Social Network Analysis

Transcription

1 Social Network Analysis Basic Concepts, Methods & Theory COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie:

2 COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie:

3 Textbooks Gloor (6): Swarm Creativity Competitive Advantage Through Collaborative Innovation Networks, Oxford: Oxford University Press. Gloor & Cooper (7): Coolhunting Chasing Down the Next Big Thing, New York: Mcgraw-Hill Professional. Hanneman & Riddle (5) Introduction to Social Network Methods, available at Wasserman & Faust (994): Social Network Analysis Methods and Applications, Cambridge: Cambridge University Press. COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 3

4 Introduction COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 4

5 Basic Concepts What is a network? COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 5

6 What is a Network? Actors / nodes / vertices / points Ties / edges / arcs / lines / links COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Motivation Folie: 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 COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Motivation Folie: 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 COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Motivation Folie: 8

9 What is a Social Network? - Relations among People Rob Steve John Paul Kai Stanley Lee Kim Patrick Peter Juan Johannes Homer George Ken COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Überblick Folie: 9

10 What is a Network? - Relations among Institutions % 7 % 6 % 3 % 9 % 4 % 3 % % 6 % % 5 % 7 % 8 % 7 % 6% % 7% 5 % % 4% 5% 9% 3% as institutions owned by, have partnership / joint venture purchases from, sells to competes with, supports through stakeholders board interlocks Previously worked for COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Überblick Folie:

11 Why study social networks? COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie:

12 Example ) Social Capital Actor s embeddedness in a social network determines opportunities and constraints an actor encounters network social capital COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie:

13 Example) HomophilyTheory I I I Male Female Male 3 95 Female Birds of a feather flock together See McPherson, Smith- Lovin & Cook () > > age / gender network COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 3

14 COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 4

15 Managerial Relevance Social Network Source: COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Vertiefung Folie: 5

16 vs. Organigram COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Vertiefung Source: Folie: 6

17 SNA A Recent Trend in Social Sciences Research Keyword search for social + network in 4 literature databases 8 6 Abstracts 4 Titles YEAR Source: Knoke, David (7) Introduction to Social Network Analysis COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 7

18 SNA A Recent Trend in IS Research Artikelanzahl EBSCO ACM ScienceDirect < Jahr COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 8

19 How to analyze Social Networks? COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 9

20 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?... COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Vertiefung Folie:

21 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?... COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Vertiefung Folie:

22 Example: Centrality Measures Who is the most prominent? Who knows the most actors Who has the sthortest distance to the other actors? Who controls knowledge flows? (Betweenness Centrality)... COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Vertiefung Folie:

23 Basic Concepts COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 3

24 Dyads, Triads and Relations actor dyad triad friendship kinship relation: collection of specific ties among members of a group COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 4

25 Strength of a Tie Ken, male, 34 Anna, female, 7 5 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 (+/-) COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 5

26 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 COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 6

27 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 COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 7

28 Walks, Trails and Paths - Repetition l n3 l4 l3 l5 n4 n l n n5 l6 n6 l7 W = n l n l n3 l4 n5 l6 n6 n n3 W = n l n l n3 l4 n5 l4 n3 W = n l n l n3 l4 n5 l5 n4 l3 n3 Path origin terminus Walk Trail COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 8

29 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 ) n l n l n3 n5 l3 l4 l5 l6 n6 n4 l7 COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 9

30 Mathematical Notation and Fundamentals COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 3

31 Three different notational schemes. Graph theoretic. Sociometric 3. Algebraic COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 3

32 . Graph Theoretic Notation N Actors {n, n,, 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-)/ number of ordered pairs in nondirectional network L collection of ordered pairs with ties {l, l,, l g } G graph descriped by sets (N, L) Simple graph has no reflexive ties, loops COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 3

33 . Sociometric Notation - From Graphs to (Adjacency/Socio)-Matrices III IV II V VI I I Binary, undirected Valued, directed I II III IV V VI I II III IV V VI I - I II - symmetrical II 4 III - III IV - IV 5 V - V 3 VI - VI 4 3 II 4 III 4 V IV 4 VI COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 33

34 . 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 COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 34

35 . Sociometric Notation From Matrices to Adjacency Lists and Arc Lists 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 Arc List 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 COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 35

36 Network Statistics COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 36

37 Different Levels of Analysis Actor-Level Dyad-Level Triad-Level Subset-level (cliques / subgraphs) Group (i.e. global) level COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 37

38 Measures at the Actor-Level: Measures of Prominence: Centrality and Prestige COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 38

39 Degree Centrality Who knows the most actors? (Degree Centrality) Who has the shortest distance to the other actors? Who controls knowledge flows?... COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Vertiefung Folie: 39

