Minimize the average mean first passage time of random walk in complex networks by genetic algorithm

We investigate the methods of rewiring a given connected network so that the average mean first passage time (AMFPT) for random walk processes can be reduced. Two rewiring mechanisms are used, which we call the zeroth order method (Z) and the higher order method (H) according to the series expansion analysis of the first passage time. We aim at finding the optimal sequence of rewiring action of k units long composed of the Z and H methods that yields the maximal reduction of average mean first passage time. We use both the simple genetic algorithm (SGA) and mutation only genetic algorithm (MOGA) and the results show in general MOGA produces networks with higher reduction of AMFPT for benchmark tests on all three major classes of complex networks (ER, BA, WS). In general, the higher order method appears more often in the networks with large clustering coefficient. The networks with small clustering coefficient will develop to regular networks through the zeroth order rewiring method, while the networks with initially large clustering coefficient have preference to reduce the number of small loops through the higher order rewiring method. We also apply our rewiring sequence of three steps on real networks (IEEE30, IEEE 57) and produce remarkable reduction on the AMFPT. This indicates that our method of analysis can be of practical importance for engineering.

[1]  Heiko Rieger,et al.  Random walks on complex networks. , 2004, Physical review letters.

[2]  Kwok Yip Szeto,et al.  Adaptive Genetic Algorithm with Mutation and Crossover Matrices , 2007, IJCAI.

[3]  Beom Jun Kim,et al.  Path finding strategies in scale-free networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Hong Shen,et al.  The maximal rates of more general complex orthogonal designs , 2005, Sixth International Conference on Parallel and Distributed Computing Applications and Technologies (PDCAT'05).

[5]  Roberto Beraldi.  Random walk with long jumps for wireless ad hoc networks , 2009, Ad Hoc Networks.

[6]  L. Gallos.  Random walk and trapping processes on scale-free networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  O Bénichou,et al.  Random walks and Brownian motion: a method of computation for first-passage times and related quantities in confined geometries. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Markus Brede,et al.  Networks—An Introduction. Mark E. J. Newman. (2010, Oxford University Press.) $65.38, £35.96 (hardcover), 772 pages. ISBN-978-0-19-920665-0. , 2012, Artificial Life.

[9]  Paul Erdös,et al.  On random graphs, I , 1959 .

[10]  M. Greicius,et al.  Default-mode network activity distinguishes Alzheimer's disease from healthy aging: Evidence from functional MRI , 2004, Proc. Natl. Acad. Sci. USA.

[11]  Kevin E. Bassler,et al.  Optimal transport on complex networks , 2007 .

[12]  Exact first-passage exponents of 1D domain growth: Relation to a reaction-diffusion model. , 1995, Physical review letters.

[13]  Michael E. Fisher,et al.  Walks, walls, wetting, and melting , 1984 .

[14]  Kim Sneppen,et al.  Modeling dynamics of information networks. , 2003, Physical review letters.

[15]  Jian Zhang,et al.  Adaptive Genetic Algorithm and Quasi-parallel Genetic Algorithm: Application to Knapsack Problem , 2005, LSSC.

[16]  Kwok Yip Szeto,et al.  Asymptotic analysis of first passage time in complex networks , 2009, 0909.0605.

[17]  Lada A. Adamic,et al.  Search in Power-Law Networks , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Henry C. Tuckwell,et al.  Introduction to theoretical neurobiology , 1988 .

[19]  J. Klafter,et al.  First-passage times in complex scale-invariant media , 2007, Nature.

[20]  J S Kim,et al.  Fractality in complex networks: critical and supercritical skeletons. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  R. Albert,et al.  Search in weighted complex networks. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  D. Watts,et al.  Small Worlds: The Dynamics of Networks between Order and Randomness , 2001 .

[23]  Soon-Hyung Yook,et al.  Diffusive capture process on complex networks. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Benjamin P. Lee.  Renormalization group calculation for the reaction kA + 0 , 1994 .

[25]  Dimitry Volchenkov,et al.  Random walks and flights over connected graphs and complex networks , 2011 .

[26]  Alhussein A. Abouzeid,et al.  Modeling and analysis of random walk search algorithms in P2P networks , 2005 .

[27]  Wen-Xu Wang,et al.  Integrating local static and dynamic information for routing traffic. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  K. Birgitta Whaley,et al.  Quantum random-walk search algorithm , 2002, quant-ph/0210064.

[29]  Bo Hu,et al.  Efficient routing on complex networks. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  A. Roberts,et al.  Generalization of the fractal Einstein law relating conduction and diffusion on networks. , 2009, Physical review letters.

[31]  E Ben-Naim,et al.  Knots and random walks in vibrated granular chains. , 2001, Physical review letters.

[32]  J. Stoyanov.  A Guide to First‐passage Processes , 2003 .

[33]  Xiao-Gai Tang,et al.  Integrating network structure and dynamic information for better routing strategy on scale-free networks , 2009 .

[34]  John Scott.  Social Network Analysis , 1988 .

[35]  Shlomo Havlin,et al.  Origins of fractality in the growth of complex networks , 2005, cond-mat/0507216.

[36]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[37]  Nong Ye,et al.  Onset of traffic congestion in complex networks. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  Benjamin P. Lee,et al.  Renormalization Group Calculation for the Reaction kA → ∅ , 1994 .

[39]  Luigi Fortuna,et al.  Dynamical network model of infective mobile agents. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  H. Diamant,et al.  Screened hydrodynamic interaction in a narrow channel. , 2002, Physical review letters.

[41]  K Sneppen,et al.  Communication boundaries in networks. , 2005, Physical review letters.

[42]  Jon M. Kleinberg,et al.  Navigation in a small world , 2000, Nature.

[43]  Yosef Sheffi,et al.  Urban Transportation Networks: Equilibrium Analysis With Mathematical Programming Methods , 1985 .