Introduction to Social Network Analysis PSU Quantitative Methods Seminar, June 15

Transcription

1 Introduction to Social Network Analysis PSU Quantitative Methods Seminar, June 15 Jeffrey A. Smith University of Nebraska-Lincoln Department of Sociology Course Website

2 The Macro Structure of the Course n Introduction to network analysis (~ 3 hours) n The what, why and how of network analysis n Lunch (1 hour) n Eating n Doing network analysis in R (~4 hours) n R basics n Dealing with network data in R n Plotting and basic measurement

3 Goals of This Course 1. Understand the unique features of network data 2. Manipulate network data in R 3. Create an attractive network plot in R 4. Understand the potential and limitations of a network approach

4 The What of Network Analysis Q) What is a network? A) Network=set of links between objects A F B E D C

5 Examples of Links Similarities Social Relations Interactions Flows Location e.g., Membership e.g., Attribute e.g., Kinship e.g., Other role e.g., Affective e.g., Cognitive e.g., e.g., Sex with e.g., Information Same spatial and temporal space Same clubs Same events etc. Same gender Same attitude etc. Mother of Sibling of Friend of Boss of Student of Competitor of Likes Hates etc. Knows Knows about Sees as happy etc. Talked to Advice to Helped Harmed etc. Beliefs Personnel Resources etc. Credit: Borgatti et al. 2011

6 Clarifying Terminology n Different words for the same thing n Objects: nodes, actors, vertices n Links: edges, ties, connections, relations A F B E D C

7 Example Networks Slide Credit: James Moody

8

9 Ties between Female Sex Workers in Liuzhou, China

10 Coauthorship between scholars in physics literature Slide Credit: Porter et al. 2009

11 The What of Network Analysis n What is network analysis? n Set of relational methods for studying the connections between actors n Based on computational methods and graphical imagery n Interested in the ties between actors rather than the actors themselves n Formally capture the complex pattern of connections amongst actors using empirical data

12 The Why of Network Analysis n 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. Blau Exchange and Power in Social Life, 1964

13 The Why of Network Analysis n 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. It is a little like a biologist putting his experimental animals through a hamburger machine and looking at every hundredth cell through a microscope; anatomy and physiology get lost, structure and function disappear If our aim is to understand behavior rather than simply record it, we want to know about primary groups, neighborhoods, organizations, social circles, and communities; about interaction, communication, role expectations, and social control. Barton 1968, quoted from Freeman 2004

14 The Why of Network Analysis n Why study social networks? n Because individuals do not live in a vacuum n Network structure matters Even if individuals are unaware of this! n Demographic groups are not always social groups

15 Example: Individuals in a Social Vacuum Gerald Ursula Smoke Does not smoke Larry

16 Example: Individuals not in a Social Vacuum Gerald Ursula Smoke Does not smoke Larry

17 Example: Network Structure Matters Slide Credit: James Moody

18 Example: Network Structure Matters Slide Credit: James Moody

19 Example: Putting Attributes into Context Setting 1: race as social group White Black Hispanic Colored by race Circled by cohesive group

20 Example: Putting Attributes into Context Setting 2: race not as social group White Black Hispanic Colored by race Circled by cohesive group

21 The How of Network Analysis n Methods of Collection n Observational n Survey n Archival n Digital records Internet Sensor data Cell phone records

22 Assumption when collecting data n Boundaries n Captured meaningful relations on population of interest n Missing data

23 Assumption when collecting data n Generally assume we have no missing data n But this is almost never true! n What happens when we do have missing data?

24 Representing Networks n Need formal, systematic means of analyzing our network data n Start with representing the network data n Need both graphical and numerical representations

25 Graphs and Matrices n Graphs and matrices offer two different ways of summarizing the same data n Graphs n Advantage: offer an intuitive representation of network n Disadvantage: not analytically tractable n Adjacency Matrix n Advantage: can calculate measure of interest from matrix n Disadvantage: inefficient way of storing data

26 Not all figures are equally good id9 id8 id7 id6 id5 id10 id4 id11 id3 id12 id2 id13 id1 id14 id24 id15 id23 id16 id22 id17 id18 id19 id20 id21

27 Not all figures are equally good id12 id11 id15 id20 id4 id18 id13 id24 id14 id10 id21 id19 id17 id6 id7 id3 id16 id23 id8 id5 id1 id22 id2 id9

28 Not all figures are equally good

29 Graphical Representations n Different types of networks yield (slightly) different representations Undirected versus directed Binary versus valued Single relation versus multiplex One mode versus two-mode

30 Undirected or Directed (0/1 tie) Undirected, Binary Directed, Binary

31 Binary or Valued (undirected) Undirected, Binary Undirected,Valued

32 Binary or Valued (directed) Directed, Binary Directed, Valued

33 Binary or Signed (directed) Directed, Binary Directed, Signed Dislike Like

34 Single Relation or Multiplex (directed) Directed, Binary, single relation Directed, Binary, Muliplex Friendship Get Advice From

35 One Mode or Two Mode Alice Two-mode network Chess Club One-mode Projection Chess Club Bob Band Band Carl Person Club One-mode Projection Alice Bob Carl Person Club

