i involved in communication between two actors n j and n k C B ( n) i jk standardized by dividing through (g-)(g-2)/2 g g jk jk n i

46 Several other Centrality Measures beyond the scope of this lecture Status or Rank Prestige, Eigenvector Centrality also reflects status or prestige of people whom actor is linked to Appropriate to identify hubs (actors adjacent to many peripheral nodes) and bridges (actors adjacent to few central actors) attention: more common, different meaning of bridge!!! Information Centrality see Wasserman & Faust (994), p. 92 ff. Random Walk Centrality see Newman (2005) 46

47 Condor Betweenness Centrality 47

48 (Actor) Contribution Index messa g essen _ t messa g esreceived _ messa g essen _ t messa g esreceived _ Sender (+) links to external networks Connector Gatekeeper coordinates and organizes tasks Contribution index Communicator Ambassador Creator Guru Collaborator Expediter Contribution frequency Receiver (-) Knowledge Expert serves as the ultimate source of explicit knowledge Maven provides the overall vision and guidance Salesman 48

49 Measures at the Group-(Global-)Level and Subgroup-Level 49

50 Diameter of a Graph and Average Geodesic Distance Diameter 2 Largest geodesic distance between any pair of nodes Average Geodesic Distance How fast can information get transmitted?

51 Density Proportion of ties in a graph High density (44%) Low density (4%) 5

52 Density III IV I II V VI L L gg ( -) / 2 g 2 In undirected graph: Proportion of ties I 2 3 II 4 III g V 5 2 g i j IV x ij gg ( -) VI In valued directed graph: Average strength of the arcs

53 Group Centralization I How equal are the individual actors centrality values? C A (n i* ) actor centrality index C A (n * ) max i C A (n i* ) g * CA n CA n i sum of difference between largest value i and observed values General centralization g index: * C A n CA n i i CA g * max CA n CA ni i 53

54 Group Centralization II C D g i * C n C n D D i ( g)( g2) C C g i C n C n ' * ' C C i ( g )( g 2) (2g 3) CB g g * ' * ' C n C n C n C n B B i B B i i i 2 ( g ) ( g 2) ( g ) 54

55 Condor Group Centralization 55

56 Subgroup Cohesion average strength of ties within the subgroup divided by average strength of ties that are from subgroup members to outsiders > ties in subgroup are stronger in g s in s ( g ) s jn s jn g ( g g ) s s s x x ij ij s I 2 II 3 4 III 4 V IV 4 VI 56

57 Connectivity of Graphs and Cohesive Subgroups 57

58 Connectivity of Graphs 58

59 Connected Graphs, Components, Cutpoints and Bridges Connectedness A graph is connected if there is a path between every pair of nodes Components Connected subgraphs in a graph Connected graph has component Two disconnected graphs are one social network!!! 59

60 Connected Graphs, Components, Cutpoints and Bridges n n n n n n2 n2 n2 n2 n5 n2 n3 n3 n3 n3 n6 n3 n4 n4 n4 n4 n4 Connectivity of pairs of nodes and graphs Weakly connected Joined by semipath Unilaterally connected Path from n j to n j or from n j to n j Strongly connected Path from n j to n j and from n j to n j Path may contain different nodes Recursively Connected Nodes are strongly connected and both paths use the same nodes and arcs in reverse order 60

61 Connected Graphs, Components, Cutpoints and Bridges Cutpoints number of components in the graph that contain node n j is fewer than number of components in subgraphs that results from deleting n j from the graph Cutsets (of size k) k-node cut Bridges / line cuts Number of components that contain line l k

62 Node- and Line Connectivity How vulnerable is a graph to removal of nodes or lines? Point connectivity / Node connectivity Minimum number of k for which the graph has a k- node cut For any value <k the graph is k-node-connected Line connectivity / Edge connectivity Minimum number λ for which for which graph has a λ-line cut 62

63 Cohesive Subgroups 63

64 Cohesive Subgroups, (n-)cliques, n-clans, n-clubs, k- Plexes, k-cores Cohesive Subgroup Subset of actors among there are relatively strong, direct, intense, frequent or positive ties Complete Graph All nodes are adjacent Clique Maximal complete subgraph of three or more nodes Cliques can overlap {, 2, 3} {, 3, 4} {2, 3, 5, 6} 64

65 Cohesive Subgroups, (n-)cliques, n-clans, n-clubs, k- Plexes, k-cores n-clique maximal subgraph in which d(i,j) n for all n i, n j 2: cliques: {2, 3, 4, 5, 6} and {, 2, 3, 4, 5} intermediaries in geodesics do not have to be n-clique members themselves! n-clan n-clique in which the d(i,j) n for the subgraph of all nodes in the n-clique 2-clan: {2, 3, 4, 5, 6} n-club maximal subgraph of diameter n 2-clubs: {, 2, 3, 4} ; {, 2, 3, 5} and {2, 3, 4, 5, 6} 65

66 Cohesive Subgroups, (n-)cliques, n-clans, n-clubs, k- Plexes, k-cores Problem: vulnerability of n-cliques k-plexes maximal subgraph in which each node is adjacent to not fewer than g s -k nodes ( maximal : no other nodes in subgraph that also have d s (i) (g s -k) ] k-cores subgraph in which each node is adjacent to at least k other nodes in the subgraph 66

