Centrality II. MGT 780 Social Network Analysis Steve Borgatti LINKS Center University of Kentucky. 11 April 2016 (c) 2016 Stephen P Borgatti 1

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1 Centrality II MGT 780 Social Network Analysis Steve Borgatti LINKS Center University of Kentucky 1

2 Experimental exchange theory Recruit subjects to play game in which pairs of people have to allocate 24 points between them at each round Game continues multiple rounds, with the goal of accumulating points People can only interact with certain others (determined by the experimenter) Objective is to understand how network position interacts with individual characteristics to determine Ysuccess X in accumulating Y points Y Z X Z Y 2

3 Expectations from centrality In both networks, X should be most powerful node Y X Y Y Z X Z Y Eigenvector scores 3

4 Experimental results According to centrality X in both networks should have most power Y X Y Y Z X Z Y X has most power Z s have most power Two principles of power Power is a function of having multiple potential trading partners Power is a is a function of connection to weak alters In other settings, power is function of access to strong alters 4

5 Counting odd/even paths Principle of power as a function of having alternatives leads an odd/even rule: Nodes that are an odd number of links away from you are a benefit Nodes that are an even number of links away are a detriment Y Z X Z Y

6 Beta centrality in competitive settings Beta can be negative E.g., if beta = -0., then R + = b 0 R 1 + b 1 R 2 + b 2 R 3 + b 3 R R R + + = (.) = 1R 1 0 R 1.R + (.) R R (.).12R 4 2 R (.) 3 R So, odd-length paths add to power, even-length paths subtract from power Y X Y Y Z X Z Y

7 Issues with beta centrality Often highly related to degree How to choose beta? 7

8 Induced centralities Suppose we take importance seriously as a definition of centrality One way to phrase this is: how different would the network be if the node weren t there? Induced centrality approach (aka vitality measures) Start with a network level statistic ( graph invariant ), such as avg distance Remove a node and recalculate the graph invariant Define the node s centrality as the difference between the two invariants IC F (x) = F(G) - F(G-{x}) 8

9 Degree centrality is an IC (Degree centrality = no. of ties a node has) Select as our invariant the number of ties in a graph Removing a node reduces the number of edges in the graph precisely by its degree 2 9

10 Is flow betweenness an IC? Freeman, Borgatti & White (1991) define an alternative to standard betweenness (Freeman, 1977) takes into account all paths between nodes and works with valued data Based on concept of maximum flow x/y gives the actual flow x and the capacity of the pipe y Flow betweenness node X is amount of total flow between nodes that passes through X 11 April 2016 (c) 2016 Stephen P Borgatti 10

11 Problem with calculating flow betweenness Flow solutions are not necessarily unique All 1 units of flow pass through X S 0 X S Max flow solution 1 1 T 10 Capacities 1 T S Max flow solution 2 10 units of flow pass through X X T 11 April 2016 (c) 2016 Stephen P Borgatti 11

12 Solution for flow betweenness For each node: Compute the maximum flow between all possible pairs of nodes and sum up Remove a node, Recalculate the maximum flow Compute the reduction in flow This is clearly an induced centrality 12

13 Are all centralities ICs? If so, provides a new way to define centrality: Centrality as a node s contribution to a graph invariant Particularly useful if the invariant is a measure of cohesion Centrality is a node s contribution to group cohesion More generally, select any group capability (ability to coordinate, ability to transmit information, ability to get things done) that can be measured with and without a node Centrality is a node s contribution to the group success B A 11 April 2016 (c) 2016 Stephen P Borgatti 13

14 No: The case of closeness Closeness centrality sum of distances from node to all others Closeness(A) = 14, closeness(b) = 13 Obvious graph invariant is sum of distances among all pairs of nodes So if we remove A and separately B and recalculate sum of distances in each case, the induced centralities should correspond to closeness scores B A 11 April 2016 (c) 2016 Stephen P Borgatti 14

15 G G - A G - B A A B B Graphs G-A and G-B are isomorphic, so they must have same score on any invariant. Therefore, the induced centrality score for A and B have to be the same But A and B have different closeness scores in the original graph, so closeness cannot be an IC Not all centralities are ICs 11 April 2016 (c) 2016 Stephen P Borgatti 1

16 Empirical example Data (courtesy of Ron Burt) consist of 331 managers at a large data management company Dependent variable is compensation Independent variable is induced centrality where the invariant is no. of pairs of nodes within 3 links of each other Node with a high score serves as a shortcut Control vars include mgr status, age, etc 16

17 Regression results IC related to compensation IC is recognized and rewarded? High performers placed in bridging positions? 17

18 An interesting class of invariants Take any off-the-shelf centrality measure and calculate centrality of each node Sum the centrality scores to get graph-level statistic. This is your invariant. E.g., information centrality Get info centrality for each node. Get sum of scores. This is F(G), the invariant Remove a node x Recalculate information centrality for each node and recalculate the sum of scores. This is F(G-x) IC(x) = F(G) - F(G-x) 18

19 Centrality decomposition Call the off-the-shelf measure of centrality being used to create the invariant the endogenous or direct centrality of the node Call the induced centrality based on it the total centrality of the node Call the difference between total centrality and endogenous centrality exogenous or indirect centrality contribution of other nodes to its centrality Total Centrality = Endogenous Centrality + Exogenous Centrality Total Centrality = Direct Centrality + Indirect Centrality 11 April 2016 (c) 2016 Stephen P Borgatti 19

20 Case of betweenness Calculate betweenness for every node This is the endogenous/direct centrality score Sum the betweenness scores to get graph-level invariant For each node x: Remove the node and recalculate all betweenness scores and sum to get new invariant. Subtract from original invariant. This induced centrality is the total centrality of x Subtract x s endogenous score from its total centrality score. This is its exogenous/indirect score Total Centrality = Endogenous Centrality + Exogenous Centrality 11 April 2016 (c) 2016 Stephen P Borgatti 20

21 Ordinary betweenness Case of betweenness 11 April 2016 Total Cent Endo Cent Exo Cent MEDICI GUADAGNI ALBIZZI SALVIATI RIDOLFI BISCHERI STROZZI 14 9 BARBADORI 14 9 TORNABUONI 13 8 CASTELLANI PERUZZI PAZZI GINORI ACCIAIUOLI LAMBERTESCHI PUCCI Salviati give more than they receive Ridolfi actually reduce the betweenness of other nodes (c) 2016 Stephen P Borgatti Have no betweenness of their own, but contribute to Guadagni s 21

22 Empirical example Data from Ron Burt collected at large data management company Positive ties among 300+ employees along with compensation and evaluation data Control variables include age, seniority, gender, office location, manager status 22

Unit 5: Centrality. ICPSR University of Michigan, Ann Arbor Summer 2015 Instructor: Ann McCranie

Unit 5: Centrality. ICPSR University of Michigan, Ann Arbor Summer 2015 Instructor: Ann McCranie 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