40 Degree Centrality I I I II III IV V I II II III V III IV IV VI V VI 3 Indegree d I (n i ) Popularity, status, deference, degree prestige ( ) ( ) Outdegree d O (n i ) Expansiveness g CDO( ni) = do( ni) = xij = x i + Total degree x number of edges g C n = d n = x = x + DI i I i ji i j j VI 3 Marginals of adjacency matrix COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 4

41 Degree Centrality II Interpretation: opportunity to (be) influence(d) Classification of Nodes Isolates - d I (n i ) = d O (n i ) = Transmitters - d I (n i ) = and d O (n i ) > Receivers - d I (n i ) > and d O (n i ) = Carriers / Ordinaries - d I (n i ) > and d O (n i ) > II V III IV VI I VII Standardization of C D to allow comparison across networks of different sizes: divide by ist maximum value ( ) C ' D ( n ) = i d n i g COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 4

42 Closeness Centrality Who knows the most actors? Who has the shortest distance to the other actors? (Closesness Centrality) Who controls knowledge flows?... COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Vertiefung Folie: 4

43 Closeness Centrality I I II III IV V VI I II II - 3 III V III IV IV VI V VI Index of expected arrival time CC( ni) = g d n, n j= ( ) i j Reciprocal of marginals of geodesic distance matrix Standardize by multiplying (g-) Problem: Only defined for connected graphs COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 43

44 Proximity Prestige I i/ (g-) P ( n ) = P i g j= /( g ) ( ) 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 average distance these actors are to n i I i d n, n / I j i i COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 44

45 Eccentricity / Association Number Largest geodesic distance between a node and any other node max j d(i,j) COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 45

46 Betweenness Centrality Who knows the most actors? Who has the shortest distance to the other actors? Who controls knowledge flows? (Betweenness Centrality) COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Vertiefung Folie: 46

47 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 3 involved in communication between two actors n j and n k C B ( n ) i = j< k ( ) standardized by dividing through (g-)(g- )/ 9 g g jk jk n i COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 47

48 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. 9 ff. Random Walk Centrality - see Newman (5) COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 48

49 Condor Betweenness Centrality COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 49

50 (Actor) Contribution Index messages _ sent messages _ sent + messages _ received messages _ received 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 COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 5

51 Measures at the Group-(Global-)Level and Subgroup-Level COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 5

52 Diameter of a Graph and Average Geodesic Distance Diameter Largest geodesic distance between any pair of nodes Average Geodesic Distance How fast can information get transmitted? COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 5

53 Density Proportion of ties in a graph High density (44%) Low density (4%) COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 53

54 Density III IV I II V VI L L Δ= gg ( -)/ = g In undirected graph: Proportion of ties I 3 II Δ= 4 III g V g i= j= IV x ij gg ( -) VI In valued directed graph: Average strength of the arcs COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 54

55 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 * sum of difference between largest value CA( n ) CA( ni) i= and observed values General centralization index: C A = g i= max g i= ( * ) ( ) C n C n A A i ( * ) ( ) C n C n A A i COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 55

56 Group Centralization II C D = g i= * ( ) ( ) C n C n D D i ( g )( g ) C C = g i= ( ) ( ) C n C n ' * ' C C i [ ] ( g )( g ) (g 3) CB g g * ' * ' ( ) ( ) ( ) ( ) CB n CB n i CB n CB n i i= i= = = ( g ) ( g ) ( g ) COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 56

57 Condor Group Centralization COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 57

58 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 i N g s i N s ( g ) s j N s j N g ( g g ) s s s x x ij ij s I II 3 4 III 4 V 5 3 IV 4 VI COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 58

59 Connectivity of Graphs and Cohesive Subgroups COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 59

60 Connectivity of Graphs COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 6

61 Connected Graphs, Components, Cutpoints and Bridges Connectedness A graphisconnectedif 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!!! COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 6

62 n n n n n Connected Graphs, Components, Cutpoints and Bridges n n n n n5 n 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 an both paths use the same nodes and arcs in reverse order COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 6

63 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 COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 63

64 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-nodeconnected COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Line connectivity / Edge connectivity Minimum number λ for which for which graph has a λ-line cut Folie: 64

65 Cohesive Subgroups COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 65

66 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 {,, 3} {, 3, 4} {, 3, 5, 6} COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 66

67 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 : cliques: {, 3, 4, 5, 6} and {,, 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 -clan: {, 3, 4, 5, 6} n-club maximal subgraph of diameter n -clubs: {,, 3, 4} ; {,, 3, 5} and {, 3, 4, 5, 6} COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 67