36 Matrix Representation n Define a matrix X ij such that the element i,j represents the existence/non-existence of a tie between i and j n Row i corresponds to ties sent by actor i n Column j corresponds to ties received by actor j

37 Alice Beth Carl Diana Ed Fred Alice Beth Carl Diana Ed Fred Beth Alice Carl Diana Fred Ed

38 Alternative Representations: Edgelist Beth Sender Receiver Alice Carl Diana Alice Beth Beth Beth Alice Carl Fred Ed Beth Carl Diana Diana Diana Carl Ed Carl

39 Alternative Representation: Nomination List Beth Alice Carl Diana Fred Ed ID Nom1 Nom2 Nom3 Alice Beth.. Beth Alice Carl Diana Carl Diana.. Diana Carl.. Ed Carl.. Fred...

40 Running through an example n Let s say we collect the following information: n Tim likes Sally, Harry, Mary n Joe likes Harry and Mary n Sally likes Tim and Harry n Harry likes Tim, Joe, and Sally n Mary likes Joe n What does the matrix look like for this network?

41 Tim likes Sally, Harry, Mary Joe likes Harry and Mary Sally likes Tim and Harry Harry likes Tim, Joe, and Sally Mary likes Joe Tim Joe Sally Harry Marry Tim Joe Sally Harry Marry

42 What would the edgelist look like? Tim likes Sally, Harry, Mary Joe likes Harry and Mary Sally likes Tim and Harry Harry likes Tim, Joe, and Sally Mary likes Joe Sender Receiver

43 Nomination List ID Nom1 Nom2 Nom3 Tim Sally Harry Mary Joe Harry Mary Sally Tim Harry. Harry Tim Joe Sally Mary Joe.. Edgelist Sender Receiver Tim Sally Tim Mary Tim Harry Joe Harry Joe Marry Sally Tim Sally Harry Harry Tim Harry Joe Harry Sally Mary Joe Sally Tim Mary Harry Joe Matrix Tim Joe Sally Harry Marry Tim NA Joe 0 NA Sally 1 0 NA 1 0 Harry NA 0 Marry NA

44 Analyzing a Network n Take the underlying matrix and calculate measures of interest n Examples: centrality, cohesion, group structure, roles/position, hierarchy, dynamics, diffusion n Measure becomes thing to predict or used as predictor of other variable of interest n Could also try to model the network (predict ties between actors) n Local tendencies like transitivity and reciprocity n Question of aggregation?

45 Measuring Network Structure and Diffusion Potential n Network structure affects diffusion n Networks measures related to diffusion n Density n Walks n Reachability n Components n Distance

46 Network Structure and Diffusion n Higher density=more diffusion potential n Higher reachability=more diffusion potential Less grouped =more global diffusion potential n Shorter paths=more likely diffusion n More paths=more likely diffusion Especially if independent paths

47 Some Simple Measures n Density: total number of edges/total possible number of edges!!! =!!"!!! degree!!! (!! 1 ) n Where X ij =the network; n=number of people in network

48 Density: 0/90=0

49 Density: 39/90=.433

50 Density: 90/90=1

51 Walks n Focus is on indirect connections between actors n What makes a network a social system n Important for diffusion n Walks: sequence of nodes and edges that connect i to j n can go over same node more than once n can over same edge more than once n Calculate if i and j are connected (indirectly) and how far

52 Sally Tim Mary Harry Joe

53 Walks of length 2, starting from Tim Tim Sally Harry Mary Sally Harry Tim Tim Joe Sally Joe Tim Mary Harry Joe

54 Walks continued n Calculate number of walks between i and j of length p by multiplying adjacency matrix by itself n X 2 tells us how many walks of length 2 there are between all i, j pairs n X p tells us how many walks of length p there are between all i, j pairs

55 Matrix Multiplication in One Slide

56 Reachability and Components n Node i and j are said to be reachable if there is at least one sequence of edges that connects them n Minimal condition for global diffusion to occur n Component is a set of nodes where everyone can reach everyone else

57 Lower Diffusion Potential Source: Potterat, Muth, Rothenberg, et. al Sex. Trans. Infect 78:

58 Higher Diffusion Potential Source: Potterat, Muth, Rothenberg, et. al Sex. Trans. Infect 78:

59 Distance n Distance (D ij ) is the shortest path between i and j n With paths, not allowed to go over same node more than once n Not allowed to go over same edge more than once

60 Finding Distance between Tim and Joe? Sally Tim Harry Mary Joe

61 Finding Distance between Tim and Joe Using Breadth First Search (BFS) Sally Tim Tim Harry Mary Sally Harry Mary Joe Harry Tim Tim Joe Sally Joe

62 Distance Matrix Sally Tim Mary Tim Joe Sally Harry Marry Tim NA Joe 2 NA Sally 1 2 NA 1 2 Harry NA 2 Marry NA Harry Joe

63 Network Structure and Diffusion n Higher density=more diffusion potential n Higher reachability=more diffusion potential Less grouped =more global diffusion potential n Shorter paths=more likely diffusion n More paths=more likely diffusion Especially if independent paths