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Systems: Essential Ideas

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  1. Networks: Basic Concepts Centrality

  2. Networks: Basic Concepts • In this discussion, we’ll outline some basic concepts of network analysis, focusing on centrality. • We’ll also fold into this discussion an overview of UCINET. UCINET is a software program that is commonly used with network analysis. While it does not handle some of the more recent ways in which networks can be analyzed (such as longitudinal or cross-sectional ERGM methods), it is a very user friendly way to obtain network-related measures, and to create visual depictions of networks. UCINET offers a free 30 day trial. • Much of the discussion of UCINET was drawn from the tutorial created by Hanneman at Riverside.

  3. Basic concepts • Recall—vertices or nodes are the units or actors in a network (or a graph or a system). • Edges are the ties or connections between nodes. • And the “ego” is the node under consideration—any particular node that you might be thinking of.

  4. Basic Concepts: Centrality • Centrality is a measure of how many connections one node has to other nodes. • Degree centrality refers to the number of ties a node has to other nodes. Actors who have more ties may have multiple alternative ways and resources to reach goals—and thus be relatively advantaged.

  5. Basic Concepts: Degree Centrality • Degree centrality for an undirected graph is straightforward—if A is connected to B, then B is by definition connected to A. • Degree centrality for a directed graph or network has one of two forms.

  6. Degree Centrality—Directed Networks • One is in-degree centrality: An actor who receives many ties, they are characterized as prominent. The basic idea is that many actors seek to direct ties to them—and so this may be regarded as a measure of importance. • The other is out-degree centrality. Actors who have high out-degree centrality may be relatively able to exchange with others, or disperse information quickly to many others. (Recall the strength of weak ties argument.) So actors with high out-degree centrality are often characterized as influential.

  7. Degree Centrality: Individual and Network • Consider the network on the left. Which nodes (actors) are more “central” than others? • 2, 5, and 7 appear relatively “central”.

  8. Degree Centrality (directed networks) • So, node 7 has an in-degree centrality absolute value of 9 (there are 9 other nodes connected to node 7). The normalized value is 100 (all possible other nodes are connected to node 7). The out-degree centrality has an absolute value of 3 (node 7 is connected out to nodes 2, 4, and 5), and a normalized value of 33.33 (3 nodes is 33.33% of the possible 9 nodes to which node 7 could extend out.) • The average outdegree is 4.9 (which means that each node has, on average, connections out to 4.9 other nodes); the average indegree is also 4.9. Normalized, both measures are 54.44 (that is, 4.9 / 9).

  9. Centrality: Network Degree Centralization • One can also calculate network indegree and outdegree centralization. These network measures represent the degree of inequality or variance in our network as a percentage of that in a perfect “star network” – the most unequal type of network. • A depiction of a star network is on the next slide—note that only one node is connected to any of the others, and that node is connected to all of the others.

  10. Star Network

  11. Degree Centrality: Bonacich • Another measure of degree centrality takes into account the problem that the power and centrality of each node (actor) depends on the power and centrality of the others. • Bonacich used an iterative estimation approach which weights each node’s centrality by the centrality of the other nodes to which it is connected. • So, node 1’s centrality depends not only on how many connections it has—but also on how many connections its neighbors have (and on how many connections its neighbors’ neighbors have, and so on.)

  12. Degree Centrality: Bonacich • When calculating out the Bonacich Power measures, the “attenuation factor” represents the weight—an “attenuation factor” that is positive (between 0 and 1) means that one’s power is enhanced by being connected to well-connected neighbors. • Alternatively, one could argue that actors who are well-connected to individuals who are not well-connected themselves are powerful, because others are “dependent” on them. In this case, one would use an negative “attenuation factor” (between 0 and -1), to compute power accordingly.

