Elementary Graph Theory & Matrix Algebra

Transcription

1 Slide 1 Elementary Graph Theory & Matrix Algebra 2008 LINKS Center Summer SNA Workshops Steve Borgatti, Rich DeJordy, & Dan Halgin

2 Slide 2 Introduction In social network analysis, we draw on three major areas of mathematics regularly: Relations Branch of math that deals with mappings between sets, such as objects to real numbers (measurement) or people to people (social relations) Matrix Algebra Tables of numbers Operations on matrices enable us to draw conclusions we couldn t just intuit Graph Theory Branch of discrete math that deals with collections of ties among nodes and gives us concepts like paths

3 Slide 3 Binary Relations The Cartesian product S1 S2 of two sets is the set of all possible ordered pairs (u,v) in which u S1 and v S2 Given sets S1 and S2, a binary relation R is a subset of their Cartesian product S1 Note: S1 and S2 could be the same set S2

4 Slide 4 Relational Terminology To indicate that u is R-related to v or u is mapped to v by the relation R, we write (u,v) R, or urv Example: If R is likes, then urv says u likes v (jim,jane) R says jim likes jane

5 Slide 5 Properties A relation is reflexive if for all u, (u,u) R E.g., suppose R is is in the same room as u is always in the same room as u, so the relation is reflexive A relation is symmetric if for all u and v, urv implies vru If u is in the same room as v, then it always true that v is in the same room as u. So the relation is symmetric A relation is transitive if for all u,v,w, urv together with vrw implies urw If u is in the same room as v, and v is in the same room as u, then u is necessarily in the same room as w So the relation is transitive A relation is an equivalence if it is reflexive, symmetric and transitive Equivalence relations give rise to partitions and vice-versa A partition of a set S is an exhaustive set of mutually exclusive classes such at each member of S belongs to one and only one class

6 Slide 6 Operations The converse or inverse of a relation R is denoted R -1. For all u and v, (u,v) R -1 if and only if (v,u) R The converse effectively reverses the direction of the mapping Example If R is represents gives advice to, then urv means u gives advice to v, and ur -1 v indicates that v gives advice to u If R is symmetric, then R = R -1 Important note: In the world of matrices, the relational converse corresponds to the matrix concept of a transpose, denoted X or X T, and not to the matrix inverse, denoted X -1. The -1 superscript and the term inverse are unfortunate false cognates.

7 Slide 7 Relational Composition If F and E are binary relations, then their composition F E is a new relation such that (u,v) F E if there exists w such that (u,w) F and (w,v) E. i.e., u is F E-related to v if there exists an intermediary w such that u is F-related to w and w is E-related to v Example: Suppose F and E are friend of & enemy of, respectively u F E v means that u has a friend who is the enemy of v Equally, v is the enemy of a friend of u

8 Slide 8 More Relational Composition Assume F is likes and E is dislikes u F F v means u likes someone who likes v (loosely, v is a friend of a friend of u) (u,v) E E means u dislikes someone who dislikes v (loosely, v is an enemy of an enemy) u E F v means u dislikes someone who likes v (loosely, v is friend of an enemy of u)

9 Slide 9 More Relational Composition Given these relations A (authored). Relates persons documents P (published in). Relates docs journals K (has keyword). Relates docs keywords Compositions AA -1. if (i,j) AA -1, then i authors a documents that is authored by j. i.e., i and j are coauthors AP. Person i authored a document that is published in journal j. so i has published in journal j AK. Person i authored a doc that has keyword j. So, i writes about topic j AKK -1 A -1. person i authored a document that has a keyword that is in a document that was authored by j. In other words, i and j write about the same topics AKK -1 A -1 AP. person i authored a document that has a keyword that is in a document that was authored by someone who has published in journal j. I.e., i has written about a topic that has appeared in journal j

10 Slide 10 Relational Equations F= F F means that ufv if and only if uf Fv Friends of friends are friends, and vice versa F=E E means that ufv if and only if ue Ev Enemies of enemies are friends, and vice-versa E=F E=E F means that uev if and only if uf Ev and ue Fv Both enemies of friends and friends of enemies are enemies, and vice-versa

11 Slide 11 Matrix Algebra In this section, we will cover: Matrix Concepts, Notation & Terminologies Adjacency Matrices Transposes Aggregations & Vectors Matrix Operations Boolean Algebra (and relational composition)

