Unit 5: Centrality ICPSR University of Michigan, Ann Arbor Summer 2015 Instructor: Ann McCranie
What does centrality tell us? We often want to know who the most important actors in a network are. Centrality is one way to do this. There are many (dozens) of different kinds of centrality measures. Here are three of the most commonly used: Degree Closeness Betweenness
Things to keep in mind Centrality is a measure of an actor, centralization is a measure of the network. It matters whether you are considering a directed or an undirected network. Consider whether the measure is on ties sent and received (centrality) or ties received (prestige) Most centrality measures work on binary/ unweighted networks. Some have been extended to weighted networks. The three ideal networks we are about to use are all undirected and binary.
Example: Star N1 N2 N3 N4 N5 N6 N7 N1 0 1 1 1 1 1 1 N2 1 0 0 0 0 0 0 N3 1 0 0 0 0 0 0 N4 1 0 0 0 0 0 0 N5 1 0 0 0 0 0 0 N6 1 0 0 0 0 0 0 N7 1 0 0 0 0 0 0
Example: Circle N1 N2 N3 N4 N5 N6 N7 N1 0 1 0 0 0 0 1 N2 1 0 1 0 0 0 0 N3 0 1 0 1 0 0 0 N4 0 0 1 0 1 0 0 N5 0 0 0 1 0 1 0 N6 0 0 0 0 1 0 1 N7 1 0 0 0 0 1 0
Example: Line N1 N2 N3 N4 N5 N6 N7 N1 0 1 1 0 0 0 0 N2 1 0 0 1 0 0 0 N3 1 0 0 0 1 0 0 N4 0 1 0 0 0 1 0 N5 0 0 1 0 0 0 1 N6 0 0 0 1 0 0 0 N7 0 0 0 0 1 0 0
Actor Degree Centrality In an undirected graph, the number of actor s ties. A highly central actor is where the action is and has connections with many other actors, relatively speaking. This person in a major conduit of whatever the content of the relationship is: information, affection, infectiousness. An actor with a low degree centrality has few ties.
Actor Degree Centrality Actor Degree Centrality (W&F, 178) Note, it is dependent on the size of the group. So...standardized, with range of 0-1:
Group Degree Centralization A centralization measure quantifies the range or variability of the individual actor indices. Most commonly used, reported in UCINET. Standardized:
Actor Degree Centrality and Group Centralization Actor Star Circle Line N 1 1.0 0.333 0.333 N 2 0.167 0.333 0.333 N 3 0.167 0.333 0.333 N 4 0.167 0.333 0.333 N 5 0.167 0.333 0.333 N 6 0.167 0.333 0.167 N 7 0.167 0.333 0.167 C D 1.0 0.0 0.067
Degree Centrality Illustrated Fig. 1 from The Privileging of Communitarian Ideas: Citation Practices and the Translation of Social Capital in Public Health Research. S Moore, A Shiell, P Hawe, VA Haines - American Journal of Public Health, 2005" " (Actually shows indegree centrality) "
Actor Closeness Centrality An actor is close to all other actors in the network. If this actor is central by this measure, they can interact with all others. If they aren t connected to everyone, the geodesics, or shortest paths, should be as short as possible. This person can get to many others quickly. An actor who has very low closeness centrality takes many more steps to get to everyone. We are calculating distance using the sum of geodesic distances, the standard Freeman measure. There are other calculations of distances (sum of reciprocal distances, average of reverse distances (Valente and Foreman), all trails, all paths.
Actor Closeness Centrality Actor Closeness Centrality (W&F, 184-185) Again, it is dependent on the size of the group. So...standardized:
Group Closeness Centralization Essentially: the sum of the differences between the maximum standardized actor centrality measure and each actor s centrality divided by the theoretical maximum of these summed differences. Range: 0-1
Actor Closeness Centrality and Group Closeness Centralization Actor Star Circle Line N 1 1.0 0.5 0.5 N 2 0.545 0.5 0.46 N 3 0.545 0.5 0.46 N 4 0.545 0.5 0.38 N 5 0.545 0.5 0.38 N 6 0.545 0.5 0.28 N 7 0.545 0.5 0.286 C C 1.0 0.0 0.277
Closeness Centrality Illustrated Blue to red hue indicates closeness centrality From Wikipedia http://en.wikipedia.org/wiki/image:graph_betweenness.svg"
Actor Betweenness Centrality An actor has a high betweenness centrality when they occupy a position in the geodesics connecting many pairs of other actors in the network. This doesn t mean they have the most ties, just the most that connect the most other people... High betweenness actors can be thought of as a cutpoint in the shortest path connecting two other nodes, therefore being a critical link between them. Think of terrorist networks. Actors who have few unique ties would not be as central. We are considering Freeman Betweenness Centrality, but there are other types.
Actor Betweenness Centrality W&F (pg 190-191) g jk (n i ) = # of geodesics between n j and n k containing n i g jk = # of geodesics connecting n j and n k sum of percentages of all geodesics which contain n i Betweenness Centrality Standardized
Actor Betweenness Centrality and Group Betweenness Centralization Actor Star Circle Line N 1 1.0 0.2 0.6 N 2 0 0.2 0.533 N 3 0 0.2 0.533 N 4 0 0.2 0.33 N 5 0 0.2 0.33 N 6 0 0.2 0 N 7 0 0.2 0 C B 1.0 0.0 0.311
Example: Betweenness If you add lines from C to D and from D to H, you remove the high betweenness centrality of E and F. "The Strength of Strong Ties: The Importance of Philos in Organizations." In N. Nohria & R. Eccles (eds.), Networks and Organizations: Structure, Form,and Action: 216-239. Boston, MA: Harvard Business School Press. ""!!
