Terrorist Networks, Network Energy and Node Removal: A New Measure of Centrality Based on Laplacian Energy

Xingqin Qi, Robert D. Duval, Kyle Christensen, Edgar Fuller, Arian Spahiu, Qin Wu, Yezhou Wu, Wenliang Tang, Cunquan Zhang

Department of Mathematics West Virginia University, Morgantown, USA.
Department of Political Science Columbus State University, Columbus, USA.
Department of Political Science West Virginia University, Morgantown, USA.
School of IOT Engineering Jiangnan University, Wuxi, China.
School of Mathematics and Statistics Shandong University (Weihai), Weihai, China; Department of Mathematics West Virginia University, Morgantown, USA.

DOI: 10.4236/sn.2013.21003                

Abstract

In this work we propose a centrality measure for networks, which we refer to as Laplacian centrality, that provides a general framework for the centrality of a vertex based on the idea that the importance (or centrality) of a vertex is related to the ability of the network to respond to the deactivation or removal of that vertex from the network. In particular, the Laplacian centrality of a vertex is defined as the relative drop of Laplacian energy caused by the deactivation of this vertex. The Laplacian energy of network G with n vertices is defined as , where  is the eigenvalue of the Laplacian matrix of G. Other dynamics based measures such as that of Masuda and Kori and PageRank compute the importance of a node by analyzing the way paths pass through a node while our measure captures this information as well as the way these paths are “redistributed” when the node is deleted. The validity and robustness of this new measure are illustrated on two different terrorist social network data sets and 84 networks in James Moody’s Add Health in school friendship nomination data, and is compared with other standard centrality measures.

Keywords

Network; Centrality; Laplacian Energy; 9/11 Hijacking; Bali Bombing; Terrorism

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Z. Maoz, D. Kuperman, L. Terris and I. Talmud, “Structural Equivalence and International Conflict: A Social Networks Analysis,” Journal of Conflict Resolution, Vol. 50, No. 5, 2006, pp. 664-689. doi:10.1177/0022002706291053
[2]	M. Sageman, “Understanding Terror Networks,” Philadelphia, University of Pennsylvania Press. 2004.
[3]	J. T. Scholz, R. Berardo and B. Kile, “Do Networks Enhance Cooperation? Credibility, Search, and Collaboration,” Journal of Politics, Vol. 70, No. 2, 2008, pp. 393-406.
[4]	J. Fowler, B. Grofman and N. Masuoka, “Social Networks in Political Science: Hiring and Placement of Ph. Ds, 1960-2002. Political Science: 2007, pp. 727-739.
[5]	S. Sreenivasan, R. Cohen, E. Lopez, Z. Toroczkai and H. E. Stanley 2007. Phys. Rev. E 75, 036105. doi:10.1103/PhysRevE.75.036105
[6]	M. O. Jackson, “Presenting and Measuring networks,” Chapter 2, Social and Economic Networks, Princeton University Press, Princeton, 2008.
[7]	T. Opsahl, F. Agneessens and J. Skvoretz, “Node Centrality in Weighted Networks: Generalizing Degree and Shortest Paths,” Social Networks, Vol. 32, No. 3, 2010, pp. 245-251. doi:10.1016/j.socnet.2010.03.006
[8]	F. Jordan and I. Scheuring, “Network Ecology: Topological Constraints on Ecosystems Dynamics,” Physics of Life Reviews, Vol. 1, No. 3, 2004, pp. 139-172. doi:10.1016/j.plrev.2004.08.001
[9]	F. Jordan, W. C. Liu and A. J. Davis, “Topological Keystone Species: Measures of Positional Importance in Food Webs,” Oikos, Vol. 112, No. 3, 2006, pp. 535-546. doi:10.1111/j.0030-1299.2006.13724.x
[10]	M. Lazic, “On the Laplacian Energy of a Graph,” Cze- choslovak Mathematical Journal, Vol. 56, No. 131, 2006, pp. 1207-1213.
[11]	S. Koschade, “A Social Network Analysis of Jemaah Islamiyah: The Applications to Counterterrorism and Intelligence,” Studies in Conflict and Terrorism, Vol. 29, No. 6, 2006, pp. 559-575. doi:10.1080/10576100600798418
[12]	V. Krebs, “Uncloaking Terrorist Networks,” First Monday, Vol. 7, No. 4, 2002, 1st April.
[13]	F. Chung, “Spectral Graph Theory,” American Mathematical Society, USA1997.
[14]	J. Magouirk, S. Atran and M. Sageman, “Connecting Terrorist Networks,” Studies in Conflict and Terrorism, Vol. 31, No. 1, 2008, pp. 1-16.
[15]	T. H. Kean et al., “The 9/11 Commission Report,” W. W. Norton and Company, New York, 2004.
[16]	M. Hamada, K. Sato and K. Asai, “Improving the Accuracy of Predicting Secondary Structure for Aligned RNA Sequences,” Nucleic Acids Research, 1-10. 2010.
[17]	M. Xu, Z. Su, “Computational Prediction of cAMP Receptor Protein (CRP) Binding Sites in Cyanobacterial Genomes,” BMC Genomics, 10-23. 2009.
[18]	S. Pemmaraju and S. Skiena, “All-Pairs Shortest Paths and Transitive Closure and Reduction. In: Computational Discrete Mathematics: Combinatorics and Graph Theory in Mathematica. Cambridge, England, Cambridge University Press, 2003. pp. 330-331 and 353-356.
[19]	B. C. Csaji, R. M. Jungers and V. D. Blondel, “Pagerank Optimization by Edge Selection,” The Computing Research Repository abs/0911.2280. 2009.