Alessandro Del Ponte, Weijia Ran PAD 637 Week 3 Summary January 31, Wasserman and Faust, Chapter 3: Notation for Social Network Data

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1 Wasserman and Faust, Chapter 3: Notation for Social Network Data Three different network notational schemes Graph theoretic: the most useful for centrality and prestige methods, cohesive subgroup ideas, as well as dyadic and triadic methods. Sociometric: often used for the study of structural equivalence and blockmodels. Algebraic: most appropriate for role and positional analyses and relational algebras *For detailed graph notations, please refer to attached document (graph theory glossary) Wasserman and Faust, Chapter 4: Graphs and Matrices 1. Why is graph theory useful? 1) Provides a vocabulary which can be used to label and denote many social structural properties; 2) Gives us mathematical operations and ideas with which many of these properties can be quantified and measured; 3) Gives us the ability to prove theorems about graphs, and hence, about representations of social structure. 2. The terminology and concepts of graph theory Basic graph theory concepts *For detailed concept definitions, please refer to attached document (graph theory glossary) Concepts Calculated from or defined based on Used to represent graph undirected relations nodes actors lines ties between actors loops usually no meaning in real life networks adjacent two actors directly connected trivial node#, line# one actor network empty node#, line# multiple actors but no ties subgraph sub network, subgroup dyads two actors and the possible tie between them triads Three actors and the possible lines among them nodal degree incident line# of each node the degree of a network actor mean nodal degree nodal degrees, node# nodal degree nodal degrees, node# graph centralization variability d-regularity a comparison of nodal degrees network uniformity 1

2 density total line#/maximum possible line # network cohesiveness complete density=1 actors directly connected with all other actors walk a sequence of directly connected actors trial a walk, all lines are distinct path a walk, all nodes/lines are distinct reachable a path exists a path exists between two actors closed walk a walk, begins and ends at the same node cycle a closed walk more than three nodes important in the study of balance and clusterability connected a path between every pair of nodes all pairs of actors are reachable component a connected subgraph a connected subnetwork or subgroup geodesics a shortest path between two nodes distance the length of a geodesic used in centrality, cohesiveness measures eccentricity the largest geodesic distance/a node used in centrality measures diameter the largest geodesic distance/a graph how far apart the farthest two actors are cutpoint node, component a critical actor: If the actor (cutpoint) is removed, the remaining network becomes two subnetworks. bridge line, component a critical tie: If the tie (bridge) is removed, the remaining network becomes two subnetworks. point-connectivity # of nodes that must be removed to network connectivity, cohesiveness make the graph disconnected line-connectivity # of lines that must be removed to make network connectivity, cohesiveness the graph disconnected isomorphic graph graphs: same node adjacency networks identical in all graph theoretic properties, used in defining positional equivalence isomorphic isomorphic graph, subgraph used in studies of dyads and triads subgraphs complement same node set, complement line set tree a graph contains no cycle bipartite graph two subsets of nodes, each line incident upon notes in both sets Undirected graph vs. Directed graph *For detailed concept definitions, please refer to attached document (graph theory glossary) Undirected graph nodes line: an unordered pair of nodes dyad nodal degree walk/path/cycle reachable Directed graph first node (sender) second node (receiver) arc: an ordered pair of nodes 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 the node (receptivity, popularity) nodal outdegree: # of arcs originating with the node isolate, transmitter, receiver, carrier(p.128) directed walk/path/cycle: all arcs pointing in the same direction semiwalk/path/cycle: the direction of the arcs is irrelevant weakly connected: joined by a semipath unilaterally connected: joined by a directed path in one direction strongly connected: joined by directed paths in both directions recursively connected: strongly connected and joined by the same nodes and arcs in both directions 2

3 connectivity geodesics/distance diameter complement no counterpart connected by one of four kinds of connectivity above the geodesics/distance from n i to n j is different from the geodesics/distance from n j to n i strongly connected or recursively connected graphs: the longest geodesic between any pair of nodes. unilaterally or weakly connected graphs: undefined complement: same node set, complement arc set converse: same node set, direction reversed arc set tournament: a set of actors competing in some event(s) and a relation indicating superior performances or beats in competition. Graph vs. Signed graph and Signed directed graph *For detailed concept definitions, please refer to attached document (graph theory glossary) Graph Signed graph Signed directed graph dyads three possible states: 1) positive line between nodes; 2) negative line between nodes; 3) no line between nodes with additional direction information triad four possible states: zero, one, two, or three positive(negative) lines are present among the three nodes with additional direction information cycle the sign of a cycle: the product of the signs of the lines included in the cycle semicycle: a cycle in which the arcs may point in either direction Valued graphs Valued: the strength or intensity of each line or arc is recorded [can refer to directional (dollar amount of exports) and nondirectional (number of interactions) data]. Integer weighted digraph: all values in the valued digraph are from the set of integers. Markov chains: set of graphs whose values are probabilities. Value of a path: is equal to the smallest value attached to any line in it. Reachability: level is set at the strongest line between the nodes. Path length: sum of the values of the lines in it. Multigraphs Multigraph or multivariate graph: allows more than one relation or set of lines. If more than one relation is measured on the same set of actors, then the graph representing this network must allow each pair of nodes to be connected in more than one way (p. 146). Hypergraphs Appropriate for membership or affiliation networks. Hypergraph: consists of a set of objects and a collection of subsets of objects, in which each object belongs to at least on subset, and no subset is empty. Points: objects. Edges: collections of objects. 3