12 Slide 12 Matrices Symbolized by a capital letter, like A Each cell in the matrix identified by row and column subscripts: a ij First subscript is row, second is column Age Gender Income Mary ,000 Bill ,000 John Larry ,000 a 12 = 1 a 43 = 8000 A

13 Slide 13 Vectors Each row and each column in a matrix is a vector Vertical vectors are column vectors, horizontal are row vectors Denoted by lowercase bold letter: y Each cell in the vector identified by subscript x i X Y Z y 3 = 2.1 z 2 = 45,000 Mary ,000 Bill ,000 John Larry ,000

14 Slide 14 Ways and Modes Ways are the dimensions of a matrix. Modes are the sets of entities indexed by the ways of a matrix Event Event Event Event EVELYN LAURA THERESA BRENDA CHARLO FRANCES ELEANOR PEARL RUTH VERNE MYRNA way, 2-mode Mary Bill John Larry Mary Bill John Larry way, 1-mode

15 Slide 15 Examples of 1-mode and 2-mode 1-mode matrices Item by item proximity matrices Correlation matrices Distance matrices Social proximity / strength of tie 2-mode matrices Profile matrices Individuals scores on a set of personality scales Participation in events; membership in groups

16 Slide 16 Proximity Matrices Proximity Matrices record degree of proximity. Proximities are usually among a single set of actor (hence, they are 1-mode), but they are not limited to 1s and 0s in the data. What constitutes the proximity is user-defined. Driving distances are one form of proximities, other forms might be number of friends in common, time spent together, number of s exchanged, or a measure of similarity in cognitive structures. Proximity matrices can contain either similarity or distance (or dissimilarity) data. Similarity data, such as number of friends in common or correlations, means a larger number represents more similarity or greater proximity Distance (or dissimilarity data) such as physical distance means a larger number represents more dissimilarity or less proximity

17 Slide 17 Profiles Typically, we use profiles to refer to the patterns of responses across a row of a matrix, generally a 2-mode matrix. We might then compare profiles across the rows to see which rows have the most similar or dissimilar 8 profiles. 7 We can also conceive of this 6 down the columns, as well. 5 In fact, when we correlate 4 variables in traditional OLS, we are actually comparing the 3 profiles of each pair of variables 2 across the respondents. 1 0 ID A B C D E A B C D E

18 Slide 18 Aggregations and Operations Unary (Intra-Matrix) Operations Row sums/marginals Column sums/marginals Matrix Sums Transpose Normalizations Dichotomization Symmetrizing Cellwise Binary (Inter-Matrix) Operations Sum Cellwise multiplication Boolean Operations Special Binary (Inter-Matrix) Operations Cross Product (Matrix Multiplication)

19 Slide 19 Summations Row sums r i x ij j Column sums c j x ij i Matrix sums m x ij i, j Row Mary Bill John Larry Marginals Mary Bill John Larry Column Marginals

20 Slide 20 Normalizing Converting to proportions Rows Columns x * ij xij ri x * ij xij c Row Mary Bill John Larry Sums Mary Bill John Larry j where r i gives the sum of row i Row Mary Bill John Larry sums Mary Bill John Larry Column Marginals Column Marginals

21 Slide 21 Normalizing Converting to z-scores (standardizing) Columns x u x * ij j where u j gives the mean of ij column j, and j j is the std deviation of column j Var 1 Var 2 Var 3 Var 4 Var 1 Var 2 Var 3 Var 4 Mary Mary Bill Bill John John Larry Larry Mean Mean Std Dev Std Dev

22 Slide 22 Transposes The transpose of a matrix is the matrix flipped on its side. The rows become columns and the columns become rows So the transpose of an m by n matrix is an n by m matrix. Here is the matrix presented on the last side, and its transpose The transpose of matrix M is indicated by M or M T A B C D E Matrix M A B C D E Its Transpose, M

23 Slide 23 Transpose (Another Example) Given Matrix M, swap the rows and columns to make Matrix M T M Tennis Football Rugby Golf Mike Ron Pat Bill Joe Rich Peg M T Mike Ron Pat Bill Joe Rich Peg Tennis Football Rugby Golf

24 Slide 24 Dichotomizing X is a valued matrix, say 1 to 10 rating of strength of tie Construct a matrix Y of ones and zeros so that y ij = 1 if x ij > 5, and y ij = 0 otherwise EVE LAU THE BRE CHA EVELYN LAURA THERESA BRENDA CHARLOTTE X EVE LAU THE BRE CHA EVELYN LAURA THERESA BRENDA CHARLOTTE Y x ij > 5

