# Week 3. Network Data; Introduction to Graph Theory and Sociometric Notation

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1 Wasserman, Stanley, and Katherine Faust Social Network Analysis: Methods and Applications, Structural Analysis in the Social Sciences. New York, NY: Cambridge University Press. Chapter III: Notation for Social Network Data Notations are a unified set of symbols to identify 1) a set of ; 2) relations themselves; and; 3) actor attributes. Table 1: Different notational schemes Graph theoretic (GT) Sociometric (S) Algebraic (A) Usefulness/most common uses Scope Centrality and prestige methods Cohesive subgroup ideas Dyadic and triadic methods Elementary way to represent and relations Structural equivalence Blockmodels Most common in SNA literature Role and positional analysis Relational algebras Study of multiple relations Based on Graph theory (Ch. 4) Sociometry Algebra Relationships to Consistent with S & A??? other notations Sociomatrices are adjacency matrices for graphs, so consistent with GT Consists of Nodes joined by lines Two-way matrices Algebraic functions Chapter IV: Graphs and matrices Relevance of graphs: 1) Graph theory provides a vocabulary which can be used to label and denote many social structural properties. 2) Graph theory gives us mathematical operations and ideas with which many of these properties can be quantified and measured. 3) Graph theory gives us the ability to prove theorems about graphs, and hence, about representations of social structure. 4) Graph theory gives us a representation of a social network as a model of social system consisting of a set of and the ties between the. Table 2: Core concepts in graphs and matrices Concept not Definition Consists of/calculated from Graph G Model for a social network with an undirected -Nodes (aka vertices, not Represent N n g Salto & Owens Week 3 Summary 1

2 dichotomous relation Subgraph G s Idem graph. Can be either node-generated or linegenerated Dyads Node-generated subgraphs consisting of a pair of nodes and the possible line between the nodes Triads Subgraphs consisting of three nodes and the possible lines among them Nodal degree Walks Trails Paths Geodesic d(n i ) Number of lines that are incident with a node, or the number of nodes adjacent to it. Sequence of nodes and connections, starting out from one node and ending at another node, passing through the connections which join the various nodes of the route made Special type of walk in which all the connection lines are distinct, but the nodes can be repeated Special case of walk in which all of the nodes and connection lines are distinct, and there can be no repetitions The shortest path between two nodes is called geodesic, and the length of this path, in number of intermediate connections, is called geodesic distance points) -Lines (aka edges, arcs) Number of lines which are connected (incident) to this node -Nodes -Lines L l L Importance of the node in the network It allows us to see how many connections and how many nodes are intermediaries in a relationship between two of a network Salto & Owens Week 3 Summary 2

3 Marsden, Peter V. (1987). Core Discussion Networks of Americans. American Sociological Review, 52(1), Purpose and objectives: This paper describes the features of core social networks in the United States, focusing on important relationships as defined by the participants. Important relationships consist of people with whom the respondents discuss important matters. Theoretical framework: Using measure of interpersonal environments as defined by previous literature on networks, this paper emphasized network size, density and heterogeneity. Size refers to the number of important relationships; density measures the closeness of relationships between these people; and, heterogeneity measures diversity of age, education, race/ethnicity and sex. Data and methods: 1985 General Social Survey (GSS) is a nationally representative sample of adults (N=1,531). The GSS gathers data on contemporary American society in order to monitor and explain trends and constants in attitudes, behaviors, and attributes; and, to examine the structure and functioning of society in general as well as the role played by relevant subgroups. Structural variables include alters important relationships; and; relationships between alters. Composition variables include sex, race/ethnicity, age, and religious preference. Respondents generated names of alters, then defined relationships between them as especially close or total stranger. Results: Networks are small (the mean and mode are 3 ranging from 0 to 6+), centered on kin (average kin/nonkin proportion 0.55), dense, and homogenous especially in terms of race/ethnicity and education. Networks are heterogeneous in terms of sex. Analysis of subgroups: Age: network range is greatest among the young and middle-aged. Density rises with age although heterogeneity in race/ethnicity and sex fall. Education: network range grows with education; density decreases. Race/ethnicity: Whites have the largest networks (μ=3.1); blacks and Hispanics smaller. Sex: Women have more kin than nonkin in network. Size of place: network range increases in urban areas; density decreases in rural or less urban places. Significance: Well-educated, young and middle-aged people living in urban areas have more diverse networks. This study represents a baseline study with GSS; investigations into network characteristics and other pertinent questions, such as social support or individual well-being, can be derived from this kind of analysis with this dataset. This article demonstrates how theoretical concepts have been applied to quantify network properties (network size, density, and heterogeneity). Rethemeyer, R. Karl. (2005). Conceptualizing and measuring collaborative networks. Public Administration Review, 65(1), Salto & Owens Week 3 Summary 3

4 Book Review: This article reviews three books which have sought to reconcile policy design and implementation analysis through networks. 1. Walter J.M. Kickert, Erik-Hans Klijn, and Joop F. M. Koppenjan, eds., Managing Complex Networks: Strategies for the Public Sector (London: Sage Publications, 1997). Managing Complex Networks looks at policy design and implementation as network management first by identifying the context of the network and then by conceptualizing collaboration as both ideas and interactions. Networks described are mostly horizontal. 2. Myrna P. Mandell, ed., Getting Results through Collaboration: Networks and Network Structures for Public Policy and Management (Westport, CT: Quorum Books, 2001). Getting Results through Collaboration builds upon the theoretical framework in Managing Complex Networks by incorporating practitioner case studies and exploring the behavioral implications of network-based policymaking. Several chapters are international in orientation. 3. Robert Agranoff, and Michael McGuire, Collaborative Public Management: New Strategies for Local Governments (Washington, DC: Georgetown University Press, 2003). Collaborative Public Management provides an empirical analysis of 267 Midwestern city management in order to examine patterns of linkage, collaborative instrument use and collaborative activity. Networks analyzed are horizontal and vertical. Jurisdiction-based public management is collaborative public management: public policies that build and use relationships to advance the interests of local jurisdictions. The study determines that only 21 percent of cities use jurisdiction-based management. Networks as normative analysis: Managing assumes networks are an integral aspect of policy. Collaborative advocates for horizontal networks as a hedge against vertical limitations of autonomy. The larger the political unit analyzed, the more likely self-organizing networks already exist to contest policy decisions. Getting Results emphasizes cultural and institutional f, varying according to national policy context. The role of public managers: Managing purports that managers are supporting, facilitators rather than leaders of networks. To control or intentionally manipulate a network is to destroy its organic foundation. Collaborative describes the public manager as a central coordinator. Jurisdiction-based management is built on the idea that public managers must pursue what is best for their polity by negotiating with vertical powers and coordinating with horizontal collaborators to achieve it (p.120). Getting Results takes the middle road, viewing managers as somewhere between a facilitator and a creator, depending on the culture and history of public management in a national context. Critique: These three books fail to use social network analysis or theoretical constructs. The structure of indirect connections can determine, for instance, whether government or private Salto & Owens Week 3 Summary 4

5 sector can control information and resource flows, which may affect the payoff to participants in collaborative endeavors. To some extent, what the authors have tried to do in these three volumes is the equivalent of trying to study a monopoly without knowing whether the monopolist sells into a market of one or one thousand. The structure of the market matters in a monopoly; the structure of indirect connections may matter in collaboration. (p120). Salto & Owens Week 3 Summary 5

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