For Directed Relations: Prestige and Rank Centrality focuses on choices made we can measure degree, closeness Prestige focuses on choices received we can measure degree, proximity, rank
Prestige Object of many ties, direct or indirect. Must be able to distinguish choices received from choices made. have to have a directed relation. Sometimes called status, deference, popularity, rank. P A (n i ) = actor level prestige index for actor i P A = group level prestige A = D, P, R Group level indices: variances
Prestige In-Degree divide the number of ties received by g-1 P D (n i ) = x +i / (g-1) Proximity like closeness, the ratio between the proportion of actors in the influence domain to the average distance of these actors to actor i. I i = # of actors who can reach n i Sum taken over all actors in n i s influence domain Between 0 (unreachable) and 1 (everyone can reach n i ) Can calculate variance for both of these to get group level index Nan Lin s index (1976)
A Rank solution: Eigenvector Centrality A node, using this measure, is important because they are connected to important others. Eigenvector centrality can be thought of as an extended form of degree centrality, in which we take into account not only how many neighbors a vertex has but also how central those neighbors themselves are. (Newman, Mark (2010) Networks: An Introduction)
Eigenvector Centrality Can be calculated in different ways, but commonly it is a matrix-based calculation in which node j s centrality is determined by the jth entry in the eigenvector corresponding to the largest positive eigenvalue
An ideal example. Hansen, Derek (2010). Analyzing Social Media Networks with NodeXL: Insights from a Connected World
An ideal example. Not center of attention But a broker of information Hansen, Derek (2010). Analyzing Social Media Networks with NodeXL: Insights from a Connected World
Padgett s Business Relation Bonacich Eigenvector Centralities 1 2 Eigenvec neigenvec --------- --------- 1 ACCIAIUOLI 0.000 0.000 2 ALBIZZI 0.000 0.000 3 BARBADORI 0.390 55.195 4 BISCHERI 0.344 48.676 5 CASTELLANI 0.391 55.300 6 GINORI 0.191 26.943 7 GUADAGNI 0.235 33.243 8 LAMBERTESCHI 0.435 61.495 9 MEDICI 0.241 34.097 10 PAZZI 0.073 10.288 11 PERUZZI 0.471 66.583 12 PUCCI -0.000-0.000 13 RIDOLFI 0.000 0.000 14 SALVIATI 0.073 10.288 15 STROZZI -0.000-0.000 16 TORNABUONI 0.073 10.288
A Directed Example Right now, UCINET doesn t implement in- out- Eigenvector, so Open KA in Netdraw Analysis->Centrality. Choose the directed version Size the nodes based on in-degree eigenvector Use the egonet viewer as shown above and step through the different ego inneighborhoods to see the differences in ego s neighbors centrality
For more information Newman (2010) provides a very technical discussion of the different ways of calculating centrality by using eigenvectors. This is a good tutorial on calculating eigenvectors. You do need to have familiarity with matrix algebra. http://www.miislita.com/information-retrieval-tutorial/matrix-tutorial-3- eigenvalues-eigenvectors.html
There are many more types of centrality Bonacich Power (Bonacich P (1987). Power and Centrality: A family of Measures. American Journal of Sociology 92, 1170-1182) Flow betweenness - Freeman L C, Borgatti S P and White D R (1991). Centrality in valued graphs: A measure of betweenness based on network flow Social Networks 13, 141-154. Information - Stephenson K and Zelen M (1991). Rethinking Centrality Social Networks 13. Random Walk Betweenness - Newman, MEJ. (2005) A measure of betweenness centrality based on random walks. Social Networks,
Two-Mode Centrality Fig. 4. Strongest weighted cosponsorship ties in the full Senate network, 1973 2004. Note: Size of each vertex is proportional to the Senator's connectedness score, the width of each arrow is proportional to the weighted quantity of bills cosponsored wij (values of wij < 10 not shown) and vertices that represent the top 20 Senators are identified by name. Figure drawn using Kamada Kawai algorithm in Pajek. From: Legislative cosponsorship networks in the US House and Senate Social Networks, Volume 28, Issue 4, October 2006, Pages 454-465 James H. Fowler
Caution! This is an active field of research with regular innovation. Be careful when you are considering centrality measures that involve stuff traveling through networks. For instance, do the things being passed on get shared like diseases or are they passed on like packages? The examples shown above are common examples of centrality, but such off the shelf solutions often don t get it right. Borgatti. 2005. Centrality and network flow. Social Networks, 27(1) Pages 55-71
A few more resources Chapter 5, Wasserman and Faust Chapter 4, Extending Centrality in Carrington, Scott, and Wasserman Borgatti and Everett. 2006. A Graphtheoretic perspective on centrality. Social Networks. 28(4): 466-484. (for undirected) Friedkin. 1999. Theoretical Foundations for Centrality Measures. The American Journal of Sociology 96(6):1478-1504
For tomorrow Assignment due Tuesday night. Next assignment due on Monday. Will require that you have worked through Lab 6.
RANK Most interesting prestige index P R (n i ) = x 1i P R (n 1 )+x 2i P R (n 2 )+ +x gi P R (n g ) or p = p X, where p = (P R (n 1 ),P R (n 2 ),,P R (n g )) or (I X )p = 0 Variety of suggestions for a solution: Katz: standardize X to have column sums of unity first (largest) eigenvalue of standardized X will be 1. corresponding eigenvector is p.