Chapter 4 Graphs and Matrices. PAD637 Week 3 Presentation Prepared by Weijia Ran & Alessandro Del Ponte

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1 Chapter 4 Graphs and Matrices PAD637 Week 3 Presentation Prepared by Weijia Ran & Alessandro Del Ponte 1

2 Outline Graphs: Basic Graph Theory Concepts Directed Graphs Signed Graphs & Signed Directed Graphs Valued Graphs & Valued Directed Graphs Generalization: Multigraphs & Hypergraphs Mathematical Relations Matrices and Basic Matrix Operations for Graphs 2

3 Why Graph Theory? a vocabulary: to label and denote social structural properties; mathematical operations and ideas: to quantify and measure these properties; ability: to prove theorems about graphs (representations of social structure). 3

4 Graphs 4

5 Graphs Graph: a model for a network with an undirected dichotomous relation; a set of nodes & a set of lines Relevant concepts: adjacent (two nodes ); incident with (a node & a line) trivial (# of nodes=1); empty (# of nodes 1 & # of lines=0) 5

6 Subgraphs of a graph (p.98) 6

7 Dyads and Triads Two possible dyadic states: adjacent or not adjacent 7

8 Nodal Degree & Density Degree of a node: 0 d(n i ) (g-1) Mean nodal degree: Variance of the degrees: Density of a graph: 8

9 Complete and empty graphs (p.102) 9

10 Walks, trails, and paths Walk not all nodes/lines some nodes/lines used more than once length of a walk: number of occurrences of lines Trail a walk in which all lines are distinct Path a walk in which all nodes and all lines are distinct Reachable two nodes are reachable if there is a path between them 10

11 Walks, trails, and paths in a graph (p.106) 11

12 Closed Walks, Tours, and Cycles(p.108) 12

13 Connected Graphs and Components (p.109) 13

14 Geodesic/Distance, Eccentricity and Diameter (p.111) 14

15 Connectivity of Graphs: Cutpoint & k-node cut (p.112) 15

16 Connectivity of Graphs: Bridge & l-line cut (p.114) 16

17 Node- and line-connectivity (p.116) 17

18 Isomorphic Graphs (p.118) 18

19 Special kinds of Graphs (p.119) Complement Tree and forest Bipartite Graphs 19

20 Directed Graphs 20

21 21

22 Undirected Directed nodes line first node (sender) second node (receiver) arc: an ordered pair of nodes dyad nodal degree null dyad: two nodes without arcs asymmetric: with an arc going in one direction mutual dyad: with two arcs nodal indegree: # of node terminating at nodal outdegree: # of arcs originating with walk/path/cycle directed walk/path/cycle: the same direction semiwalk/path/cycle: the direction is irrelevant 22

23 Undirected reachable connectivity geodesics/ distance diameter complement not available Directed weakly connected; unilaterally connected; strongly connected; recursively connected; one of four kinds of connectivity above sometimes, d (n i, n j ) d (n j, n i ) strongly connected or recursively connected graphs: the longest geodesic ; unilaterally or weakly connected graphs: undefined. complement: same node set, complement arc set; converse: same node set, direction reversed arc set. tournament: a relation indicating superior performances or beats in competition. 23

24 Signed Graphs 24

25 25

26 Graph Signed graph Signed directed graph dyad triad cycle three possible states: 1) positive line; 2) negative line; 3) no line. four possible states: zero, one, two, or three positive(negative) lines are present among the three nodes the sign of a cycle: the product of the signs of the lines included in the cycle with additional direction information with additional direction information semicycle: a cycle in which the arcs may point in either direction 26

27 Valued Graphs 27

28 Valued Graphs & Valued Directed Graphs To represent: Valued relations (e.g., the frequency of interaction among pair of people) Possible values: Integers integer weighted digraph Probabilities Markov chains (transition/stochastic matrices) Non-numerical network Example of a valued directed graph (P.142) 28

29 Graphs nodal degree dyad density Valued Graphs average the values over all lines/arcs valued directed graphs: two different values, usually to compare these two values = Vk/g(g-1) path value of a path: min(v1,v2,,vk), a path at level c reachability: two nodes are reachable at level c path length: Vk 29

30 Graph Generalization Multigraph A simple graph or digraph that allows more than one set of lines Hypergraph To represent an affiliation network/a two-mode network (a set of actors + a set of events) 30

31 Example of a hypergraph (p.147) 31

32 Relations 32

33 Mathematical Relations A set Cartesian product of sets N and N, the collection of all ordered pairs of actors for whom substantive ties could be present. A subset of the Cartesian product N X N, all ordered pairs <ni, nj> for whom the substantive tie from i to j is present. 33

34 Properties of Relations Reflexivity reflexive: all possible <ni, ni> ties are present in R; irreflexive: no iri is not reflexive: neither reflexive nor irreflexive Symmetry symmetric: <ni, nj> є R if and only if <nj, ni> є R (all dyads are mutual or null) antisymmetric: irj implies that not jri (beats relation/tournament) asymmetric: neither symmetric nor antisymmetric Transitivity transitive: if whenever irj and jrk, then irk (a friend of a friend is a friend) 34

35 Matrices 35

36 Sociomatrix (adjacency matrix): X 36

37 Incidence Matrix : I 37

38 Matrices for Digraphs 38

39 Matrices for More Graphs Matrices for Valued Graphs: X Xij : the value associated with the line/arc Matrices for Two-Mode Networks N={n1,n2,,ng}, M={m1,m2,,mh}: a g x h matrix Matrices for Hypergraphs: A incidence matrix point set N={n1,n2,,ng}, edge set M={m1,m2,,mh} A={aij}, aij=1 if point ni is in edge Mj 39

40 Basic Matrix Operations 40

41 Permutation Rearrangement of rows, and simultaneously of columns Example of matrix permutation (p.156) 41

42 Transpose Interchanging the rows and columns of the original matrix a tie from row actor i to column actor j row actor i received a tie from column actor j Example of matrix transpose (p.157) 42

43 Addition and Subtraction X,Y: both of size g by h Z=X+Y where zij=xij+yij; Z=X-Y where zij=xij-yij Matrix Multiplication used to study walks and reachability Y: size of g by h; W: size of h by k Z=YW where 43

44 Powers of a Matrix X p (X to the pth power): the matrix product of X times itself, p times Boolean Matrix Multiplication used to study walks and reachability 44

45 Computing Simple Network Properties 45

46 Walks and Reachability X 2 The number of walks of length 2 between nodes n i and n j, for all k. The number of walks of length P between nodes ni and nj, for all k. The total number of walks from ni to nj, of any length less than or equal to g-1 Pairs of nodes that are not reachable : a 0 in cell (i, j ) 46

47 Geodesic and Distance (Geodesic) Distance the length of a shortest path between two notes: the first power p for which the (i, j) element is non-zero: d(i,j) =min p X ij [p] >0 Diameter the largest geodesic in the graph or digraph 47

48 Powers of a sociomatrix for a directed graph (p.162) 48

49 Computing Nodal Degrees and Density Nodal degree Graph Directed Graph Density 49

50 Properties of Graphs, Relations, and Matrices Reflexivity reflexive: x ii = 1 irreflexive: x ii undefined is not reflexive: has some, but not all, values of x ii = 1 Symmetry x ij = x ji Transitivity if there is a walk ni nk nj for at least one node nk: X ij [2] 1 transitive: whenever X ij [2] 1, then x ij must equal 1. 50

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