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Introduction to Social Network Analysis Lluís Coromina Departament d’Economia. Universitat de Girona Girona, 18/01/2005

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Social network analysis is: a set of relational methods for systematically understanding and identifying connections among actors Introduction

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Network analysis assumes that: How actors behave depends in large part on how they are linked together Example: Adolescents with peers that smoke are more likely to smoke themselves. The success or failure of organizations may depend on the pattern of relations within the organization Example: The ability of companies to survive strikes depends on how product flows through factories… Patterns of relations reflect the power structure of a given setting, and clustering may reflect coalitions within the group Example: Overlapping voting patterns in a coalition government Introduction

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Origins of network analysis: Beginning in the 1930s, a systematic approach to theory and research, based on the notion that relations matter, began to emerge In 1934 Jacob Moreno introduced the ideas and tools of sociometry At the end of World War II, Alex Bavelas founded the Group Networks Laboratory at M.I.T. Basic concepts

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From the outset, the network analysis has been: a. guided by formal theory organized in mathematical terms, and b. grounded in the systematic analysis of empirical data In the 1970s, when modern discrete combinatorics (esp. graph theory) developed rapidly and powerful computers became readily available that the study of social networks began to flourish Basic concepts

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Actors (nodes, points, vertices): - Individuals, Organizations, Events … Relations (lines, arcs, edges, ties): between pairs of actors. - Undirected (symmetric) / Directed (asymmetric) - Binary / Valued Basic concepts Network Components

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1) Egocentered Networks Data on a respondent (ego) and the people they are connected to. Measures: Size Types of relations Basic concepts Types of network data:

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2) Complete Networks Connections among all members of a population. Data on all actors within a particular (relevant) boundary. Never exactly complete (due to missing data), but boundaries are set Ex: Friendships among workers in a company. Measures: Graph properties Density Sub-groups Positions Background Types of network data:

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The unit of interest in a network are the combined sets of actors and their relations. We represent actors with points and relations with lines. Example: Social Network data a b ce d

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In general, a relation can be: Undirected / Directed Binary / Valued a b ce d Undirected, binary Directed, binary a b ce d a b ce d Undirected, Valued Directed, Valued a b ce d 13 4 2 1 Social Network data

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From pictures to matrices Undirected, binaryDirected, binary abcde a b c d e 1 1 1 1 1 1 1 abcde a b c d e 1 1 1 1 1 1 1 1 1 1 Basic Data Structures Social Network data a b ce d a b ce d

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From matrices to lists abcde a b c d e 1 1 1 1 1 1 1 1 1 1 a b b a c c b d e d c e e c d a b b a b c c b c d c e d c d e e c e d Adjacency List Arc List Basic Data Structures Social Network data

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d e c Indirect connections are what make networks systems. One actor can reach another if there is a path in the graph connecting them. a b ce d f bf a Connectivity Measuring Networks

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Basic elements: A path is a sequence of nodes and edges starting with one node and ending with another, tracing the indirect connection between the two. On a path, you never go backwards or revisit the same node twice. Example: a b c d A walk is any sequence of nodes and edges, and may go backwards. Example: a b c b c d A cycle is a path that starts and ends with the same node. Example: a b c a Measuring Networks Connectivity

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If you can trace a sequence of relations from one actor to another, then the two are connected. If there is at least one path connecting every pair of actors in the graph, the graph is connected and is called a component. Intuitively, a component is the set of people who are all connected by a chain of relations. Measuring Networks Connectivity

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Distance is measured by the (weighted) number of relations separating a pair, Using the shortest path. Actor “a” is: 1 step from 4 2 steps from 5 3 steps from 4 4 steps from 3 5 steps from 1 Distance & number of paths Measuring Networks a

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Paths are the different routes one can take. Node-independent paths are particularly important. There are 2 independent paths connecting a and b. There are many non- independent paths Distance & number of paths Measuring Networks a b

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An information network: Email exchanges within the Reagan white house, early 1980s (source: Blanton, 1995) Measuring Networks

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Power positions and potential influence Measuring Networks

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Centrality refers to (one dimension of) location, identifying where an actor resides in a network. Centrality Measuring Networks Centrality is fairly straight forward: we want to identify which nodes are in the ‘center’ of the network. In the sense that they have many and important connections. Three standard centrality measures capture a wide range of “importance” in a network: Degree Closeness Betweenness

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The most intuitive notion of centrality focuses on degree. Degree is the number of lines, and the actor with the most lines is the most important: Centrality Measuring Networks

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Centrality Measuring Networks Relative measure of Degree Centrality: Degree Centrality:

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A second measure is closeness centrality. An actor is considered important if he/she is relatively close to all other actors. Closeness is based on the inverse of the distance of each actor to every other actor in the network. Closeness Centrality: Relative Closeness Centrality Centrality Measuring Networks

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Closeness Centrality Centrality Measuring Networks

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Betweenness Centrality: Model based on communication flow: A person who lies on communication paths can control communication flow, and is thus important. Betweenness centrality counts the number of shortest paths between i and k that actor j resides on. b a C d e f g h Centrality Measuring Networks

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Centrality Measuring Networks Betweenness centrality can be defined in terms of probability (1/g ij ), C B (p k ) = i ij (p k ) = = g ij = number of geodesics that bond actors p i and p j. g ij (p k )= number of geodesics which bond p i and p j and content p k. i ij (p k ) = probability that actor p k is in a geodesic randomly chosen among the ones which join p i and p j. Betweenness centrality is the sum of these probabilities (Freeman, 1979). Normalizad: C’ B (p k ) = CB(pk) / [(n-1)(n-2)/2]

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Betweenness Centrality: Centrality Measuring Networks

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Comparing across centrality values Generally, the 3 centrality types will be positively correlated When they are not correlated, it probably tells you something interesting about the network. Low Degree Low Closeness Low Betweenness High Degree Embedded in cluster that is far from the rest of the network Ego's connections are redundant - communication bypasses him/her High Closeness Key player tied to important important/active alters Probably multiple paths in the network, ego is near many people, but so are many others High Betweenness Ego's few ties are crucial for network flow Very rare cell. Would mean that ego monopolizes the ties from a small number of people to many others. (hidden) Centrality Measuring Networks

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If we want to measure the degree to which the graph as a whole is centralized, we look at the dispersion of centrality: Freeman’s general formula for centralization (which ranges from 0 to 1): Centralization Measuring Networks

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Degree Centralization Scores Freeman: 1.0 Freeman:.02 Freeman: 0.0 Centralization Measuring Networks

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Density Measuring Networks The more actors are connected to one another, the more dense the network will be. Undirected network: n(n-1)/2 = 2n-1 possible pairs of actors. Δ = Directed network: n(n-1)*2/2 = 2n-2possible lines. Δ D =

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Freeman:.25 Freeman:.23 Freeman: 0.25 Density Measuring Networks

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UCINET The Standard network analysis program, runs in Windows Good for computing measures of network topography for single nets Input-Output of data is a special 2-file format, but is now able to read PAJEK files directly. Not optimal for large networks Available from: Analytic Technologies Social Network Software

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PAJEK Program for analyzing and plotting very large networks Intuitive windows interface Started mainly a graphics program, but has expanded to a wide range of analytic capabilities Can link to the R statistical package Free Available from: http://vlado.fmf.uni-lj.si/pub/networks/pajek/ Social Network Software

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NetDraw Also very new, but by one of the best known names in network analysis software. Free Social Network Software

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