Abstract
Given a finite, undirected graph G (possibly with multiple edges), we assume that the vertices are operational, but the edges are each independently operational with probability p. The (all-terminal) reliability, \(\operatorname{Rel}(G,p)\), of G is the probability that the spanning subgraph of operational edges is connected. It has been conjectured that reliability functions have at most one point of inflection in (0,1). We show that the all-terminal reliability of almost every simple graph of order n has a point of inflection, and there are indeed infinite families of graphs (both simple and otherwise) with more than one point of inflection.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Barbeau, E. J. (1989). Polynomials. New York: Springer.
Bataineh, S., & Odet, A. (1998). Reliability of mesh and torus topologies in the presence of faults. Telecommunications Systems, 10, 389–408.
Boesch, F. T. (1986). On unreliability polynomials and graph connectivity in reliable network synthesis. Journal of Graph Theory, 10, 339–352.
Brown, J. I., Koç, Y., & Kooij, R. E. (2011). Reliability polynomials crossing more than twice. In Proceedings of the 3rd international workshop on Reliable Network Design and Modeling (RNDM’11), Budapest, Hungary.
Colbourn, C. J. (1987). The combinatorics of network reliability. New York: Oxford University Press.
Colbourn, C. J. (1993). Some open problems for reliability polynomials. Congressus Numerantium, 93, 187–202.
Ghamry, W. K., & Elsayed, K. M. F. (2012). Network design methods for mitigation of intentional attacks in scale-free networks. Telecommunication Systems, 49, 313–327.
Graves, C. (2011). Inflection points of coherent reliability polynomials. Australasian Journal of Combinatorics, 49, 111–126.
Moore, E. F., & Shannon, C. E. (1956). Reliable circuits using less reliable relays. Journal of the Franklin Institute, 262, 191–208.
Secci, S., & Sansó, B. (2011). Survivability and reliability of a composite-star transport network with disconnected core switches. Telecommunication Systems, 46, 43–59.
Sperner, E. (1928). Ein Sats über Untermengen einer endlichen Menge. Mathematische Zeitschrift, 27, 544–548.
Zou, W., Janic, M., Kooij, R., & Kuipers, F. (2007). On the availability of networks, broadband Europe. In Proceedings of BroadBand Europe 200, Antwerp, Belgium.
Acknowledgement
The first author would like to acknowledge the support of the Natural Sciences and Engineering Research Council of Canada.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Brown, J.I., Koç, Y. & Kooij, R.E. Inflection points for network reliability. Telecommun Syst 56, 79–84 (2014). https://doi.org/10.1007/s11235-013-9820-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11235-013-9820-0