Abstract
In this paper, we design a spin model to optimize the search in complex networks using two random walkers by reducing the wasted time in revisiting nodes. A good measure of the global performance of the searching process is the Mean Cover Time (MCT), which is the time needed from the start of the searching process to the end that all sites in the network are reached by at least one walker. We use a three-state spin model to minimize the MCT for two random walkers. In the model, each site in the network is described by a three-state spin, with unvisited sites having a spin state defined by white color, and the visited sites in spin states with non-white colors (red and blue in the case of two walkers). The visit of a site by a walker changes the state of the spin. We introduce a repulsive interaction between spins to model the interactions between walkers. Numerical results using the Erdős-Rényi (ER) network, the Watts-Strogatz (WS) network and three small-world real network datasets show satisfactory results in reducing the MCT. For small-world networks, both the artificial WS network and real-world network datasets show the existence of the critical repulsion strength which minimizes the MCT. We also provide a heuristic explanation for the presence of the critical repulsion that minimizes the MCT for the WS network and its absence in the ER network. Our model provides guidance to future research on multiple random walkers over complex networks, with potential applications for efficient information spreading in social networks.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ahn, Y.Y., Bagrow, J.P., Lehmann, S.: Link communities reveal multiscale complexity in networks (2009). arXiv preprint arXiv:0903.3178
Almaas, E., Kulkarni, R., Stroud, D.: Scaling properties of random walks on small-world networks. Phys. Rev. E 68(5), 056,105 (2003)
Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: Synchronization in complex networks. Phys. Rep. 469(3), 93–153 (2008)
Bonaventura, M., Nicosia, V., Latora, V.: Characteristic times of biased random walks on complex networks. Phys. Rev. E 89(1), 012,803 (2014)
Castellano, C., Fortunato, S., Loreto, V.: Statistical physics of social dynamics. Rev. Mod. Phys. 81(2), 591 (2009)
Clauset, A., Moore, C., Newman, M.E.: Structural inference of hierarchies in networks. In: Statistical Network Analysis: Models, Issues, and New Directions, pp. 1–13. Springer (2007)
Condamin, S., Bénichou, O., Moreau, M.: First-passage times for random walks in bounded domains. Phys. Rev. Lett. 95(26), 260,601 (2005)
Condamin, S., Bénichou, O., Moreau, M.: Random walks and brownian motion: a method of computation for first-passage times and related quantities in confined geometries. Phys. Rev. E 75(2), 021,111 (2007)
Ding, M.C., Szeto, K.Y.: Selection of random walkers that optimizes the global mean first-passage time for search in complex networks. Procedia Comput. Sci. 108, 2423–2427 (2017)
Evans, J.W., Nord, R.: Random walks on finite lattices with multiple traps: application to particle-cluster aggregation. Phys. Rev. A 32(5), 2926 (1985)
Fogel, J., Nehmad, E.: Internet social network communities: risk taking, trust, and privacy concerns. Comput. Hum. Behav. 25(1), 153–160 (2009)
Haldane, A.G., May, R.M.: Systemic risk in banking ecosystems. Nature 469(7330), 351 (2011)
Hughes, B.: Random Walks and Random Environments: Random Walks, vol. 1. Oxford Science Publications, Clarendon Press (1995)
Leskovec, J., Lang, K.J., Dasgupta, A., Mahoney, M.W.: Community structure in large networks: natural cluster sizes and the absence of large well-defined clusters. Internet Math. 6(1), 29–123 (2009)
Patel, A., Rahaman, M.A.: Search on a hypercubic lattice using a quantum random walk. I. d \(>\) 2. Phys. Rev. A 82(3), 032,330 (2010)
Power, J.D., Cohen, A.L., Nelson, S.M., Wig, G.S., Barnes, K.A., Church, J.A., Vogel, A.C., Laumann, T.O., Miezin, F.M., Schlaggar, B.L., et al.: Functional network organization of the human brain. Neuron 72(4), 665–678 (2011)
Redner, S.: A guide to first-passage processes. Cambridge University Press (2001)
Rieser, M., Nagel, K.: Network breakdown “at the edge of chaos” in multi-agent traffic simulations. Eur. Phys. J. B Condens. Matter Complex Syst. 63(3), 321–327 (2008)
Rosvall, M., Bergstrom, C.T.: Maps of random walks on complex networks reveal community structure. Proc. Natl. Acad. Sci. 105(4), 1118–1123 (2008)
Telesford, Q.K., Joyce, K.E., Hayasaka, S., Burdette, J.H., Laurienti, P.J.: The ubiquity of small-world networks. Brain Connect. 1(5), 367–375 (2011)
Volchenkov, D., Blanchard, P.: Random walks along the streets and canals in compact cities: Spectral analysis, dynamical modularity, information, and statistical mechanics. Phys. Rev. E 75(2), 026,104 (2007)
Watts, D., Strogatz, S.: Collective dynamics of small-worldnetworks. Nature 393, 440–442 (1998). https://doi.org/10.1038/30918
Weng, T., Zhang, J., Small, M., Hui, P.: Multiple random walks on complex networks: a harmonic law predicts search time. Phys. Rev. E 95(5), 052,103 (2017)
Zhong, M., Shen, K.: Popularity-biased random walks for peer-to-peer search under the square-root principle. In: Proceedings of the 5th International Workshop on Peer-to-Peer Systems (2006)
Acknowledgements
W. Guo and J. Wang acknowledge the support of the Hong Kong University of Science and Technology through the Undergraduate Research Opportunity Program.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this paper
Cite this paper
Guo, W., Wang, J., Szeto, K.Y. (2018). Spin Model of Two Random Walkers in Complex Networks. In: Cherifi, C., Cherifi, H., Karsai, M., Musolesi, M. (eds) Complex Networks & Their Applications VI. COMPLEX NETWORKS 2017. Studies in Computational Intelligence, vol 689. Springer, Cham. https://doi.org/10.1007/978-3-319-72150-7_45
Download citation
DOI: https://doi.org/10.1007/978-3-319-72150-7_45
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-72149-1
Online ISBN: 978-3-319-72150-7
eBook Packages: EngineeringEngineering (R0)