25 Slide 25 Symmetrizing When matrix is not symmetric, i.e., x ij x ji Symmetrize various ways. Set y ij and y ji to: Maximum(x ij, x ji ) {union rule} Minimum (x ij, x ji ) {intersection rule} Average: (x ij + x ji )/2 Lowerhalf: choose x ij when i > j and x ji otherwise etc

26 Slide 26 Symmetrizing Example X is non-symmetric (and happens to be valued) Construct matrix Y such that y ij (and y ji ) = maximum of x ij and x ji X ROM BON AMB BER PET LOU ROMUL_ BONAVEN_ AMBROSE_ BERTH_ PETER_ LOUIS_ ROM BON AMB BER PET LOU ROMUL_ BONAVEN_ AMBROSE_ BERTH_ PETER_ LOUIS_ Symmetrized by Maximum

27 Slide 27 Cellwise Binary Operators Sum (Addition) C = A + B where c ij = a ij + b ij Cellwise (Element) Multiplication C = A * B where c ij = a ij * b ij Boolean operations C = A B (Logical And) where c ij = a ij b ij C = A B (Logical Or) where c ij = a ij b ij

28 Slide 28 Matrix Multiplication Notation: Definition: Example: c ij C = AB k a ik b kj Note: matrix products are not generally commutative. i.e., AB does not usually equal BA Mary Bill John Larry Mary Bill John Larry Mary Bill John Larry Mary Mary Mary Bill Bill Bill John John John Larry Larry Larry A B C=AB

29 Slide 29 Matrix Multiplication C = AB or C = A x B Only possible when the number of columns in A is the same as the number of rows in B, as in m A k and kb n These are said to be conformable Produces m C n It is calculated as: c ij = a ik * b kj for all k

30 Slide 30 A Matrix Product Example Skills Math Verbal Analytic Kev Jeff Lisa Kim Given a Skills and Items matrix calculate the affinity that each person has for each question Kev for Question 1 is: = 1.00 * * *.40 = = Lisa for Question 3 is: =.75 * * *.1 = = Items Q1 Q2 Q3 Q4 Math Verbal Analytic Affin Q1 Q2 Q3 Q4 Kev Jeff Lisa Kim

31 Slide 31 Matrix Inverse and Identity The inverse of a matrix X is a Matrix X -1 such that XX -1 = I, where I is the identity matrix Inverse matrices can be very useful for solving matrix equations that underlie some network algorithms X X -1 = I Note: (XX -1 = X -1 X = I) Non square matrices do not have an inverse*

32 Slide 32 Boolean matrix multiplication Values can be 0 or 1 for all matrices Products are dichotomized to conform: Would have been a 2 in regular matrix multiplication Mary Bill John Larry Mary Bill John Larry Mary Bill John Larry Mary Mary Mary Bill Bill Bill John John John Larry Larry Larry A B AB

33 Slide 33 Relational Composition If we represent binary relations as binary adjacency matrices, boolean matrix products correspond to relational composition F E corresponds to FE Likes Has conflicts with Likes someone who has conflicts with Mary Bill John Larry Mary Bill John Larry Mary Bill John Larry Mary Mary Mary Bill Bill Bill John John John Larry Larry Larry F E FE

34 Slide 34 Products of matrices & their transposes X X = pre-multiplying X by its transpose ( X ' X ) ij akibkj k Computes sums of products of each pair of columns (cross-products) The basis for most similarity measures Mary Bill John Larry

35 Slide 35 Products of matrices & their transposes XX = product of matrix X by its transpose ( XX ') ij aikb jk k Computes sums of products of each pair of rows (cross-products) Similarities among rows Mary Bill John Larry Mary Bill John Larry Mary Bill John Larry

36 Slide 36 E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 EVELYN LAURA THERESA BRENDA CHARLOTTE FRANCES ELEANOR PEARL RUTH VERNE MYRNA KATHERINE SYLVIA NORA HELEN DOROTHY OLIVIA FLORA Multiplying a matrix by its transpose EV LA TH BR CH FR EL PE RU VE MY KA SY NO HE DO OL FL E E E E E E E E E E E E E E EVE LAU THE BRE CHA FRA ELE PEA RUT VER MYR KAT SYL NOR HEL DOR OLI FLO EVELYN LAURA THERESA BRENDA CHARLOTTE FRANCES ELEANOR PEARL RUTH VERNE MYRNA KATHERINE SYLVIA NORA HELEN DOROTHY OLIVIA FLORA