  13. Degree Centrality: Bonacich • Recall the graph presented above, in which actors #5 and #2 were the most central. Calculating out Bonacich measures suggests that actors #8 and #10 are also central—they don’t have many connections, but they have the “right” connections. • However, taking the second approach (using a negative attenuation factor) identifies actors 3, 7, and 9 as being strong – because they have weak neighbors (who are “dependent” on them).

  14. Degree Centrality: Bonacich • As with all quantitative methods, it’s important to think about what you as a researcher are trying to measure before using the methods. In your particular context, are actors connected with other well-connected actors the most powerful? Or is it actors that are connected with those who are very dependent on them who are more powerful?

  15. Centrality: Closeness Centrality • Closeness is a measure of the degree to which an individual is near all other individuals in a network. It is the inverse of the sum of the shortest distances between each node and every other node in the network. • Closeness is the reciprocal of farness. • Nearness can also be standardized by norming it against the minimum possible nearness for a graph of the same size and connection.

  16. Centrality: Closeness Centrality • Closeness can also be calculated as a measure of inequality in the distribution of distances across the actors. • These measures rely on the sum of the geodesic distances from each actor to all the others. However, in complicated graphs, this can be misleading. • An actor can be very close to a relatively closed subset of a network—or moderately close to every actor in a large network—and receive the same closeness score. In reality, the two are very different.

  17. Centrality: Eigenvector Closeness • The Eigenvector approach to measuring closeness uses a factor analytic procedure to discount closeness to small local subnetworks.

  18. Closeness: Influence Measures • Another way to think of closeness is to move away from thinking just about the geodesic or most efficient (shortest) path from one node to another—but to also think about all connections of ego (that is, the one node in question) to all the others.

  19. Closeness: Influence Measures • There are several such measures: Hubbell, Katz, Taylor, Stephenson, and Zelen. • Hubbell and Katz methods count the total number of connections between actors (and do not distinguish between directed and non-directed data), but use an attenuation factor to discount longer paths. The two measures are very similar; the Katz measure uses an identity matrix (each node is connected to itself) while the Hubbell measure does not.

  20. Closeness: Influence Measures • The Taylor measure also uses an attenuation factor, but is more useful for measuring the balance of in- versus out-ties in directed graphs. Positive values of closeness indicate relatively more out-ties than in-ties.

  21. Centrality: Actor Betweenness • Betweenness—Betweenness is a measure of the extent to which a node is connected to other nodes that are not connected to each other. It’s a measure of the degree to which a node serves as a bridge. • This measure can be calculated in absolute value, as well as in terms of a normed percentage of the maximum possible betweenness that an actor or node could have had.

  22. Centrality: Edge Betweenness • In addition to calculating betweenness measures for actors, we can also calculate betweenness measures for edges. • Edge betweenness is the degree to which an edge makes other connections possible. • Recall the Knoke example we used earlier, and look at the edge from 3 to 6.

  23. Centrality: Edge Betweenness

  24. Centrality: Edge Betweenness • That edge from 3 to 6 makes many other edges possible—without that edge, 6 would be relatively isolated.

  25. Centrality: Levels of Hierarchy • One can also identify levels of hierarchy. If one eliminates all the actors with no betweenness (that is, the “subordinates”), some of the remaining actors will then have 0 betweenness—they are at the second level of the hierarchy. We can continue to remove actors, and measure the # of levels of hierarchy exist in the network or system. • Note that the Knoke data presented above is not very hierarchical.

  26. Centrality: Flow Betweenness • What if two actors want to have a relationship, but the path between them is blocked by a reluctant intermediary? Another pathway—even if it is longer—means another alternative / resource. The flow approach to centrality assumes that actors will use all the pathways that connect them. For each actor, the measure reflects the # of times the actor is in a flow (any flow) between all other pairs of actors (generally, as a ratio of the total flow betweenness that does not involve the actor).

  27. Basic Concepts • This has been an overview of various perspectives on centrality, largely drawn from the UCINET tutorial. The UCINET tutorial also has a number of very useful review questions.