An optimal approach for the critical node problem using semidefinite programming

Detecting critical nodes in complex networks (CNP) has great theoretical and practical significance in many disciplines. The existing formulations for CNP are mostly, as we know, based on the integer linear programming model. However, we observed that, these formulations only considered the sizes but neglected the cohesiveness properties of the connected components in the induced network. To solve the problem and improve the performance of CNP solutions, we construct a novel nonconvex quadratically constrained quadratic programming (QCQP) model and derive its approximation solutions via semidefinite programming (SDP) technique and heuristic algorithms. Various types of synthesized and real-world networks, in the context of different connectivity patterns, are used to validate and verify the effectiveness of the proposed model and algorithm. Experimental results show that our method improved the state of the art of the CNP.

[1]  P. Grindrod.  Range-dependent random graphs and their application to modeling large small-world Proteome datasets. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Sheng Liu,et al.  An Efficient Approach to Solve the Large-Scale Semidefinite Programming Problems , 2012 .

[3]  Reuven Cohen,et al.  Efficient immunization strategies for computer networks and populations. , 2002, Physical review letters.

[4]  Pierre Hansen,et al.  A branch and cut algorithm for nonconvex quadratically constrained quadratic programming , 1997, Math. Program..

[5]  Stephen P. Boyd,et al.  Graph Implementations for Nonsmooth Convex Programs , 2008, Recent Advances in Learning and Control.

[6]  Panos M. Pardalos,et al.  Detecting critical nodes in sparse graphs , 2009, Comput. Oper. Res..

[7]  Xin Zhou.  A novel critical node in generalized networking , 2014, 2014 20th International Conference on Automation and Computing.

[8]  Ee-Peng Lim,et al.  DelayFlow centrality for identifying critical nodes in transportation networks , 2013, 2013 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM 2013).

[9]  Timothy A. Davis,et al.  The university of Florida sparse matrix collection , 2011, TOMS.

[10]  Zhou Tao,et al.  Epidemic dynamics on complex networks , 2006 .

[11]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[12]  My T. Thai,et al.  Detecting Critical Nodes in Interdependent Power Networks for Vulnerability Assessment , 2013, IEEE Transactions on Smart Grid.

[13]  Alain Billionnet,et al.  Improving the performance of standard solvers for quadratic 0-1 programs by a tight convex reformulation: The QCR method , 2009, Discret. Appl. Math..

[14]  Yinyu Ye,et al.  Interior point algorithms: theory and analysis , 1997 .

[15]  Han Zhao,et al.  Identifying influential nodes in complex networks with community structure , 2013, Knowl. Based Syst..

[16]  Massimo Marchiori,et al.  Error and attacktolerance of complex network s , 2004 .

[17]  Yicheng Zhang,et al.  Identifying influential nodes in complex networks , 2012 .

[18]  Hua Yu,et al.  The node importance in actual complex networks based on a multi-attribute ranking method , 2015, Knowl. Based Syst..

[19]  Stephen P. Borgatti,et al.  Identifying sets of key players in a social network , 2006, Comput. Math. Organ. Theory.

[20]  Clayton W. Commander,et al.  Identifying Critical Nodes in Protein-Protein Interaction Networks , 2009 .

[21]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[22]  Mark S. Granovetter.  The Strength of Weak Ties , 1973, American Journal of Sociology.

[23]  Mario Ventresca,et al.  A derandomized approximation algorithm for the critical node detection problem , 2014, Comput. Oper. Res..

[24]  Massimo Marchiori,et al.  How the science of complex networks can help developing strategies against terrorism , 2004 .