68 Cohesive Subgroups, (n-)cliques, n-clans, n-clubs, k-plexes, k-cores Problem: vulnerability of n-cliques 3 4 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 COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 68

69 Analyzing Affiliation Networks COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 69

70 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 Peter lines between nodes belonging to different subsets Pat Jack Ann Kim Mary Football Ballet Tennis COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 7

71 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 COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 7

72 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 - COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 7

73 Affiliation Matrix, Bipartite Graph and Hypergraph, Rate of Participation, Size of Events Homogenous pairs and heterogenous pairs X rn (g g), X rm (h h), X N,M r (g h), X N,M r (h g) Peter Pat Jack Ann Kim Mary 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 Football Ballet Tennis COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 73

74 Affiliation Matrix, Bipartite Graph and Hypergraph, Rate of Participation, Size of Events Event Overlap / Interlocking Matrix Football Ballet Tennis Football 3 Ballet 3 Tennis 4 Peter Pat Jack Ann Kim Football Ballet Tennis Mary COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 74

75 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 co-members of at least q+ events COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 75

76 When is which Centrality Measure Appropriate? Source: Borgatti, Stephen P. (5) Centrality and Network Flow, Social Networks 7, p COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 76

77 Assumptions of Centrality Measures Which things flow through a network and how do they flow? Transfer Serial Parallel Walks Money exchange Emotional support Attitude influencing Trails Used Book Gossip broadcast Paths Mooch Viral infection Internet nameserver Geodesics Package Delivery Mitotic reproduction <no process> COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 77

78 Assumptions of Centrality Measures Example: Betweenness Centrality Information travels along the shortest route Probability of all geodesics being chosen is equal Transfer Serial Parallel Walks Random Walk Betweenness? Closeness Degree Eigenvector Trails?? Closeness Degree Paths?? Closeness Degree Geodesics Closeness Betweenness Closeness? COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 78

79 Adequacy of Centrality Measures Transfer Serial Parallel Walks Money exchange Emotional support Attitude influencing Trails Used Book Gossip broadcast Paths Mooch Viral infection Internet nameserver Geodesics Package Delivery Mitotic reproduction <no process> Source: Borgatti, Stephen P. (5) Centrality and Network Flow, Social Networks 7, p COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 79

80 How to Calculate Geodesic Distance Matrices? COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 8

81 From Adjacency Matrices to (Geodesic) Distance Matrices I (Reachability) Repetition: Matrix Multiplication XY = Z 3 h n= z ij = x in y nj X Y Z g h h k g k 3 3 COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 8

82 I II V VI From Adjacency Matrices to (Geodesic) Distance Matrices II (Reachability) III IV I II III IV V VI I II III IV V VI X Power Matrix: Multiplying adjacency matrices x ik x kj = only if lines (n i,n k ) and (n k,n j ) are present, i.e. X [,3,4] counts the number of walks (n i n k n j ) of length [,3,4] between nodes n i and n j I II III IV V VI I II III IV V VI I I II II III III 3 IV IV 3 V V 3 VI VI COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 8

83 From Adjacency Matrices to (Geodesic) Distance Matrices II (Reachability) x ij >? two nodes can be connected by paths of length (g-) Calculate X [Σ] = X + X + X X g- X [Σ] shows total number of walks from n i to n i X I II III IV V VI III IV I II III 3 I II V VI IV V VI 3 3 COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 83

84 From Graphs to (Geodesic Distance)- Matrices (Reachability) Geodesic Distance observer power matrices first power p for which the (i,j) element is nonzero gives the shortest path III d(i,j) = min p x [p] ij > IV I II X III IV V VI I I II II X III IV V V VI VI I I II II III III 3 IV IV 3 V V 3 VI VI COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 84

85 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 COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 85

86 Modelling the Co-Evolution of Network and Behavior COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 86

87 What is Science? What is Science? Proving Causalities Independent Variable Dependent Variable Example: Social Capital: network determines outcomes Homophily: actor attributes determine network Endogeneity problem of SNA-studies network can be dependent and independent variable at the same time Advanced statistical methodologies: Modelling the co-evolution of network and behaviour ( to be cont.) COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 87

88 Example 3) Co-Evolution of Network and Behavior study by Steglich, Snijders and West (6) 6 students of a school cohort in Glasgow (Scotland) three waves at intervals of one year, starting in 995 (when students were 3 years old) ending in 997 (when students were 5 years olds) friendship demographic variables; music taste, self-reported smoke and alcohol consumption COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 88

89 COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 89

90 Example 3) Co-Evolution of Network and Behavior Results I COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 9

91 Example 3) Co-Evolution of Network and Behavior Results II COIN-Seminar 8/9: Be a Bee! Pohligstr. ; 5969 Cologne (Köln), Germany 3..8 Folie: 9