67 Analyzing Affiliation Networks 67

68 Affiliation Matrix, Bipartite Graph and Hypergraph, Rate of Participation, Size of Events Two-mode network / affiliation network / membership network / hypernetwork nodes can be partitioned in two subsets N (for example g persons) M (for example h clubs) depicted in Bipartite Graph lines between nodes belonging to different subsets Peter Pat Jack Ann Football Ballet Kim Tennis Mary 68

69 Affiliation Matrix, Bipartite Graph and Hypergraph, Rate of Participation, Size of Events Affiliation Matrix (Incidence Matrix) Connections among members of one of the modes based on linkages established through second mode g actors, h events A= {a ij } (g h) Actor Peter Pat Jack Ann Kim Mary Football Event Ballet Tennis rate of participation size of event

70 Affiliation Matrix, Bipartite Graph and Hypergraph, Rate of Participation, Size of Events Sociomatrix [ (g+h) (g+h) ] Peter Pat Jack Ann Kim Mary Football Ballet Tennis Peter Pat Jack Ann Kim Mary Football Ballet Tennis

71 Affiliation Matrix, Bipartite Graph and Hypergraph, Rate of Participation, Size of Events Homogenous pairs and heterogenous pairs X r N (g g), X r M (h h), X r N,M (g h), X r N,M (h g) One-mode sociomatrices X N [and X M ] rows, colums: actors [events]; x ij : co-membership [number of actors in both events] (main diagonal meaningful, e.g. total events attended by an actor) Peter Pat Jack Ann Kim Mary Peter Pat Jack Ann Kim Mary Peter Pat Jack Ann Kim Mary Football Ballet Tennis 7

72 Affiliation Matrix, Bipartite Graph and Hypergraph, Rate of Participation, Size of Events Event Overlap / Interlocking Matrix Football Ballet Tennis Football Ballet 0 3 Tennis 2 4 Peter Pat Jack Ann Kim Football Ballet Tennis Mary 72

73 Cohesive Subsets of Actors or Events clique at level c (cf. also k-plexes, n-cliques etc.) subgraph in which all pairs of events share at least c members connected at level q subset in which all actors in the path are comembers of at least q+ events 73

74 HOLDOUT I 74

75 When is Which Centrality Measure Appropriate? Source: Borgatti, Stephen P. (2005) Centrality and Network Flow, Social Networks 27, p

76 Assumptions of Centrality Measures Which things flow through a network and how do they flow? Transfer Serial Parallel Walks Money exchange Emotional support Trails Used Book Gossip Paths Mooch Viral infection Geodesics Package Delivery Mitotic reproduction Attitude influencing broadcast Internet nameserver <no process> Source: Borgatti, Stephen P. (2005) Centrality and Network Flow, Social Networks 27, p

77 Assumptions of Centrality Measures Example: Betweenness Centrality Information travels along the shortest route Probability of Transfer all geodesics being Serial chosen is Parallel equal Walks Random Walk Betweenness? Closeness Degree Eigenvector Trails?? Paths?? Closeness Degree Closeness Degree Geodesics Closeness Betweenness Closeness? 77

78 Adequacy of Centrality Measures Transfer Serial Parallel Walks Money exchange Emotional support Trails Used Book Gossip Paths Mooch Viral infection Geodesics Package Delivery Mitotic reproduction Attitude influencing broadcast Internet nameserver <no process> Source: Borgatti, Stephen P. (2005) Centrality and Network Flow, Social Networks 27, p

79 How to Calculate Geodesic Distance Matrices? 79

80 From Adjacency Matrices to (Geodesic) Distance Matrices I (Reachability) Repetition: Matrix Multiplication XY = Z 3 X Y Z g h h k g k 3 23 z ij = h n x in y nj 80

81 From Adjacency Matrices to (Geodesic) Distance Matrices II (Reachability) I II III IV V VI I II III IV V VI I I I II V VI III IV II III IV V VI X II III IV V VI Power Matrix: Multiplying adjacency matrices I II III IV V VI x ik x kj = only if lines (n i,n k ) and (n k,n j ) are present, i.e. X [2,3,4] counts the number of walks (n i n k n j ) of length [2,3,4] between nodes n i and n j I II III IV V VI

82 From Adjacency Matrices to (Geodesic) Distance Matrices II (Reachability) x ij >0? two nodes can be connected by paths of length (g-) Calculate X [Σ] = X + X 2 + X X g- X [Σ] shows total number of walks from n i to n i I II III V IV VI X 2 I II III IV V VI I II III IV V VI

83 From Graphs to (Geodesic Distance)-Matrices (Reachability) Geodesic Distance observer power matrices first power p for which the (i,j) element is non-zero gives the shortest path d(i,j) = min p x ij [p] > 0 III IV X I II III IV V VI I II III IV V VI I II V X 2 I II III IV V VI I II III IV V VI VI 83

84 From Graphs to (Geodesic Distance)-Matrices (Reachability) Geodesic Distance III IV I II V VI I II III IV V VI Binary, undirected I II III IV V VI

85 MIT OpenCourseWare Workshop in IT: Collaborative Innovation Networks Fall 20 For information about citing these materials or our Terms of Use, visit: