Social Network Notation

Transcription

1 Social Network Notation Wasserman & Faust (1994) Chapters 3 & 4 [pp ] Marsden (1987) Core Discussion Networks of Americans Junesoo, Xiaohui & Stephen Monday, February 8th, 2010

2 Wasserman & Faust (1994) Chapter 3: Notation for Social Network Data o Graph Theoretic Notation o Sociometric Notation o Algebraic Notation

3 Mathematical Descriptions of Social Network Data Graph Theoretic Centrality, Prestige Models, Cohesive Subgroups, Dyadic & Triadic Methods Sociometric Study of Structural Equivalence & Blockmodels Algebraic Role & Positional Analysis, Relational Algebras

4 Graph Theoretic Notation Notation Description N = {n 1, n 2,,n g } Set N contains a total of g actors. N = {Lunix, Junesoo, Steve} Set N contains a total of 3 actors where n 1 = Lunix, n 2 = Junesoo, and n 3 = Steve < n i, n j > or < Lunix, Junesoo > An ordered pair of actors; n i relates to n j as Lunix relates to Junesoo; order matters n i n j or Lunix Junesoo n i has a tie to n j as Lunix has a directed tie to Junesoo (two nodes connected by a line) L = {l 1, l 2,, l L } A set of directed lines (L ) that includes lines l 1, l 2,, l L [L = total number of lines] G = (N, L ) A graph ( G ) consists of a set of nodes ( N ) and a set of lines (L ) < n i, n j > vs. ( n i, n j ) < > indicates an ordered pair whee order matters; ( ) indicates relations that are nondirectional, therefore order doesn t L = {l 1, l 2, l 3 } where l 1 = <Junesoo, Lunix>, l 2 = <Steve, Lunix>, l 3 = <Lunix, Junesoo> matter A set (L )of 3 directed lines (l) where Junesoo has a relation to Lunix, Steve has a relation to Lunix, and Lunix has a relation to Junesoo. Therefore, Junesoo Lunix or (Junesoo, Lunix), Steve Lunix.

5 Lunix Junesoo Steve Sociometric Notation Relational data are often presented in two-way matrices termed sociomatrices. The two dimensions of a sociomatrix are indexed by the sending actor (the rows) and the receiving actors (the columns). Thus, if we have a one-mode network, the sociomatrix will be square (p. 77). Lunix Junesoo 1-0 Steve 1 0 -

6 Sociometric Notation Notation Description N = {n 1, n 2,,n g } X X X ij Set N contains a total of g actors. Single-valued, directional relation in N The resultant sociomatrix The value of the tie from n i to n j on relation X X ijr The value of any tie from n i to n j on relation X r

7 Sociometric Notation Friendship at the Beginning of Year Allison Drew Eliot Keith Ross Sarah Allison Drew Eliot Keith Ross Sarah Friendship at End of Year Allison Drew Eliot Keith Ross Sarah Allison Drew Eliot Keith Ross Sarah Lives Near Allison Drew Eliot Keith Ross Sarah Allison Drew Eliot Keith Ross Sarah

8 Algebraic Notation In order to present algebraic methods and models for multiple relations it is useful to employ a notation that is different from, though consistent with, the sociometric and graph theoretic notations that we have just discussed (p. 84).

9 Two Sets of Actors (Two-Mode Networks) o The first actor is the sender, the second the receiver in both sets o They are designated differently in the two transactions: N is the first set of actors, and M is the second set o Actors (dyads, triads and the like) in both sets (N & M) can also interact in the other set of actors, thereby creating separate and distinct sociomatrices.

10 Two Sets of Actors (Two-Mode Networks) M Mr. Jones Ms. Smith Ms. Davis Mr. White Allison Drew N Eliot Keith Ross Sarah

11 Graph Theory Wasserman & Faust (1994) Chapter 4: Graphs & Matrices

12 Why Graph Theory? provides a vocabulary, which can be used to label and denote many social structural properties gives us mathematical operations and ideas with which many of these properties can be quantified and measured gives us the ability to prove theorems about graphs, and hence, about representation of social structure gives us a representation of a social network as a model of a social system

13 Graph A graph G = (N, L) consists of a set of nodes: N = {n 1, n 2,, n g }, and a set of lines, L = {l 1, l 2,, l m } between pairs of nodes. (i.e. l k = (n i, n j )). N = {1, 2, 3, 4, 5, 6} L ={(1,4) (2,4), (3,4), (4,5), (5,6)} Graph 2

14 Related Concepts A loop is a line between a node and itself. A simple graph has no loops and includes no more than one line between a pair of nodes. A multigraph allows more than one set of lines. A trivial graph has only one node and empty line set. An empty graph has at least one node and no lines. 1 Loop a b c Simple graph 1 2 Multigraph 3 a Trivial graph Empty graph

15 Related Concepts A hypergraph consists of a set of objects and a collection of subsets of objects, in which each object belongs to at least one subset, and no subset is empty. If any two nodes in a graph are adjacent, the graph is called complete graph or clique. Two nodes are adjacent if there is a line connects them. A node is incident with a line, and the line is incident with the node, if the line connects the nodes. B 1 a 2 a 3 B 2 B a l 1 b a 1 Hypergraph a Clique a, b are adjacent a is incident with l 1 l 1 is incident with a and b

16 Subgraph A graph G s (N s, L s ) is a subgraph of G(N, L) if N s N, and L s L. Node-generated subgraph: a subgraph with its line set L s includes all lines from L that are between pairs of nodes in N s. Line-generated subgraph: a subgraph with its node set N s includes all nodes from N that are incident with lines in L s. l 3 1 l l 4 l 2 3 N = {1, 2, 3, 4, 5} L = {l 1, l 2, l 3, l 4 } l 1 = (1, 2) l 3 = (1, 5) l 2 = (1, 3) l 4 = (3, 4) 4 l Subgraph N s = {1, 3, 4} L s = {l 2 } 4 l 4 l 2 3 Node-gen. subgraph by nodes 1, 3, 4 N s = {1, 3, 4} L s = {l 2, l 4 } 1 l 3 l Line-gen. subgraph by lines l 1, l 3 N s = {1, 3, 5} L s = {l 1, l 3 }

17 Nodal Degree The degree of a node n, denoted by d(n), is the number of lines that are incident with n, or, the number of nodes adjacent to n. The mean degree is the average degree of the nodes in the graph., where m = number lines in the graph Variance of the degree: b e a l l 1 3 l 2 l d 4 l 5 l 6 l 7 c f Nodal degree: d(a) = 3 d(d) = 4 d(c) = 2 Mean degree: d(a)+d(b)+d(c)+d(d)+d(e)+d(f) = 14 = 2*7 14/6 = 2.33

18 Density Density : the ratio of the number of lines in a graph to the maximum number of lines that can be present: where m is the number of lines in the graph, g is the number of nodes in the graph.

19 Walk, Trail and Path Walk: a sequence of nodes and lines, starting and ending with nodes, in which each node is incident with the lines following and preceding it in the sequence. Trail: a walk in which all of the lines are distinct. Path: a walk in which all nodes and all lines are distinct. Repeated node allowed? Repeated line allowed? Walk YES YES Trail YES NO Path NO NO Every path is a trail and every trail is a walk. l a 1 b d l 4 l 2 c l 3 A walk: a, l 1, b, l 2, c, l 2, b A trail: a, l 1, b, l 2, c, l 3, d, l 4, b A path: a, l 1, b, l 2, c, l 3, d

20 Related Concepts Reachable: If there is a path between nodes n i and n j, then n i and n j are said to be reachable. The number of lines in a walk (resp. trail, path) is the length of the walk (resp. trail, path). Closed walk: begins and ends at the same node. Cycle: a closed walk of at least three nodes in which all lines are distinct, and all nodes except the beginning and ending nodes are distinct. Tour: a closed walk in which each line in the graph is used at least once Tour: 3, 2, 4, 3, 5, 1, 4, 3 Cycle: 5, 1, 4, 3, 5 Closed walk: 5, 1, 4, 3, 2, 4, 1, 5

21 Connected Graph & Component Connected graph: there is a path between all pairs of nodes in the graph. A component of a graph is a maximal connected subgraph. Only one component in a graph => the graph is connected. More than one component in a graph => the graph is disconnected Connected graph Disconnected graph

22 Geodesics, Distance, and Diameter Geodesic: a shortest path between two nodes. Distance between two nodes n i and n j, denoted by d(i, j), is the length of the shortest path between the two nodes. Eccentricity of a node n i is the largest geodesic distance (max j d(i, j)) between that node and any other nodes. The diameter of a connected graph is the length of the largest geodesic (max i max j d(i, j)) between any pair of nodes Geodesic distance: d(1, 2) = 1 d(1, 3) = 1 d(1, 4) = 2 d(1, 5) = 3 d(2, 3) = 1 d(2, 4) = 1 d(2, 5) = 2 d(3, 4) = 1 d(3, 5) = 2 d(4, 5) = 1 Diameter: max i max j d(i, j) = d(1, 5) = 3 Eccentricity: max j d(3, j) = d(3, 5) = 2

23 Connectivity of Graphs A node n i in a connected graph is called a cutpoint, if its removal disconnects the graph Node 1 is a cutpoint Graph without node 1

24 Connectivity of Graphs A line l i in a connected graph is called a bridge, if its removal disconnects the graph Line(2, 3) is a bridge After removing line(2,3)

25 Connectivity of Graphs The node-connectivity of a graph, K(G), is the minimum number K of nodes that must be removed to make the graph disconnected, or to leave a trivial graph. The line-connectivity of a graph, λ(g), is the minimum number λ of lines that must be removed to make the graph disconnected, or to leave a trivial graph.

26 Isomorphic Graphs Two graphs G = (N, L) and G = (N, L ) are called isomorphic, if there is a one-to-one mapping function f from N onto N such that any two nodes n 1, n 2 N are adjacent iff f(n 1 ) and f(n 2 ) are adjacent in G. If two graphs are isomorphic, they are identical on all graph theoretic properties a d b c Isomorphic graphs

27 Special Kinds of Graphs The complement G = (N, L ) of a graph G = (N, L) has the same node set as its node set and two nodes are adjacent in G iff they are NOT adjacent in G Complement G G

28 Special Kinds of Graphs A graph that is connected and acyclic is called a tree A tree

29 Special Kinds of Graphs If the nodes in a graph G = (N, L) can be partitioned into two subsets, N = N 1 N 2, and any two adjacent nodes belong to different subsets, then G is a bipartite graph. A bipartite graph with any two nodes from different partitions are adjacent is called complete bipartite graph Bipartite graph Complete bipartite graph

30 Directed Graph A directed graph, or digraph, G = (N, L) is a graph such that each arc is an ordered pair of distinct nodes. l k = <n i, n j > is different from l k = <n j, n i > <1,4>!= <4,1> <5,6>!= <6,5> 6 5

31 Graph vs Digraph Graph Digraph line/arc (1,2) = (2, 1) <1,2>!= <2,1> dyad non-directional directional triad non-directional directional nodal degree non-directional directional out/in walk non-directional directional path non-directional directional semipath NA may have different directions cycle non-directional directional semicycle NA may have different directions geodesic non-directional directional diameter non-directional directional

32 Related Concepts A node n i in a digraph is a(n) isolate if d I (n i ) = d o (n i ) = 0, transmitter if d I (n i ) = 0 and d o (n i ) > 0, receiver if d I (n i ) > 0 and d o (n i ) = 0, carrier or ordinary if d I (n i ) > 0 and d o (n i ) > 0. b a d e c Isolated node: e Transmitter: a Receiver: d Carrier: b, c

