Unit 2: Graphs and Matrices. ICPSR University of Michigan, Ann Arbor Summer 2015 Instructor: Ann McCranie

Transcription

1 Unit 2: Graphs and Matrices ICPSR University of Michigan, Ann Arbor Summer 2015 Instructor: Ann McCranie

2 Four main ways to notate a social network There are a variety of ways to mathematize a social network, reflecting the wide range of disciplines which have contributed to the social network analysis. Graph theoretic Sociometric Algebraic Dyadic We will focus on the first two ways now, and the third later.

3 Graph Theoretic Notation One relation N : A set of actors, represented by nodes N = {n 1,n 2,,n g } g actors (or nodes) in N L : A set of lines or arcs representing relational ties between pairs of actors (nodes) L = {l 1,l 2,,l L } L lines (or arcs) in L

4 Graph Theoretic Notation Multiple relations just require an extension. R relations R sets of lines (arcs): L r for r = 1, 2,, R.

5 Sociometric Notation One relation N = {n 1,n 2,,n g } X refers to a single relation x ij is the strength or value of the tie from actor i to actor j Sociomatrix, X, of size g x g Rows and columns of this sociomatrix index the actors, in identical order.

6 Sociogram Points/Nodes represent Actors Edges (Lines or Arcs) drawn between nodes if there is a line/arc in the set L

7 Intro to Graph Theory A few graph theoretic terms have been very important in social network analysis Adjacent nodes n i and n j are adjacent if l k = (n i, n j ) є L Incident nodes n i and n j are incident with a line if l k = (n i, n j ) Sociogram points depict nodes a line is drawn between two points if there is a line between the corresponding two nodes in the set of lines, L.

8 Walks, Trails, and Paths Walks in a graph A walk is 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. Length of a walk The length of a walk = number of instances of lines in it. Trails and Paths A trail is a walk in which all of the lines are distinct, though some node(s) may be included more than once. A path is a walk in which all nodes and all lines are distinct. The length of a path is the number of lines in it.

9 Paths in Valued Graphs Value of a path the value of a path (semipath) is equal to the smallest value attached to any line (arc) in it (Peay, 1980). Reachability in a valued graph, two nodes are reachable at level c if there is a path at level c between them. Path length The length of a path in a valued graph is equal to the sum of the values of the lines included in the path.

10 Some Other Important Concepts Nodal Degree The degree of a node, denoted d(n i ), is the number of lines that are incident with it. Complete graphs If all lines are present, then all nodes are adjacent, and the graph is said to be complete. Density of a subgraph We can also define the density of a subgraph, Δ s

11 Some Other Important Concepts Density of Graphs and Subgraphs The density of a graph is the proportion of possible lines that are actually present in the graph. The density of a graph, which we denote by Δ, is calculated as:

12 Undirected Graph: Florentine Marriages

13 Same principles apply for directed graphs fifth grade network

14 These walks and trails and paths lead us to some very important graph theoretic 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. Connected Graphs and Components A graph is connected if there is a path between every pair of nodes in the graph. Components The connected subgraphs in a graph are called components. A component of a graph is a maximal connected subgraph. Geodesics and Distance A shortest path between two nodes is referred to as a geodesic. d(i, j) is the geodesic distance between n i and n j.

15 Geodesic Distances for Florentine Families - Marriage

16 Eccentricity of a node The eccentricity of a node is the largest geodesic distance between that node and any other node. The eccentricity of node n i in a connected graph is equal to the maximum distance d(i, j), for all j. Diameter of a graph The diameter of a connected graph is the largest geodesic within the graph between any pair of nodes (equivalently, the largest nodal eccentricity). Diameter is the maximum d(i, j), for all i and j. Diameter of a subgraph The diameter of a subgraph is the length of the largest geodesic within the subgraph between any pair of nodes within the subgraph.

17 Connectivity of Graphs The connectivity of a graph is a function of whether a graph remains connected when nodes and/or lines are deleted. Cutpoint A node, n i, is a cutpoint if the number of components in the graph that contains n i is fewer than the number of components in the subgraph that results from deleting n i from the graph.

18 Bridge A bridge is a line such that the graph containing the line has fewer components than the subgraph that is obtained after the line is removed. Node- and Line-Connectivity The point-connectivity or node-connectivity of a graph, k(g), is the minimum number, k, of nodes that must be removed to make the graph disconnect, or to leave a trivial graph. The line-connectivity or edge-connectivity of a graph, l(g), is the minimum number l, of lines that must be removed to disconnect the graph or leave a trivial graph.

19 Nodal Indegree and Outdegree indegree The indegree of a node, d I (n i ), is the number of nodes that are adjacent to n i, or the number of arcs terminating at n i. The indegree of node n i is equal to the number of arcs of the form l k = <n j, n i >, for all l k in L, and for all n j in N. outdegree The outdegree of a node, d O (n i ), is the number of nodes adjacent from n i, or the number of arcs originating with node n i. The outdegree of node n i is equal to the number of arcs of the form l k = <n i, n j >, for all l k in L, and for all n j in N.

20 Isomorphic Graphs and Subgraphs Graphs Two graphs, G and G* are isomorphic if there is a one-to-one mapping from the nodes of G to the nodes of G* that preserves the adjacency of nodes. Subgraphs Two subgraphs, G s and G s * are isomorphic if there is a one-to-one mapping from the nodes of G s to the nodes of G s * that preserves the adjacency of nodes. Isomorphism class Subgraphs that are isomorphic belong to the same isomorphism class.

21 Why are these concepts important? Basic structures and concepts you will need for understanding many other more sophisticated measurements. Centrality Subgroup identification Clustering Statistical modeling ergm/p*