Numerical optimization of generative network parameters

We address the design of complex, large-scale systems by viewing them as random networks, and optimizing network structure over generative parameters. We do not seek specific topologies, but rather classes of optimal or near optimal networks which correspond to desirable statistical behavior, while also allowing flexibility to accommodate unmodeled constraints. This approach is a computationally feasible forward design path for large-scale systems. A numerical example is given in which a network's degree distribution is optimized for combined efficiency and robustness in a cascading failure scenario; the work has application to electric distribution and other systems.

[1]  M. Newman,et al.  The structure of scientific collaboration networks. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[2]  S. Havlin,et al.  Optimization of robustness of complex networks , 2004, cond-mat/0404331.

[3]  Edward A. Bender,et al.  The Asymptotic Number of Labeled Graphs with Given Degree Sequences , 1978, J. Comb. Theory A.

[4]  Béla Bollobás,et al.  Random Graphs , 1985 .

[5]  Alice E. Smith,et al.  Local search genetic algorithm for optimal design of reliable networks , 1997, IEEE Trans. Evol. Comput..

[6]  K. Goh,et al.  Universal behavior of load distribution in scale-free networks. , 2001, Physical review letters.

[7]  Hiroki Sayama,et al.  Optimization of robustness and connectivity in complex networks. , 2003, Physical review letters.

[8]  Robert Hooke,et al.  `` Direct Search'' Solution of Numerical and Statistical Problems , 1961, JACM.

[9]  F. Chung,et al.  Connected Components in Random Graphs with Given Expected Degree Sequences , 2002 .

[10]  Thomas L. Magnanti,et al.  Network Design and Transportation Planning: Models and Algorithms , 1984, Transp. Sci..

[11]  F. Chung,et al.  The average distances in random graphs with given expected degrees , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[12]  S. Havlin,et al.  Optimization of network robustness to waves of targeted and random attacks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  L. L. Garver,et al.  Transmission Network Planning Using Linear Programming , 1985, IEEE Power Engineering Review.

[14]  B. Bollobás.  The evolution of random graphs , 1984 .

[15]  Chonghui Guo,et al.  Entropy optimization of scale-free networks’ robustness to random failures , 2005, cond-mat/0506725.

[16]  H. Stanley,et al.  Optimization of network robustness to random breakdowns , 2005, cond-mat/0507249.

[17]  Leonard M. Freeman,et al.  A set of measures of centrality based upon betweenness , 1977 .

[18]  Sharon L. Milgram,et al.  The Small World Problem , 1967 .

[19]  M. Newman,et al.  Random graphs with arbitrary degree distributions and their applications. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  V. E. Lynch,et al.  Critical points and transitions in an electric power transmission model for cascading failure blackouts. , 2002, Chaos.

[21]  Chonghui Guo,et al.  Optimization of network structure to random failures , 2006 .

[22]  Konstantinos E. Parsopoulos,et al.  PARTICLE SWARM OPTIMIZER IN NOISY AND CONTINUOUSLY CHANGING ENVIRONMENTS , 2001 .

[23]  A. Motter,et al.  Ensemble averageability in network spectra. , 2007, Physical review letters.

[24]  Réka Albert,et al.  Structural vulnerability of the North American power grid. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  T. C. Hu.  Optimum Communication Spanning Trees , 1974, SIAM J. Comput..

[26]  Adilson E Motter,et al.  Cascade-based attacks on complex networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  Jing J. Liang,et al.  Comprehensive learning particle swarm optimizer for global optimization of multimodal functions , 2006, IEEE Transactions on Evolutionary Computation.

[28]  Adilson E Motter,et al.  Introduction: optimization in networks. , 2007, Chaos.

[29]  M. A. Muñoz,et al.  Entangled networks, synchronization, and optimal network topology. , 2005, Physical review letters.

[30]  R. Jan,et al.  Topological optimization of a communication network subject to a reliability constraint , 1993 .

[31]  Bruce A. Reed,et al.  A Critical Point for Random Graphs with a Given Degree Sequence , 1995, Random Struct. Algorithms.

[32]  R. Albert,et al.  The large-scale organization of metabolic networks , 2000, Nature.

[33]  Béla Bollobás,et al.  The Diameter of a Scale-Free Random Graph , 2004, Comb..

[34]  Eduardo G. Carrano,et al.  Electric distribution network multiobjective design using a problem-specific genetic algorithm , 2006, IEEE Transactions on Power Delivery.

[35]  H.M. Khodr,et al.  Ant colony system algorithm for the planning of primary distribution circuits , 2004, IEEE Transactions on Power Systems.

[36]  Riccardo Poli,et al.  Particle swarm optimization , 1995, Swarm Intelligence.

[37]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..

[38]  R. Boorstyn,et al.  Large-Scale Network Topological Optimization , 1977, IEEE Trans. Commun..

[39]  Michalis Faloutsos,et al.  On power-law relationships of the Internet topology , 1999, SIGCOMM '99.

[40]  Benjamin F. Hobbs,et al.  Optimization methods for electric utility resource planning , 1995 .

[41]  R. Solé,et al.  Optimization in Complex Networks , 2001, cond-mat/0111222.

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

[43]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.