33 Reachability and Connectivity in Digraphs If there is a directed path from n i to n j in a digraph, then node n j is reachable from node n i. A pair of nodes, n i and n j, is: weakly connected: joined by a semipath; unilaterally connected: joined by a path from n i to n j, OR a path from n j to n i ; strongly connected: joined by a path from n i to n j, AND a path n j to n i ; recursively connected: strongly connected and the two paths use the same nodes and arcs Weak Unilateral Strong Recursive

34 Special Kinds of Digraphs The complement G = (N, L ) of a digraph G = (N, L) has the same node set as its node set, and <n i, n j > L iff <n i, n j > L. The converse G* = (N, L*) of a digraph G = (N, L), has the same set of nodes as G, and arc <n i, n j > L* only if the arc < n j, n i > L. A tournament is a digraph in which every pair of nodes is connected by a single arc Digraph G Converse Complement Tournament

35 Signed Graphs A signed graph (resp. digraph) is a graph (resp. digraph) whose lines carry the additional information of a valence: a positive (+) or negative (-) sign. The sign of a cycle is the product of the signs of the lines in the cycle. (+)(+) = (+) (-)(-) = (+) (-)(+) = (+)(-) = (-) Signed graph Signed digraph

36 Valued Graphs A valued graph (resp. digraph) is a graph (resp. digraph) in which each line is associated with a value

37 Density and Path in Valued Graph The density,, in a valued graph (resp. digraph) is the ratio of the values attached to the lines (resp. arcs) to the maximum number of lines (resp. arcs) that may present. The value of a path (semipath) is the smallest value attached to any lines (arcs) in it. The length of a path in a valued graph is the sum of the values of the lines included in the path Path Length Value 1,5, ,2,3,4 6 1

38 Relations A Cartesian product of two sets, M x N, is the collection of all ordered pairs <m, n>, such that m M and n N. M = {1, 2}, N = {a, b, c} M x N = {<1,a>, <1,b>, <1,c>, <2,a>, <2,b>, <2,c>} A relation, R, on the set N is a subset of the Cartesian product N x N.

39 Properties of Relations A relation is reflexive if for all i, <n i, n i > ties are present in R. A relation is symmetric if it has the property that irj iff jri, for all i and j. A relation is transitive if whenever irj and jrk, then irk, for all i, j and k. a, b, c N, relation = is reflexive, symmetric and transitive: a = a is true, a = b iff b = a a = b, and b = c, then a = c

40 Matrices A matrix is a rectangular array of numbers, such as An adjacency matrix, or sociomatrix, X, is a matrix in which each entry indicates whether two nodes are adjacent (1) or not (0). An incidence matrix, I, is a g x m matrix, such that i ij = 1 if the node n i and line l j are incident and 0 otherwise. The sociomatrix, X, of a digraph has elements x ij = 1 if <n i, n j > L and 0 otherwise. The sociomatrix, X, for a valued graph has entries, x ij = v k if there is a line or arc l k associated with v k connect node n i and n j.

41 Matrices for Graphs X a b c d a a b c l1 l3 d d I b l1 l2 l3 a b c l2 Graph c d 0 0 0

42 Matrices for Graphs X a b c d a d a b c d b c Directed graph X a b c d a a d b c d b 0.9 c Valued graph

43 Basic Matrix Operations Matrix Permutation: simultaneously rearrange rows and columns. Transpose: interchanging the rows and columns of the original matrix. Addition and Subtraction: summation/subtraction of the elements in the corresponding cells of the matrices. Z = X+Y: z ij = x ij + y ij Z = X-Y: z ij = x ij - y ij Matrices X and Y have to be of the same size.

44 Permutation and Transpose n1 n2 n3 n4 n5 n n n n n permutation n5 n1 n3 n2 n4 n n n n n transpose

45 Basic Matrix Operations Matrix Multiplication Z = YW: The number of columns in Y must equal the number of the rows in W. Powers of a Matrix X p = the matrix product of X times itself, p times. X has to be a square matrix. Boolean Matrix Multiplication

46 Multiplication and Power X multiplication a b c d e f a b c d e f X -> X 5 a b c d e f a b c d e f

47 Computing Simple Network Properties Walk: Entries of matrix X * + = X + X 2 + X X g-1, give the total number of walks from n i to n j. Reachable: None-zero elements of X * + indicate reachability. Geodesic: Starting with p = 2, compute power of the matrix, X [p]. If x ij [p-1] = 0 and x ij [p] > 0, then there is a shortest path of length p between n i and n j. Nodal Degree: Incidence matrix:,where m is the number of lines.