37 Slide 37 Graph Theoretic Concepts In this section we will cover: Definitions Reachability/Connectedness Terminology Connectivity, flows Adjacency Isolates, Pendants, Centers Density concepts Components, bi-components E.g, Completeness Walk Lengths, distance Walks, trails, paths Geodesic distance Cycles, Trees Independent paths Cutpoints, bridges

38 Slide 38 Undirected Graphs An undirected graph G(V,E) (often referred to simply as a graph) consists of Set of nodes vertices V representing actors Set of lines links edges E representing ties among pairs of actors An edge is an unordered pair of nodes (u,v) Nodes u and v adjacent if (u,v) E So E is subset of set of all pairs of nodes Drawn without arrow heads Sometimes with dual arrow heads Used to represent logically symmetric social relations In communication with; attending same meeting as

39 Slide 39 Graphical representation of a graph

40 Slide 40 Adjacency matrix of a graph HOLLY BRAZEY CAROL PAM PAT JENNIE PAULINE ANN MICHAEL BILL LEE DON JOHN HARRY GERY STEVE BERT RUSS

41 Slide 41 Directed vs. Undirected Ties Undirected relations Attended meeting with Communicates daily with Directed relations Lent money to Logically vs empirically directed ties Empirically, even undirected relations can Bob be non-symmetric due to measurement error Biff Bonnie Betty Betsy

42 Slide 42 Directed Graphs (Digraphs) Digraph G(V,E) consists of Set of nodes V Set of directed arcs E An arc is an ordered pair of nodes (u,v) (u,v) E indicates u sends arc to v (u,v) E does not imply that (v,u) E Ties drawn with arrow heads, which can be in both directions Represent logically non-symmetric or anti-symmetric social relations Lends money to

43 Slide 43 Transpose Adjacency matrix Interchanging rows/columns of adjacency matrix effectively reverses the direction of ties Mary Bill John Larry Mary Bill John Larry Mary Mary Bill Bill John John Larry Larry Gives money to Gets money from mary bill mary bill larry john larry john

44 Slide 44 Valued Digraphs (vigraphs) A valued digraph G(V,E,W) consists of Set of nodes V Set of directed arcs E An arc is an ordered pair of nodes (u,v) (u,v) E indicates u sends arc to v (u,v) E does not imply that (v,u) E Mapping W of arcs to real values Values can represent such things as Strength of relationship Information capacity of tie Rates of flow or traffic across tie Distances between nodes Probabilities of passing on information Frequency of interaction

45 Slide 45 Valued Adjacency Matrix Dichotomized Jim Jill Jen Joe Jim Jill Jen Joe Distances btw offices Jim Jill Jen Joe Jim Jill Jen Joe The diagram below uses solid lines to represent the adjacency matrix, while the numbers along the solid line (and dotted lines where necessary) represent the proximity matrix. In this particular case, one can derive the adjacency matrix by dichotomizing the proximity matrix on a condition of p ij <= 3. Jen Jill 9 Joe 3 2 Jim

46 Slide 46 Used to represent 2-mode data Nodes can be partitioned into two sets (corresponding to modes) Ties occur only between sets, not within Bipartite graphs

47 Slide 47 LEE BERT BRAZEY RUSS Node-related concepts STEVE JOHN GERY PAULINE ANN CAROL EVANDER MICHAEL HARRY PAM PAT JENNIE HOLLY BILL DON Degree The number of ties incident upon a node In a digraph, we have indegree (number of arcs to a node) and outdegree (number of arcs from a node) Pendant A node connected to a component through only one edge or arc A node with degree 1 Example: John Isolate A node which is a component on its own E.g., Evander

48 Slide 48 Density and Completeness A graph is complete if all possible edges are present. The density of a graph is the number of edges present divided by the number that could have been BERT BRAZEY STEVE LEE GERY RUSS

49 Slide 49 Density Number of ties, expressed as percentage of the number of ordered/unordered pairs Low Density (25%) Avg. Dist. = 2.27 High Density (39%) Avg. Dist. = 1.76

50 Slide 50 Density Number of ties divided by number possible Undirected Directed Ties to Self Allowed T 2 n / 2 T 2 n No ties to self T n( n 1) / 2 T n( n 1) T = number of ties in network n = number of nodes

