Abstract
The number of spanning trees of a network is an important measure related to topological and dynamic properties of the network, such as its reliability, communication aspects, and so on. However, obtaining the number of spanning trees of networks and the study of their properties are computationally demanding, in particular for complex networks. In this paper, we introduce a family of small-world networks denoted G k,n , characterized by dimension k, we present its topological construction and we examine its structural properties. Then, we propose the decomposition method to find the exact formula for the number of spanning trees of our small world network. This result allows the calculation of the spanning tree entropy which depends on the network structure, indicating that the entropy of low dimensional network is higher than that of high dimensional network.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Amaral, L.A.N., Scala, A., Barthelemy, M., Stanley, H.E.: Classes of small-world networks. Proceedings of the national academy of sciences 97(21), 11,149–11,152 (2000)
Anishchenko, A., Blumen, A., M¨ulken, O.: Enhancing the spreading of quantum walks on star graphs by additional bonds. Quantum Information Processing 11(5), 1273–1286 (2012)
Chaiken, S., Kleitman, D.J.: Matrix tree theorems. Journal of combinatorial theory, Series A 24(3), 377–381 (1978)
Chang, S.C., Chen, L.C., Yang, W.S.: Spanning trees on the sierpinski gasket. Journal of Statistical Physics 126(3), 649–667 (2007)
Colbourn, C.J., Colbourn, C.: The combinatorics of network reliability, vol. 200. Oxford University Press New York (1987)
Cooper, C., Frieze, A.: A general model of web graphs. Random Structures & Algorithms 22(3), 311–335 (2003)
Hillery, M., Reitzner, D., Buˇzek, V.: Searching via walking: How to find a marked clique of a complete graph using quantum walks. Physical Review A 81(6), 062,324 (2010)
Jespersen, S., Sokolov, I., Blumen, A.: Relaxation properties of small-world networks. Physical Review E 62(3), 4405 (2000)
Lyons, R.: Asymptotic enumeration of spanning trees. Combinatorics, Probability and Computing 14(04), 491–522 (2005)
Marti, K.H., Bauer, B., Reiher, M., Troyer, M., Verstraete, F.: Complete-graph tensor network states: a new fermionic wave function ansatz for molecules. New Journal of Physics 12(10), 103,008 (2010)
Song, C., Havlin, S., Makse, H.A.: Self-similarity of complex networks. Nature 433(7024), 392–395 (2005)
Standish, R.K.: Complexity of networks (reprise). Complexity 17(3), 50–61 (2012)
Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. nature 393(6684), 440–442 (1998)
Wu, B.Y., Chao, K.M.: Spanning trees and optimization problems. CRC Press (2004)
Zhang, Z., Wu, S., Li, M., Comellas, F.: The number and degree distribution of spanning trees in the tower of hanoi graph. Theoretical Computer Science 609, 443–455 (2016)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Mokhlissi, R., Lotfi, D., Debnath, J., Marraki, M.E. (2017). Complexity Analysis of “Small-World Networks” and Spanning Tree Entropy. In: Cherifi, H., Gaito, S., Quattrociocchi, W., Sala, A. (eds) Complex Networks & Their Applications V. COMPLEX NETWORKS 2016 2016. Studies in Computational Intelligence, vol 693. Springer, Cham. https://doi.org/10.1007/978-3-319-50901-3_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-50901-3_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-50900-6
Online ISBN: 978-3-319-50901-3
eBook Packages: EngineeringEngineering (R0)