48 Computing Simple Network Properties Nodal Degree: for sociomatrix Nodal Degree: for digraph Density:

49 Properties of Graphs, Relations, and Reflexivity Matrices A simple graph is irreflexive since no <n i, n i > are present. In corresponding sociomatrix of the graph, main diagonal are undefined. If all loops are present, the graph represents a reflexive relation. Main diagonal of the sociomatrix, x ii = 1 for all i. If for some i but not all, values x ii =1, then the graph represents a not reflexive relation.

50 Properties of Graphs, Relations, and Symmetry Matrices A nondirectional relation is always symmetric. In a directed graph symmetry means <n i, n j > L implies <n j, n i > L. Transitivity In order for the relation to be transitive, whenever x ij [2] 1, then x ij must equal 1.

51 Core Discussion Networks for Americans [Marsden, P.V. (1987) ] Purpose Descriptive overview of features of core social networks of Americans with network concepts Data source 1985 General Social Survey (GSS) 1,531 people: Nationally representative sample of Americans A first establishment of a standardized instrument to collect social network data Before 1985 GSS, the absence of standardized instruments for collecting data complicated comparison, replication of findings, and accumulation knowledge.

52 Operational definitions Names of alters matter!! Name-eliciting device (name generator) sets operational boundaries on the interpersonal environment GSS used a name generator: discuss important matters GSS asked each respondent about those persons with whom a respondent discuss important matters But the definition of what was considered important was left to respondents; intimacy or positive affect Moderately intense content which represents a middle ground between acquaintanceship and kinship Less ambiguous in its meaning than friendship

53 Measures of the structure of interpersonal environments Individual attributes Gender, race/ethnicity, education, age, and religious preference of each alters Social attributes Respondents were asked to name all those people with whom they discussed important matters within the past six months All sub-questions focused on the first five names mentioned GSS data presents data on the following three items:

54 Three Social Network Measurements Network size: # of alters Network density: Availability of social support Potential strength of normative pressures toward conformity Capacity of alters to collectively influence the respondent 1: especially close, 0: total strangers, 0.5: otherwise Heterogeneity of alters: 4 types of diversity Diversity of persons an individual can contact within his or her interpersonal environment Age and education Quantitative variable Measured as the standard deviations of alter characteristics Race/ethnicity, and gender Qualitative variable Measured as the Index of Qualitative Variation (IOV)

55 The Average American Discussion Network: An Overview Comparatively large % ^ Kin environment is slightly larger than nonkin Difficult to judge whether this is high or low (1: especially close, 0:stranger)

56 The Average American Discussion Network: An Overview Mean: heterogeneity (diversity) within each network (0: no diversity) ^ ^ ^ Diversity within each network is smaller than that of population ^

57 The Average American Discussion Network: An Overview Standardized regression is not necessary unless we compare coefficients each other All p-values show significance Nonkin < Kin > < Nonkin exhibits larger diversity only in race. Independent variable: Proportion Kin (% of kin in each network)

58 The Average American Discussion Network: Subgroups Independent variables Negative relation Positive relation Negative Relation white people have broader network More kin dependent variables Less nonkin Influence of urbanization Proportion kin age Nonkin network grows faster

59 The Average American Discussion Network: Subgroups Independent variables Positive relation dependent variables Negative relation Negative Positive relation Positive Relation Negative Relation Influence of urbanization

60 Conclusions and Future Research Americans social environment in general Small, kin-centered, dense, and homogeneous Sub-groups with more diversity Young and middle-aged, the well-educated, and those in larger places Future research based on this analysis Development of standardized protocols for the collection of information on the structure and composition of interpersonal environments Construction of high-quality and reliable measures of network characteristics

61 Discussion