Actor Allegiance and Blockmodel Strength J. T. Rigsby and Dr. J. L. - PowerPoint PPT Presentation

Actor Allegiance and Blockmodel Strength J. T. Rigsby and Dr. J. L. Solka rigsbyjt@nswc.navy.mil;solkajl@nswc.navy.mil Naval Surface Warfare Center Dahlgren Division Interface 2004 – p.1/24

Agenda Blockmodel Necessity Allegiance Toy Example Real Example Conclusions Interface 2004 – p.2/24

Acknowledgments This work is supported by the In-house Laboratory Independent Research (ILIR) program. We wish to acknowledge helpful discussions with Dr. David Marchette of NSWCDD. Interface 2004 – p.3/24

Blockmodel D - directed graph E - edges(ties) V - vertexes(actors) P = { P 1 , ..., P k } - Partitions B = { B 1 , 1 , B 1 , 2 , ..., B n,n − 1 , B n,n } - Blockmodel B i,j , when i � = j - ties from actors in partition i to partition j B i,j , when i = j - internal ties between actors in partition i Interface 2004 – p.4/24

Necessity Measure blockmodel strength change Estimate the true number of partitions Turn the time separated graphs into a single graph Interface 2004 – p.5/24

Allegiance is the measure of how much an actor is helping his block Help: Having internal edges to other block members H int ( actor ) = � i = N Pi E ( actor ) Not having external edges to non-blockmembers H ext ( actor ) = N − N P i − � i � = N Pi E ( actor ) Interface 2004 – p.6/24

Allegiance Equation A ( i ) = α [ H int ( i )] + (1 − α )[ H ext ( i )] α ∈ [0 , 1] - weight function Interface 2004 – p.7/24

Toy Example Interface 2004 – p.8/24

Equivalence Cluster Interface 2004 – p.9/24

k = 4 Interface 2004 – p.10/24

Measure Change in Allegiance DA ( i,k ) = A i,k − A i,k − 1 for k > 1 At k = 1 DA ( i, 1) = A i, 1 Out degree of actor i Interface 2004 – p.11/24

Plot of Change in Allegiance Interface 2004 – p.12/24

Sum of Allegiance Change SDA k = � n i =1 DA ( i,k ) Used to estimate k : the number of partitions Interface 2004 – p.13/24

Plot of SDA k Interface 2004 – p.14/24

Real Example 5 users Oct 2002 - Feb 2004 3000 total servers accessed over 14 months 31 total clients 19 LAN computers Interface 2004 – p.15/24

Equivalence Cluster Interface 2004 – p.16/24

Individual allegiance Change Interface 2004 – p.17/24

Total Allegiance Change Interface 2004 – p.18/24

k = 4 Interface 2004 – p.19/24

k = 5 Interface 2004 – p.20/24

k = 6 Interface 2004 – p.21/24

k = 7 Interface 2004 – p.22/24

k = 8 Interface 2004 – p.23/24

Conclusions Allegiance Can be used to estimate number of partitions Used to estimate maximum number of partitions at a minimum Gives a measure for block strength change as partition size changes Interface 2004 – p.24/24