51 Slide 51 Graph traversals Walk Any unrestricted traversing of vertices across edges (Russ-Steve-Bert-Lee-Steve) BERT Trail A walk restricted by not repeating an edge or arc, although vertices can be revisited (Steve-Bert-Lee-Steve-Russ) Path A trail restricted by not revisiting any vertex (Steve- Lee-Bert-Russ) Geodesic Path The shortest path(s) between two vertices (Steve- Russ-John is shortest path from Steve to John) Cycle A cycle is in all ways just like a path except that it ends where it begins Aside from endpoints, cycles do not repeat nodes E.g. Brazey-Lee-Bert-Steve-Brazey LEE BRAZEY RUSS STEVE JOHN PAULINE ANN GERY CAROL PAM MICHAEL PAT JENNIE HOLLY BILL HARRY DON

52 Slide 52 Length & Distance Length of a path (or any walk) is the number of links it has The Geodesic Distance (aka graph-theoretic distance) between two nodes is the length of the shortest path Distance from 5 to 8 is 2, because the shortest path (5-1-8) has two links

53 Slide 53 Geodesic Distance Matrix a b c d e f g a b c d e f g

54 Slide 54 Powers of the adjacency matrix If you multiply an adjacency matrix X by itself, you get XX or X 2 A given cell x 2 ij gives the number of walks from node i to node j of length 2 More generally, the cells of X k give the number of walks of length exactly k from each node to each other

55 Slide 55 Matrix powers example Note that shortest path from 1 to 5 is three links, so x 1,5 = 0 until we get to X X X 2 X 3 X 4

56 Slide 56 Subgraphs Set of nodes Is just a set of nodes A subgraph Is set of nodes together with ties among them An induced subgraph Subgraph defined by a set of nodes Like pulling the nodes and ties out of the original graph a a b f b c e c d d f e Subgraph induced by considering the set {a,b,c,f,e}

57 Slide 57 Components Maximal sets of nodes in which every node can reach every other by some path (no matter how long) A graph is connected if it has just one component It is relations (types of tie) that define different networks, not components. A network that has two components remains one (disconnected) network.

58 Slide 58 Components in Directed Graphs Strong component There is a directed path from each member of the component to every other Weak component There is an undirected path (a weak path) from every member of the component to every other Is like ignoring the direction of ties driving the wrong way if you have to

59 Slide 59 A network with 4 weak components Who you go to so that you can say I ran it by, and she says... Recent acquisition Older acquisitions Original company Data drawn from Cross, Borgatti & Parker 2001.

60 Slide 60 Strong components BILL HARRY DON MICHAEL HOLLY LEE BRAZEY STEVE GERY CAROL PAT PAM JENNIE RUSS BERT JOHN PAULINE ANN

61 Slide 61 A tree is a connected graph that contains no cycles In a tree, there is exactly one path from any node to any other Trees

62 Slide 62 Cutpoints and Bridges Cutpoint A node which, if deleted, would b increase the number of d components a Bridge A tie that, if removed, would increase the number of components e c f If a tie is a bridge, at least one of its endpoints must be a cutpoint g r h i n p j o m k l q s

63 Slide 63 Local Bridge of Degree K A tie that connects nodes that would otherwise be at least k steps apart A B

64 Slide 64 Cutsets Vertex cut sets (aka cutsets) A set of vertices S = {u,v, } of minimal size whose removal would increase the number of components in the graph Edge cut sets A set of edges S = {(u,v),(s,t) } of minimal size whose removal would increase the number of components in the graph

65 Slide 65 Independent Paths A set of paths is node-independent if they share no nodes (except beginning and end) They are line-independent if they share no lines S T 2 node-independent paths from S to T 3 line-independent paths from S to T

66 Slide 66 Connectivity Node connectivity κ(s,t) is minimum number of nodes that must be removed to disconnect s from t Line connectivity (s,t) is the minimum number of lines that must be removed to disconnect s from t S T

67 Slide 67 Bi-Components (Blocks) A bicomponent is a maximal subgraph such that every node can reach every other by at least two node-independent paths Bicomponents contain no cutpoints There are four bicomponents in this graph: { }, {6 15}, {15 7}, and { }

68 Slide 68 Menger s Theorem Menger proved that the number of line independent paths between s and t equals the line connectivity (s,t) And the number of node-independent paths between s and t equals the node connectivity κ(u,v)

69 Slide 69 Maximum Flow If ties are pipes with capacity of 1 unit of flow, what is the maximum # of units that can flow from s to t? Ford & Fulkerson show this was equal to the number of line-independent paths S T