Abstract
We propose and analyze a potential induced random walk and its modification called random teleportation on finite graphs. The transition probability is determined by the gaps between potential values of adjacent and teleportation nodes. We show that the steady state of this process has a number of desirable properties. We present a continuous time analogue of the random walk and teleportation, and derive the lower bound on the order of its exponential convergence rate to stationary distribution. The efficiency of proposed random teleportation in search of global potential minimum on graphs and node ranking are demonstrated by numerical tests. Moreover, we discuss the condition of graphs and potential distributions for which the proposed approach may work inefficiently, and introduce the intermittent diffusion strategy to overcome the problem and improve the practical performance.
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Notes
An exponential random variable \(T\sim \exp (\alpha )\) for \(\alpha >0\) has probability density function \(p(t)=\alpha e^{-\alpha t}\) for \(t>0\).
References
Abdullah, M., Cooper, C., Frieze, A.: Cover time of a random graph with given degree sequence. Discrete Math. 312(21), 3146–3163 (2012)
Aggarwal, C.C.: Social Network Data Analytics. Springer, New York (2011)
Aldous, D.: Applications of random walks on finite graphs. In: Selected Proceedings of the Sheffield Symposium on Applied Probability (Sheffield, 1989), volume 18 of IMS Lecture Notes Monograph Series, pp. 12–26. Institute of Mathematics and Statistics, Hayward, CA, (1991)
Aldous, D., Fill, J.: Reversible Markov Chains and Random Walks on Graphs. Monograph in Preparation (1999). Book in preparation
Berg, B.: Introduction to Markov Chain Monte Carlo Simulations and Their Statistical Analysis (2004). arXiv:cond-mat/0410490
Billera, L., Diaconis, P.: A geometric interpretation of the Metropolis–Hastings algorithm. Stat. Sci. 16(4), 335–339 (2001)
Bobkov, S., Tetali, P.: Modified logarithmic sobolev inequalities in discrete settings. J. Theor. Probab. 19(2), 289–336 (2006)
Burioni, R., Cassi, D.: Random walks on graphs: ideas, techniques and results. J. Phys. A Math. Gen. 38(8), R45 (2005)
Casella, G., George, E.: Explaining the Gibbs sampler. Am. Stat. 46(3), 167–174 (1992)
Chandra, A., Raghavan, P., Ruzzo, W., Smolensky, R., Tiwari, P.: The electrical resistance of a graph captures its commute and cover times. Comput. Complex. 6(4), 312–340 (1996)
Coppersmith, D., Doyle, P., Raghavan, P., Snir, M.: Random walks on weighted graphs and applications to on-line algorithms. J. ACM 40(3), 421–453 (1993)
Doyle, P., Snell, L.: Random Walks and Electric Networks (Carus Mathematical Monographs). Mathematical Assn of America, first printing edition (1984)
Doyle, P.G., Snell, J. L.: Random Walks and Electric Networks, 3 (2000). arXiv:math/0001057
Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern. Anal. Mach. Intell. 6(6), 721–741 (1984)
Gjoka, M., Kurant, M., Butts, C., Markopoulou, A.: A Walk in Facebook: Uniform Sampling of Users in Online Social Networks (2011). arXiv:0906.0060
Hastings, W.: Monte carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97–109 (1970)
Kempe, D., Kleinberg, J., Tardos, É.: Maximizing the spread of influence through a social network. In: Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 137–146. ACM, New York (2003)
Klein, D., Palacios, J., Randic, M., Trinajstic, N.: Random walks and chemical graph theory. J. Chem. Inf. Comput. Sci. 44(5), 1521–1525 (2004)
Langville, A., Meyer, C.: Deeper inside pagerank. Internet Math. 1(3), 335–380 (2004)
Langville, A.N., Meyer, C.D.: Google’s PageRank and Beyond: The Science of Search Engine Rankings. Princeton University Press, Princeton, NJ (2009)
Lawler, G.F., Limic, V.: Random Walk: A Modern Introduction, vol. 123. Cambridge University Press, Cambridge (2010)
Lovász, L.: Random walks on graphs: a survey. Comb. Paul Erdős Eighty 2(1), 1–46 (1993)
Meer, K.: Simulated annealing versus metropolis for a TSP instance. Inf. Process. Lett. 104(6), 216–219 (2007)
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087 (1953)
Nourani, Y., Andresen, B.: A comparison of simulated annealing cooling strategies. J. Phys. A Math. Gen. 31, 8373–8385 (1998)
Richey, M.: The evolution of Markov chain Monte Carlo methods. Am. Math. Mon. 117(5), 383–413 (2010)
Rosvall, M., Bergstrom, C.: Maps of random walks on complex networks reveal community structure. Proc. Natl. Acad. Sci. 105(4), 1118–1123 (2008)
Sarkar, P., Moore, A.W.: Random walks in social networks and their applications: a survey. In: Aggarwal, C.C. (ed.) Social Network Data Analytics, pp. 43–77 (2011). Springer, US
Suman, B., Kumar, P.: A survey of simulated annealing as a tool for single and multiobjective optimization. J. Oper. Res. Soc. 57(10), 1143–1160 (2006)
Tetali, P.: Random walks and the effective resistance of networks. J. Theor. Probab. 4(1), 101–109 (1991)
van Kampen, N.: Stochastic Processes in Physics and Chemistry, 3rd edn. North Holland, Amsterdam (2007)
Wegener, I.: Simulated annealing beats metropolis in combinatorial optimization. In: Automata, Languages and Programming, p. 61 (2005)
Weiss, G.: Aspects and Applications of the Random Walk (Random Materials and Processes S.). North-Holland, Amsterdam (1994)
Weinan, E., Li, T., Vanden-Eijnden, E.: Optimal partition and effective dynamics of complex networks. Proc. Natl. Acad. Sci. USA 105(23), 7907–7912 (2008)
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This work was supported in part by NSF Grants Faculty Early Career Development (CAREER) Award DMS-0645266 and DMS-1042998, and ONR Award N000141310408.
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Chow, SN., Ye, X. & Zhou, H. Potential induced random teleportation on finite graphs. Comput Optim Appl 61, 689–711 (2015). https://doi.org/10.1007/s10589-015-9727-7
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DOI: https://doi.org/10.1007/s10589-015-9727-7