4 Relations Relations: ordered pairs of actors in a network between whom a substantive tie is present. Reflexivity: all possible ties are present in a relation. Symmetric: all the dyads are either mutual or null. Transitive: essentially, if j chooses k as a friend, and k chooses l, then j will choose l as a friend. 3. Matrices and basic matrix operations Sociomatrix (X): record which pairs of nodes are adjacent (contains only 1 s and 0 s for a graph, and is symmetric for a nondirectional relation). Incident matrix (I): records which lines are incident with which nodes (nodes index rows, lines columns). Martices for hypergraphs: codes which points are incident with which edges. Size (order): the number of rows and columns in the matrix. Cell: each entry. Main Diagonal: entries are self-choices or loops, may be undefined. Matrix Permutation: any reordering of the objects (helpful in seeing patterns). Transpose: interchanging the rows and columns (change direction of ties between actors). Addition: sum of the elements in the corresponding cells (same size matrices). Subtraction: difference between elements. Multiplication: see p Powers of a matrix: the matrix times itself. Boolean: the multiplication of matrices is referred to as having a value of 1 or 0. Marsden: Core Discussion Networks of Americans Overview. The paper Core Discussion Networks of Americans (1987) provides an overview of features of core social networks of Americans, exploiting data from the 1985 General Social Survey. In particular, it focuses on the networks in which Americans discuss important matters. The results are presented on the size, kin/nonkin composition, density, and heterogeneity of discussion networks for the entire population and for subgroups defined by age, education, race/ethnicity, sex, and size of place (p. 122). Methods. Social network analysis (network size and density) and Regression 4

5 Setting Boundaries. The data concern persons with whom respondents have discussed important matters, such as family, finances, health, politics, recreation, etc in the previous six months. The definition of what has to be considered as important is left to respondents. Findings. 1) The average American discussion network is small: nearly 24% of respondents affirmed that they discussed important matters with no one or maximum one person. The mean and mode are three people. 2) Networks draw heavily on kinship as a source of relationship: the average network has a kin/nonkin proportion of ) Alters tend to be densely linked. 4) Networks tend to be highly homogeneous in terms of education (σ <1 year) and race/ethnicity, and heterogeneous in terms of sex. Caveat: Homogeneity is underestimated due to high kin composition of the networks. Subgroup Differences in Network Form. Differences are greatest for subgroups defined by age and education (to a lesser extent by race and size of place, but still appreciable). Age: Overall network size drops with age at an increasing rate, as well as race/ethnicity and sex heterogeneity, while density rises. Education: Mean network size among those holding a college degree is nearly 1.8 times larger than among those who did not finish high school (p.129). Moreover, education brings larger, more varied networks. Race/Ethnicity: Whites have the largest networks (μ=3.1); blacks the smallest (μ=2.25). Sex: Women contain more kin than nonkin, compared to men. Size of Place: Persons in larger places cite more nonkin and fewer kin, in both absolute and relative terms. Network density falls with size, while race/ethnic heterogeneity rises (p. 129). 5

6 Synthesis Chapter 3 in Wasserman and Faust introduces three different network notational schemes including graph theoretic notation, sociometric notation and algebraic notation. These notation schemes can be used to describe, quantify and analyze various social network features. Chapter 4 in Wasserman and Faust is an introduction to graph theory and presents basic matrix operations used in social network analysis. Graph theory is important for social network analysis since it provides mathematical operations and ideas to quantify and measure network properties and graph theorems to investigate network structures. The Marsden article demonstrates how graph theoretic concepts have been applied to quantify network properties (network size and density). This article provides an overview of features of core social networks of Americans, exploiting data from the 1985 General Social Survey through social network analysis and regression. 6

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

Chapter 4 Graphs and Matrices. PAD637 Week 3 Presentation Prepared by Weijia Ran & Alessandro Del Ponte Chapter 4 Graphs and Matrices PAD637 Week 3 Presentation Prepared by Weijia Ran & Alessandro Del Ponte 1 Outline Graphs: Basic Graph Theory Concepts Directed Graphs Signed Graphs & Signed Directed Graphs